“Diamond Theory” « Search Results

“… I called the belief that

(1) Diamonds– informative, certain truths about the world– exist

the ‘Diamond Theory’ of truth. I said that for 2200 years the strongest evidence for the Diamond Theory was the widespread perception that

(2) The theorems of Euclidean geometry are diamonds….

As the news about non-Euclidean geometry spread– first among mathematicians, then among scientists and philosophers– the Diamond Theory began a long decline that continues today.

Factors outside mathematics have contributed to this decline. Euclidean geometry had never been the Diamond Theory’s only ally. In the eighteenth century other fields had seemed to possess diamonds, too; when many of these turned out to be man-made, the Diamond Theory was undercut. And unlike earlier periods in history, when intellectual shocks came only occasionally, received truths have, since the eighteenth century, been found wanting at a dizzying rate, creating an impression that perhaps no knowledge is stable.

Other factors notwithstanding, non-Euclidean geometry remains, I think, for those who have heard of it, the single most powerful argument against the Diamond Theory*– first, because it overthrows what had always been the strongest argument in favor of the Diamond Theory, the objective truth of Euclidean geometry; and second, because it does so not by showing Euclidean geometry to be false, but by showing it to be merely uncertain.” —The Non-Euclidean Revolution, p. 255

H. S. M. Coxeter, 1987, introduction to Trudeau’s book—

“There is a pleasantly discursive treatment of Pontius Pilate’s unanswered question ‘What is truth?’.”

As noted here on Oct. 8, 2008 (A Yom Kippur Meditation), Coxeter was aware in 1987 of a more technical use of the phrase “diamond theory” that is closely related to…

A kind
of cross:

Diamond formed by four diagonally-divided two-color squares

See both
Theme and
Variations
and some more
poetic remarks,

Mirror-Play
of the Fourfold.

* As recent Log24 entries have pointed out, diamond theory (in the original 1976 sense) is a type of non-Euclidean geometry, since finite geometry is not Euclidean geometry– and is, therefore, non-Euclidean, in the strictest sense (though not according to popular usage).

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Thursday, May 28, 2009

Thursday May 28, 2009

Filed under: General,Geometry — m759 @ 9:00 pm

Spelling

At right below, an image from the opening of Fox Studios Australia in Sydney on November 7, 1999. The Fox ceremonies included, notably, Kylie Minogue singing “Diamonds are a Girl’s Best Friend.”

Red Windmill

Windmill image from diamond theory

Kylie Minogue

Kylie Minogue

For the mathematical properties of the red windmill (moulin rouge) figure at left, see Diamond Theory.

“There comes a time when you have
learned enough to decide whether
the way of the Craft is for you….

First you will need to
prepare your sacred space….

Calling the Corners (or Quarters)
is something you will always do.”

— “Becoming a Witch” webpage

In related news:

CBS Evening News-- 'New York's Newest Ballpark'

Happy birthday, Kylie.

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Friday, April 10, 2009

Friday April 10, 2009

Filed under: General,Geometry — Tags: Eightfold Cube — m759 @ 8:00 am

Pilate Goes
to Kindergarten

“There is a pleasantly discursive
treatment of Pontius Pilate’s
unanswered question
‘What is truth?’.”
— H. S. M. Coxeter, 1987,
introduction to Trudeau’s
remarks on the “Story Theory“
of truth as opposed to the
“Diamond Theory” of truth in
The Non-Euclidean Revolution

Consider the following question in a paper cited by V. S. Varadarajan:

E. G. Beltrametti, “Can a finite geometry describe physical space-time?” Universita degli studi di Perugia, Atti del convegno di geometria combinatoria e sue applicazioni, Perugia 1971, 57–62.

Simplifying:

“Can a finite geometry describe physical space?”

Simplifying further:

“Yes. Vide ‘The Eightfold Cube.'”

Froebel's 'Third Gift' to kindergarteners: the 2x2x2 cube, in 'Paradise of Childhood'

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Sunday, March 29, 2009

Sunday March 29, 2009

Filed under: General,Geometry — Tags: Change Logos, Strange Change — m759 @ 7:48 pm

Getting All
the Meaning In

Webpage heading for the
2009 meeting of the
American Comparative
Literature Association:

The mysterious symbols on
the above map suggest the
following reflections:

From A Cure of the Mind: The Poetics of Wallace Stevens, by Theodore Sampson, published by Black Rose Books Ltd., 2000–

Page x:

"… if what he calls 'the spirit's alchemicana' (CP [Collected Poems] 471) addresses itself to the irrational element in poetry, to what extent is such an element dominant in his theory and practice of poetry, and therefore in what way is Stevens' intricate verbal music dependent on his irrational use of language– a 'pure rhetoric of a language without words?' (CP 374)?"

Related material:

From "'When Novelists Become Cubists:' The Prose Ideograms of Guy Davenport," by Andre Furlani:

Laurence Zachar argues that Davenport's writing is situated "aux frontieres intergeneriques" where manifold modes are brought into concord: "L'etonnant chez Davenport est la facon don't ce materiau qui parait l'incarnation meme du chaos– hermetique, enigmatique, obscur, avec son tropplein de references– se revele en fait etre construit, ordonne, structure. Plus l'on s'y plonge, et plus l'on distingue de cohesion dans le texte." 'What astonishes in Davenport is the way in which material that seems the very incarnation of chaos– hermetic, enigmatic, obscure, with its proliferation of allusions– in fact reveals itself to be constructed, organized, structured. The more one immerses oneself in them the more one discerns the texts' cohesion.' (62).

Davenport also works along the intergeneric border between text and graphic, for he illustrates many of his texts. (1) "The prime use of words is for imagery: my writing is drawing," he states in an interview (Hoeppfner 123). Visual imagery is not subordinated to writing in Davenport, who draws on the assemblage practice of superimposing image and writing. "I trust the image; my business is to get it onto the page," he writes in the essay "Ernst Machs Max Ernst." "A page, which I think of as a picture, is essentially a texture of images. […] The text of a story is therefore a continuous graph, kin to the imagist poem, to a collage (Ernst, Willi Baumeister, El Lissitzky), a page of Pound, a Brakhage film" (Geography 374-75).

Note:

(1.) Davenport is an illustrator of books (such as Hugh Kenner's The Stoic Comedians and The Counterfeiters) and journals (such as The Kenyon Review, Parnassus, and Paideuma). His art is the subject of Erik Anderson Reece's monograph, A Balance of Quinces, which reveals the inseparable relationship between Davenport's literary and pictorial work.

References:

Davenport, Guy. The Geography of the Imagination. San Francisco: North Point Press, 1981. Rpt. New York: Pantheon, 1992.

Hoepffner, Bernard. "Pleasant Hill: An Interview with Guy Davenport." Conjunctions 24 (1995): 118-24.

Reece, Erik Anderson. A Balance of Quinces: The Paintings and Drawings of Guy Davenport. New York: New Directions, 1996.

Zachar, Laurence. "Guy Davenport: Une Mosaique du genres." Recherches Anglaises et Nord-Americaines 21 (1994): 51-63.

"… when novelists become Cubists; that is, when they see the possibilities of making a hieroglyph, a coherent symbol, an ideogram of the total work. A symbol comes into being when an artist sees that it is the only way to get all the meaning in."

— Guy Davenport, The Geography of the Imagination

See also last night's
commentary on the
following symbols:

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Saturday, March 28, 2009

Saturday March 28, 2009

Filed under: General,Geometry — Tags: Innermost Kernel, Springer — m759 @ 11:07 pm

The Rest
of the Story

Today's previous entry discussed the hermeneutics of the midday NY and PA lottery numbers.

The rest of the story:

The Revelation Game
(continued from 7/26, 2008)

Lotteries on Reba's birthday, 2009	Pennsylvania (No revelation)	New York (Revelation)
Mid-day (No belief)	No belief, no revelation 726	Revelation without belief 378
Evening (Belief)	Belief without revelation 006	Belief and revelation 091

Interpretations of the evening numbers–

The PA evening number, 006, may be viewed as a followup to the PA midday 726 (or 7/26, the birthday of Kate Beckinsale and Carl Jung). Here 006 is the prestigious "00" number assigned to Beckinsale.

Will: Do you like apples?
Clark: Yeah.
Will: Well, I got her number.
How do you like them apples?

— "Good Will Hunting"

Kate Beckinsale in 'Underworld: Evolution'

The NY evening number, 091, may be viewed as a followup to the NY midday 378 (the number of pages in The Innermost Kernel by Suzanne Gieser, published by Springer, 2005)–

Page 91: The entire page is devoted to the title of the book's Part 3– "The Copenhagen School and Psychology"–

Page 91 of 'The Innermost Kernel' by Suzanne Gieser, Springer 2005

The next page begins: "With the crisis of physics, interest in epistemological and psychological questions grew among many theoretical physicists. This interest was particularly marked in the circle around Niels Bohr."

A particularly
marked circle
from March 15:

The circle above is
marked with a version of
the classic Chinese symbol
adopted as a personal emblem
by Danish physicist Niels Bohr,
leader of the Copenhagen School.

"Two things of opposite natures seem to depend
On one another, as a man depends
On a woman, day on night, the imagined

On the real. This is the origin of change.
Winter and spring, cold copulars, embrace
And forth the particulars of rapture come."

-- Wallace Stevens,
  "Notes Toward a Supreme Fiction,"
   Canto IV of "It Must Change"

The square above is marked
with a graphic design
related to the four-diamond
figure of Jung's Aion.

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Tuesday, March 17, 2009

Tuesday March 17, 2009

Filed under: General,Geometry — Tags: Logos and Branding — m759 @ 11:07 am

Deep Structures

The traditional 'Square of Opposition'

The Square of Oppositon
at Stanford Encylopedia of Philosophy

The Square of Opposition diagram in its earliest known form

The Square of Opposition
in its original form

"The diagram above is from a ninth century manuscript of Apuleius' commentary on Aristotle's Perihermaneias, probably one of the oldest surviving pictures of the square."

— Edward Buckner at The Logic Museum

From the webpage "Semiotics for Beginners: Paradigmatic Analysis," by Daniel Chandler:

The Semiotic Square

"The structuralist semiotician Algirdas Greimas introduced the semiotic square (which he adapted from the 'logical square' of scholastic philosophy) as a means of analysing paired concepts more fully (Greimas 1987,* xiv, 49). The semiotic square is intended to map the logical conjunctions and disjunctions relating key semantic features in a text. Fredric Jameson notes that 'the entire mechanism… is capable of generating at least ten conceivable positions out of a rudimentary binary opposition' (in Greimas 1987,* xiv). Whilst this suggests that the possibilities for signification in a semiotic system are richer than the either/or of binary logic, but that [sic] they are nevertheless subject to 'semiotic constraints' – 'deep structures' providing basic axes of signification."

* Greimas, Algirdas (1987): On Meaning: Selected Writings in Semiotic Theory (trans. Paul J Perron & Frank H Collins). London: Frances Pinter

Another version of the semiotic square:

Rosalind Krauss's version of the semiotic square, which she calls the Klein group

Krauss says that her figure "is, of course, a Klein Group."

Here is a more explicit figure representing the Klein group:

The Klein Four-Group, illustration by Steven H. Cullinane

There is also the logical
diamond of opposition —

The Diamond of Opposition (figure from Wikipedia)

A semiotic (as opposed to logical)
diamond has been used to illustrate
remarks by Fredric Jameson,
a Marxist literary theorist:

"Introduction to Algirdas Greimas, Module on the Semiotic Square," by Dino Felluga at Purdue University–

The semiotic square has proven to be an influential concept not only in narrative theory but in the ideological criticism of Fredric Jameson, who uses the square as "a virtual map of conceptual closure, or better still, of the closure of ideology itself" ("Foreword"* xv). (For more on Jameson, see the [Purdue University] Jameson module on ideology.)

Greimas' schema is useful since it illustrates the full complexity of any given semantic term (seme). Greimas points out that any given seme entails its opposite or "contrary." "Life" (s₁) for example is understood in relation to its contrary, "death" (s₂). Rather than rest at this simple binary opposition (S), however, Greimas points out that the opposition, "life" and "death," suggests what Greimas terms a contradictory pair (-S), i.e., "not-life" (-s₁) and "not-death" (-s₂). We would therefore be left with the following semiotic square (Fig. 1):

As Jameson explains in the Foreword to Greimas' On Meaning, "-s₁ and -s₂"—which in this example are taken up by "not-death" and "not-life"—"are the simple negatives of the two dominant terms, but include far more than either: thus 'nonwhite' includes more than 'black,' 'nonmale' more than 'female'" (xiv); in our example, not-life would include more than merely death and not-death more than life.

* Jameson, Fredric. "Foreword." On Meaning: Selected Writings in Semiotic Theory. By Algirdas Greimas. Trans. Paul J. Perron and Frank H. Collins. Minneapolis: U of Minnesota P, 1976.

"The Game in the Ship cannot be approached as a job, a vocation, a career, or a recreation. To the contrary, it is Life and Death itself at work there. In the Inner Game, we call the Game Dhum Welur, the Mind of God."

— The Gameplayers of Zan, by M.A. Foster

"For every kind of vampire,
there is a kind of cross."
— Thomas Pynchon,
Gravity's Rainbow

Crosses used by semioticians
to baffle their opponents
are illustrated above.

Some other kinds of crosses,
and another kind of opponent:

Monday, July 11, 2005

Logos
for St. Benedict's Day

Click on either of the logos below for religious meditations– on the left, a Jewish meditation from the Conference of Catholic Bishops; on the right, an Aryan meditation from Stormfront.org.

Both logos represent different embodiments of the "story theory" of truth, as opposed to the "diamond theory" of truth. Both logos claim, in their own ways, to represent the eternal Logos of the Christian religion. I personally prefer the "diamond theory" of truth, represented by the logo below.

See also the previous entry
(below) and the entries
of 7/11, 2003.

Sunday, July 10, 2005

Mathematics
and Narrative

Click on the title
for a narrative about

Nikolaos K. Artemiadis,
(co-) author of

From Artemiadis's website:
1986:	Elected Regular Member of the Academy of Athens
1999:	Vice President of the Academy of Athens
2000:	President of the Academy of Athens

Seal of the American Mathematical Society with picture of Plato's Academy

"First of all, I'd like to
thank the Academy…"

— Remark attributed to Plato

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Sunday, March 15, 2009

Sunday March 15, 2009

Filed under: General,Geometry — Tags: Change Logos, Strange Change — m759 @ 5:24 pm

The Origin of Change

A note on the figure
from this morning's sermon:

"Two things of opposite natures seem to depend
On one another, as a man depends
On a woman, day on night, the imagined

On the real. This is the origin of change.
Winter and spring, cold copulars, embrace
And forth the particulars of rapture come."

-- Wallace Stevens,
  "Notes Toward a Supreme Fiction,"
   Canto IV of "It Must Change"

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Sunday March 15, 2009

Filed under: General,Geometry — Tags: Change Logos, Strange Change — m759 @ 11:00 am

Angels, Demons,
"Symbology"

L'Osservatore Romano:

"On Monday morning, 9 March, after visiting the Mayor of Rome and the Municipal Council on the Capitoline Hill, the Holy Father spoke to the Romans who gathered in the square outside the Senatorial Palace…

'… a verse by Ovid, the great Latin poet, springs to mind. In one of his elegies he encouraged the Romans of his time with these words:

"Perfer et obdura: multo graviora tulisti."

"Hold out and persist:
you have got through
far more difficult situations."

(Tristia, Liber V, Elegia XI, verse 7).'"

This journal
on 9 March:

Note the color-interchange
symmetry of each symbol
under 180-degree rotation.

Related material:
The Illuminati Diamond:

IMAGE- Illuminati Diamond, pp. 359-360 in 'Angels & Demons,' Simon & Schuster Pocket Books 2005, 448 pages, ISBN 0743412397

Dan Brown's novel Angels & Demons introduced in the year 2000 the fictional academic discipline of "symbology" and a fictional Harvard professor of that discipline, Robert Langdon (named after ambigram* artist John Langdon).

Fictional Harvard professor of symbology Robert Langdon, as portrayed by Tom Hanks

Tom Hanks as Robert Langdon

A possible source for Brown's term "symbology" is a 1995 web page, "The Rotation of the Elements," by one "John Opsopaus." (Cf. Art History Club.)

"The four qualities are the key to understanding the rotation of the elements and many other applications of the symbology of the four elements." –John Opsopaus

* "…ambigrams were common in symbology…." —Angels & Demons

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Monday, March 9, 2009

Monday March 9, 2009

Filed under: General,Geometry — Tags: Aion Art, Change Logos, Four Color, Strange Change — m759 @ 12:00 pm

Humorism

"Always with a
little humor."
— Dr. Yen Lo

Diamond diagram of the four humors, the four qualities, the four elements, the four seasons, and four colors

From Temperament: A Brief Survey

For other interpretations
of the above shape, see
The Illuminati Diamond.

from Jung's Aion:

"From the circle and quaternity motif is derived the symbol of the geometrically formed crystal and the wonder-working stone. From here analogy formation leads on to the city, castle, church, house, room, and vessel. Another variant is the wheel. The former motif emphasizes the ego’s containment in the greater dimension of the self; the latter emphasizes the rotation which also appears as a ritual circumambulation. Psychologically, it denotes concentration on and preoccupation with a centre…." –Jung, Collected Works, Vol. 9, Part II, paragraph 352

As for rotation, see the ambigrams in Dan Brown's Angels & Demons (to appear as a film May 15) and the following figures:

Diamond Theory version of 'The Square Inch Space' with yin-yang symbol for comparison

Click on image
for a related puzzle.
For a solution, see
The Diamond Theorem.

A related note on
"Angels & Demons"
director Ron Howard:

Click image for details.

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Friday, January 30, 2009

Friday January 30, 2009

Filed under: General,Geometry — m759 @ 11:07 am

Two-Part Invention

This journal on
October 8, 2008,
at noon:

“There is a pleasantly discursive treatment of Pontius Pilate’s unanswered question ‘What is truth?'”

— H. S. M. Coxeter, introduction to Richard J. Trudeau’s remarks on the “story theory” of truth as opposed to the “diamond theory” of truth in The Non-Euclidean Revolution

Trudeau’s 1987 book uses the phrase “diamond theory” to denote the philosophical theory, common since Plato and Euclid, that there exist truths (which Trudeau calls “diamonds”) that are certain and eternal– for instance, the truth in Euclidean geometry that the sum of a triangle’s angles is 180 degrees.

Insidehighered.com on
the same day, October 8, 2008,
at 12:45 PM EDT

“Future readers may consider Updike our era’s Mozart; Mozart was once written off as a too-prolific composer of ‘charming nothings,’ and some speak of Updike that way.”

— Comment by BPJ

“Birthday, death-day–
what day is not both?”
— John Updike

Updike died on January 27.
On the same date,
Mozart was born.

Requiem

Mr. Best entered,
tall, young, mild, light.
He bore in his hand
with grace a notebook,
new, large, clean, bright.

— James Joyce, Ulysses,
Shakespeare and Company,
Paris, 1922, page 178

Related material:

Dec. 5, 2004 and

Inscribed carpenter's square

Jan. 27-29, 2009

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Sunday, December 14, 2008

Sunday December 14, 2008

Filed under: General,Geometry — m759 @ 4:00 pm

Epigraphs

The New York Times of Sunday, May 6, 2007, on a writer of pulp fiction:

His early novels, written in two weeks or less, were published in double-decker Ace paperbacks that included two books in one, with a lurid cover for each. “If the Holy Bible was printed as an Ace Double,” an editor once remarked, “it would be cut down to two 20,000-word halves with the Old Testament retitled as ‘Master of Chaos’ and the New Testament as ‘The Thing With Three Souls.'”

Epigraph for Part One:

“Ours is a very gutsy religion, Cullinane.“

— James A. Michener

Lurid cover:
The Pussycat

Epigraph for Part Two:

“Beware lest you believe that you can comprehend the Incomprehensible….“

— Saint Bonaventure

Lurid cover:
The Owl

Click on the image for a
relevant Wallace Stevens poem.

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Thursday, November 6, 2008

Thursday November 6, 2008

Filed under: General,Geometry — Tags: Crichton Time, on081106 — m759 @ 10:07 am

Death of a Classmate

Michael Crichton,
Harvard College, 1964

Authors Michael Crichton and David Foster Wallace in NY Times obituaries, Thursday, Nov. 6, 2008

Authors Michael Crichton and
David Foster Wallace in today’s
New York Times obituaries

The Times’s remarks above
on the prose styles of
Crichton and Wallace–
“compelling formula” vs.
“intricate complexity”–
suggest the following works
of visual art in memory
of Crichton.

“Crystal”—

“Dragon”
(from Crichton’s
Jurassic Park)–

For the mathematics
(dyadic harmonic analysis)
relating these two figures,
see Crystal and Dragon.

Some philosophical
remarks related to
the Harvard background
that Crichton and I share–

Hitler’s Still Point
and
The Crimson Passion.

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Sunday, October 12, 2008

Sunday October 12, 2008

Filed under: General,Geometry — m759 @ 3:28 pm

Confidence Game

Paul Newman and Robert Redford in 'The Sting'

The Winners:

European leaders in Paris agree on plan to aid banks

Related material:
Dec. 16, 2003—

Moulin Bleu

Juliette Binoche in 'Blue' Animated 2x2 kaleidoscope figures from Diamond Theory

Kaleidoscope turning…
Shifting pattern
within unalterable structure…
— Roger Zelazny, Eye of Cat

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Wednesday, October 8, 2008

Wednesday October 8, 2008

Filed under: General,Geometry — Tags: Fields — m759 @ 12:00 pm

Serious Numbers

A Yom Kippur
Meditation

"When times are mysterious
Serious numbers
Will always be heard."
— Paul Simon,
"When Numbers Get Serious"

"There is a pleasantly discursive treatment of Pontius Pilate's unanswered question 'What is truth?'"

— H. S. M. Coxeter, introduction to Richard J. Trudeau's remarks on the "story theory" of truth as opposed to the "diamond theory" of truth in The Non-Euclidean Revolution

Trudeau's 1987 book uses the phrase "diamond theory" to denote the philosophical theory, common since Plato and Euclid, that there exist truths (which Trudeau calls "diamonds") that are certain and eternal– for instance, the truth in Euclidean geometry that the sum of a triangle's angles is 180 degrees. As the excerpt below shows, Trudeau prefers what he calls the "story theory" of truth–

"There are no diamonds. People make up stories about what they experience. Stories that catch on are called 'true.'"

(By the way, the phrase "diamond theory" was used earlier, in 1976, as the title of a monograph on geometry of which Coxeter was aware.)

Richard J. Trudeau on the 'Story Theory' of truth

Excerpt from
The Non-Euclidean Revolution

What does this have to do with numbers?

Pilate's skeptical tone suggests he may have shared a certain confusion about geometric truth with thinkers like Trudeau and the slave boy in Plato's Meno. Truth in a different part of mathematics– elementary arithmetic– is perhaps more easily understood, although even there, the existence of what might be called "non-Euclidean number theory"– i.e., arithmetic over finite fields, in which 1+1 can equal zero– might prove baffling to thinkers like Trudeau.

Trudeau's book exhibits, though it does not discuss, a less confusing use of numbers– to mark the location of pages. For some philosophical background on this version of numerical truth that may be of interest to devotees of the Semitic religions on this evening's High Holiday, see Zen and Language Games.

For uses of numbers that are more confusing, see– for instance– the new website The Daily Beast and the old website Story Theory and the Number of the Beast.

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Sunday, August 3, 2008

Sunday August 3, 2008

Filed under: General,Geometry — Tags: Cubehenge, EGBDF, Geometric Theology, Geometry, Symmetric Generation — m759 @ 3:00 pm

Kindergarten
Geometry

Preview of a Tom Stoppard play presented at Town Hall in Manhattan on March 14, 2008 (Pi Day and Einstein's birthday):

The play's title, "Every Good Boy Deserves Favour," is a mnemonic for the notes of the treble clef EGBDF.

The place, Town Hall, West 43rd Street. The time, 8 p.m., Friday, March 14. One single performance only, to the tinkle– or the clang?– of a triangle. Echoing perhaps the clang-clack of Warsaw Pact tanks muscling into Prague in August 1968.

The “u” in favour is the British way, the Stoppard way, "EGBDF" being "a Play for Actors and Orchestra" by Tom Stoppard (words) and André Previn (music).

And what a play!– as luminescent as always where Stoppard is concerned. The music component of the one-nighter at Town Hall– a showcase for the Boston University College of Fine Arts– is by a 47-piece live orchestra, the significant instrument being, well, a triangle.

When, in 1974, André Previn, then principal conductor of the London Symphony, invited Stoppard "to write something which had the need of a live full-time orchestra onstage," the 36-year-old playwright jumped at the chance.

One hitch: Stoppard at the time knew "very little about 'serious' music… My qualifications for writing about an orchestra," he says in his introduction to the 1978 Grove Press edition of "EGBDF," "amounted to a spell as a triangle player in a kindergarten percussion band."

— Jerry Tallmer in The Villager, March 12-18, 2008

Review of the same play as presented at Chautauqua Institution on July 24, 2008:

"Stoppard's modus operandi– to teasingly introduce numerous clever tidbits designed to challenge the audience."

— Jane Vranish, Pittsburgh Post-Gazette, Saturday, August 2, 2008

"The leader of the band is tired
And his eyes are growing old
But his blood runs through
My instrument
And his song is in my soul."

— Dan Fogelberg

"He's watching us all the time."

— Lucia Joyce

Finnegans Wake,
Book II, Episode 2, pp. 296-297:

I'll make you to see figuratleavely the whome of your eternal geomater. And if you flung her headdress on her from under her highlows you'd wheeze whyse Salmonson set his seel on a hexengown.¹ Hissss!, Arrah, go on! Fin for fun!

¹ The chape of Doña Speranza of the Nacion.

Log 24, Sept. 3, 2003:

Reciprocity

From my entry of Sept. 1, 2003:

"…the principle of taking and giving, of learning and teaching, of listening and storytelling, in a word: of reciprocity….

… E. M. Forster famously advised his readers, 'Only connect.' 'Reciprocity' would be Michael Kruger's succinct philosophy, with all that the word implies."

— William Boyd, review of Himmelfarb, a novel by Michael Kruger, in The New York Times Book Review, October 30, 1994

Last year's entry on this date:

Today's birthday:
James Joseph Sylvester

"Mathematics is the music of reason."
— J. J. Sylvester

The picture above is of the complete graph K₆… Six points with an edge connecting every pair of points… Fifteen edges in all.

Diamond theory describes how the 15 two-element subsets of a six-element set (represented by edges in the picture above) may be arranged as 15 of the 16 parts of a 4×4 array, and how such an array relates to group-theoretic concepts, including Sylvester's synthematic totals as they relate to constructions of the Mathieu group M₂₄.

If diamond theory illustrates any general philosophical principle, it is probably the interplay of opposites…. "Reciprocity" in the sense of Lao Tzu. See

Reciprocity and Reversal in Lao Tzu.

For a sense of "reciprocity" more closely related to Michael Kruger's alleged philosophy, see the Confucian concept of Shu (Analects 15:23 or 24) described in

Shu: Reciprocity.

Kruger's novel is in part about a Jew: the quintessential Jewish symbol, the star of David, embedded in the K₆ graph above, expresses the reciprocity of male and female, as my May 2003 archives illustrate. The star of David also appears as part of a graphic design for cubes that illustrate the concepts of diamond theory:

Click on the design for details.

Those who prefer a Jewish approach to physics can find the star of David, in the form of K₆, applied to the sixteen 4×4 Dirac matrices, in

A Graphical Representation
of the Dirac Algebra .

The star of David also appears, if only as a heuristic arrangement, in a note that shows generating partitions of the affine group on 64 points arranged in two opposing triplets.

Having thus, as the New York Times advises, paid tribute to a Jewish symbol, we may note, in closing, a much more sophisticated and subtle concept of reciprocity due to Euler, Legendre, and Gauss. See

The Jewel of Arithmetic and

The Golden Theorem.

FinnegansWiki:

Salmonson set his seel:

"Finn MacCool ate the Salmon of Knowledge."

Wikipedia:

"George Salmon spent his boyhood in Cork City, Ireland. His father was a linen merchant. He graduated from Trinity College Dublin at the age of 19 with exceptionally high honours in mathematics. In 1841 at age 21 he was appointed to a position in the mathematics department at Trinity College Dublin. In 1845 he was appointed concurrently to a position in the theology department at Trinity College Dublin, having been confirmed in that year as an Anglican priest."

Related material:

Kindergarten Theology,

Kindergarten Relativity,

Arrangements for
56 Triangles.

For more on the
arrangement of
triangles discussed
in Finnegans Wake,
see Log24 on Pi Day,
March 14, 2008.

Happy birthday,
Martin Sheen.

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Friday, July 11, 2008

Friday July 11, 2008

Filed under: General — Tags: Logos and Branding — m759 @ 1:00 pm

AND MORE LOGOS:

"Serious numbers will
always be heard."
— Paul Simon

The HSBC Logo Designer —

Henry Steiner

He is an internationally recognized corporate identity consultant. Based in Hong Kong, his work for clients such as HongkongBank, IBM and Unilever is a major influence in Pacific Rim design.

Born in Austria and raised in New York, Steiner was educated at Yale under Paul Rand and attended the Sorbonne as a Fulbright Fellow. He is a past President of Alliance Graphique Internationale. Other professional affiliations include the American Institute of Graphic Arts, Chartered Society of Designers, Design Austria, and the New York Art Directors' Club.

His Cross-Cultural Design: Communicating in the Global Marketplace was published by Thames and Hudson (1995).

— Yaneff.com

Related material
from the past —

Fly from Fly Bottle:

Graphic structures from Diamond Theory and from Kyocera logo

Charles Taylor,
"Epiphanies of Modernism,"
Chapter 24 of Sources of the Self
(Cambridge U. Press, 1989, p. 477) —

"… the object sets up
a kind of frame or space or field
within which there can be epiphany."

Saturday, June 21, 2008

Saturday June 21, 2008

Filed under: General,Geometry — m759 @ 6:00 am

The Kyoto Prize

for lifetime achievement
in arts and philosophy
this year goes to
Charles Taylor,

Charles Margrave Taylor, professor emeritus of philosophy at McGill University

Montreal philosophy professor.

“The Kyoto Prize has been given in three domains since 1984:
advanced technology, basic sciences, and the arts and philosophy.
It is administered by the Inamori Foundation, whose president,
Kazuo Inamori, is founder and chairman emeritus of Kyocera and
KDDI Corporation, two Japanese telecommunications giants.”

— Montreal Gazette, June 20, 2008

“The Kyocera brand symbol is composed of a corporate mark
and our corporate logotype. The mark represents the initial
‘K’ (for Kyocera) encircling a ‘C’ (for ceramics). It was
introduced in October 1982 when the company name was
changed from ‘Kyoto Ceramic’ to ‘Kyocera.'”

— global.kyocera.com

Related material —

Wittgenstein and Fly from Fly-Bottle

Fly from Fly Bottle:

Graphic structures from Diamond Theory and from Kyocera logo

Charles Taylor,
“Epiphanies of Modernism,”
Chapter 24 of Sources of the Self
(Cambridge U. Press, 1989, p. 477) —

“… the object sets up
a kind of frame or space or field
within which there can be epiphany.”

Saturday, April 19, 2008

Saturday April 19, 2008

Filed under: General,Geometry — Tags: Lichtung, Pieces of April — m759 @ 5:01 am

A Midrash for Benedict

On April 16, the Pope’s birthday, the evening lottery number in Pennsylvania was 441. The Log24 entries of April 17 and April 18 supplied commentaries based on 441’s incarnation as a page number in an edition of Heidegger’s writings. Here is a related commentary on a different incarnation of 441. (For a context that includes both today’s commentary and those of April 17 and 18, see Gian-Carlo Rota– a Heidegger scholar as well as a mathematician– on mathematical Lichtung.)

From R. D. Carmichael, Introduction to the Theory of Groups of Finite Order (Boston, Ginn and Co., 1937)– an exercise from the final page, 441, of the final chapter, “Tactical Configurations”–

“23. Let G be a multiply transitive group of degree n whose degree of transitivity is k; and let G have the property that a set S of m elements exists in G such that when k of the elements S are changed by a permutation of G into k of these elements, then all these m elements are permuted among themselves; moreover, let G have the property P, namely, that the identity is the only element in G which leaves fixed the n – m elements not in S. Then show that G permutes the m elements S into

n(n -1) … (n – k + 1)
____________________

m(m – 1) … (m – k + 1)

sets of m elements each, these sets forming a configuration having the property that any (whatever) set of k elements appears in one and just one of these sets of m elements each. Discuss necessary conditions on m, n, k in order that the foregoing conditions may be realized. Exhibit groups illustrating the theorem.”

This exercise concerns an important mathematical structure said to have been discovered independently by the American Carmichael and by the German Ernst Witt.

For some perhaps more comprehensible material from the preceding page in Carmichael– 440– see Diamond Theory in 1937.

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Tuesday, March 4, 2008

Tuesday March 4, 2008

Filed under: General,Geometry — Tags: March Fourth Death — m759 @ 1:00 pm

… And for a
Swiftly Tilting
Shadowed Planet …

Wm. F. Buckley as Archimedes, moving the world with a giant pen as lever. The pen's point is applied to southern South America.
John Trever, Albuquerque Journal, 2/29/08

The pen's point:

Log24, Dec. 11, 2006

SINGER, ISAAC:
"Are Children the
Ultimate Literary Critics?"
— Top of the News 29
(Nov. 1972): 32-36.

"Sets forth his own aims in writing for children and laments 'slice of life' and chaos in children's literature. Maintains that children like good plots, logic, and clarity, and that they have a concern for 'so-called eternal questions.'"

— An Annotated Listing
of Criticism
by Linnea Hendrickson

"She returned the smile, then looked across the room to her youngest brother, Charles Wallace, and to their father, who were deep in concentration, bent over the model they were building of a tesseract: the square squared, and squared again: a construction of the dimension of time."

— A Swiftly Tilting Planet,
by Madeleine L'Engle

Cover of 'A Swiftly Tilting Planet' and picture of tesseract

For "the dimension of time,"
see A Fold in Time,
Time Fold, and
Diamond Theory in 1937.

A Swiftly Tilting Planet is a fantasy for children set partly in Vespugia, a fictional country bordered by Chile and Argentina.

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Saturday, February 16, 2008

Saturday February 16, 2008

Filed under: General,Geometry — Tags: Fields — m759 @ 9:29 am

Bridges
Between Two Worlds

From the world of mathematics…

“… my advisor once told me, ‘If you ever find yourself drawing one of those meaningless diagrams with arrows connecting different areas of mathematics, it’s a good sign that you’re going senile.'”

— Scott Carnahan at Secret Blogging Seminar, December 14, 2007

Carnahan’s remark in context:

“About five years ago, Cheewhye Chin gave a great year-long seminar on Langlands correspondence for GL_r over function fields…. In the beginning, he drew a diagram….

If we remove all of the explanatory text, the diagram looks like this:

CheeWhye Diagram

I was a bit hesitant to draw this, because my advisor once told me, ‘If you ever find yourself drawing one of those meaningless diagrams with arrows connecting different areas of mathematics, it’s a good sign that you’re going senile.’ Anyway, I’ll explain roughly how it works.

Langlands correspondence is a ‘bridge between two worlds,’ or more specifically, an assertion of a bijection….”

Compare and contrast the above…

… to the world of Rudolf Kaehr:

Rudolf Kaehr on 'Diamond Structuration'

The above reference to “diamond theory” is from Rudolf Kaehr‘s paper titled Double Cross Playing Diamonds.

Another bridge…

Carnahan’s advisor, referring to “meaningless diagrams with arrows connecting different areas of mathematics,” probably did not have in mind diagrams like the two above, but rather diagrams like the two below–

From the world of mathematics…

Relationship of diamond theory to other fields

“A rough sketch of
how diamond theory is
related to some other
fields of mathematics”
— Steven H. Cullinane

… to the world of Rudolf Kaehr:

Relationship of PolyContextural Logic (PCL) to other fields

Related material:

For further details on
the “diamond theory” of
Cullinane, see

Finite Geometry of the
Square and Cube.

For further details on
the “diamond theory” of
Kaehr, see

Rudy’s Diamond Strategies.

Those who prefer entertainment
may enjoy an excerpt
from Log24, October 2007:

“Do not let me hear
Of the wisdom
of old men,
but rather of
their folly”

— Four Quartets

Anthony Hopkins in 'Slipstream'

Anthony Hopkins
in the film
“Slipstream“

Anthony Hopkins
in the film “Proof“–

“Goddamnit, open
the goddamn book!
Read me the lines!“

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Wednesday, July 25, 2007

Wednesday July 25, 2007

Filed under: General,Geometry — Tags: Modernism Rally, Symmetry Framed — m759 @ 9:00 am

The Comedy of
George Tabori

From AP “Obituaries in the News”–
Filed with The New York Times
at 11:16 p.m. ET July 24, 2007–

George Tabori

“BERLIN (AP) — Hungarian-born playwright and director George Tabori, a legend in Germany’s postwar theater world whose avant-garde works confronted anti-Semitism, died Monday [July 23, 2007]. He was 93.

Tabori, who as recently as three years ago dreamed of returning to stage to play the title role in Shakespeare’s ‘King Lear,’ died in his apartment near the theater, the Berliner Ensemble said Tuesday, noting that friends and family had accompanied him through his final days. No cause of death was given.

Born into a Jewish family in Budapest on May 24, 1914, Tabori fled in 1936 to London, where he started working for the British Broadcasting Corp., and became a British citizen. His father, and other members of his family, were killed at Auschwitz.

Tabori moved to Hollywood in the 1950s, where he worked as a scriptwriter, most notably co-writing the script for Alfred Hitchcock’s 1953 film, ‘I Confess.’

He moved to Germany in the 1970s and launched a theater career that spanned from acting to directing to writing. He used sharp wit and humor in his plays to examine the relationship between Germany and the Jews, as well as attack anti-Semitism.

Among his best-known works are ‘Mein Kampf,’ set in the Viennese hostel where Adolf Hitler lived from 1910-1913, and the ‘Goldberg Variations,’ both dark farces that poke fun at the Nazis.”

From Year of Jewish Culture:

“The year 2006 marks the 100th anniversary of the establishment of the Jewish Museum in Prague.”

From the related page Programme (October-December):

“Divadlo v Dlouhé
George Tabori: GOLDBERGOVSKÉ VARIACE / THE GOLDBERG VARIATIONS, 19 October, 7 p.m. A comedy on creation and martyrdom.”

Variations on
Birth and Death

From Log24 on the date of
the Prague production of the
Tabori “Goldberg Variations,”
an illustration in honor of
Sir Thomas Browne, who
was born, and died,
on that date:

The above is from
Variable Resolution 4–k Meshes:
Concepts and Applications (pdf),
by Luiz Velho and Jonas Gomes.

See also Symmetry Framed
and The Garden of Cyrus.

Variations on
the Afterlife

From Log24
on the date of
Tabori’s death:

Theme

(Plato, Meno)

and Variations:

Click on “variations” above
for some material on
the “Goldberg Variations”
of Johann Sebastian Bach.

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Monday, July 23, 2007

Monday July 23, 2007

Filed under: General,Geometry — Tags: on070723 — m759 @ 7:59 am

Today’s Birthday:
Daniel Radcliffe
(“Harry Potter”)

Harry Potter and the Philosopher's Stone DVD

Theme

(Plato, Meno)

and Variations:

Click on picture for details.

“A diamond jubilance
beyond the fire,
That gives its power
to the wild-ringed eye”

— Wallace Stevens,
“The Owl in the Sarcophagus”

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Thursday, March 1, 2007

Thursday March 1, 2007

Filed under: General,Geometry — Tags: Wrinkle Tesseract — m759 @ 6:29 am

Senior Honors

Notes in Memory of
a Father, a Son, and a Holy Ghost

From the obituary in today's New York Times of historian Arthur M. Schlesinger Jr.–

"Mr. Schlesinger, partly through his appreciation of history, fully realized his good fortune. 'I have lived through interesting times and had the luck of knowing some interesting people,' he wrote.

A huge part of his luck was his father, who guided much of his early research, and even suggested the topic for his [Harvard] senior honors: Orestes A. Brownson, a 19th-century journalist, novelist and theologian. It was published by Little, Brown in 1938 as 'Orestes A. Brownson: A Pilgrim's Progress.'"

— Douglas Martin

From The Catholic Encyclopedia:

"It is sufficient for true knowledge that it affirm as real that which is truly real."

— Article on Ontologism

From The Diamond Theory of Truth:

"Was there really a cherubim waiting at the star-watching rock…?
Was he real?
What is real?

— Madeleine L'Engle, A Wind in the Door, Farrar, Straus and Giroux, 1973, conclusion of Chapter Three, "The Man in the Night"

"Oh, Euclid, I suppose."

— Madeleine L'Engle, A Wrinkle in Time, Farrar, Straus and Giroux, 1962, conclusion of Chapter Five, "The Tesseract"

Related material: Yesterday's first annual "Tell Your Story Day" at Harvard and yesterday's entry on Euclid.

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Tuesday, February 27, 2007

Tuesday February 27, 2007

Filed under: General,Geometry — m759 @ 9:25 pm

Suggested by today’s
New York Times story
on a Harvard student’s
research on pattern in
Islamic art —

and in memory of
George Sadek —

From Log24 in July 2005:

Intersections

A Trinity Sunday sermon
quotes T. S. Eliot:

“… to apprehend
The point of intersection of the timeless
With time, is an occupation for the saint.”

Monday, December 11, 2006

Monday December 11, 2006

Filed under: General,Geometry — m759 @ 7:20 am

Geometry and Death

J. G. Ballard on “the architecture of death“:

“… a huge system of German fortifications that included the Siegfried line, submarine pens and huge flak towers that threatened the surrounding land like lines of Teutonic knights. Almost all had survived the war and seemed to be waiting for the next one, left behind by a race of warrior scientists obsessed with geometry and death.”

— The Guardian, March 20, 2006

Edward Hirsch on Lorca:

“For him, writing is a struggle both with geometry and death.”

— “The Duende,” American Poetry Review, July/August 1999

“Rosenblum writes with
absolute intellectual honesty,
and the effect is sheer liberation….
The disposition of the material is
a model of logic and clarity.”

— Harper’s Magazine review
quoted on back cover of
Cubism and Twentieth-Century Art,
by Robert Rosenblum
(Abrams paperback, 2001)

SINGER, ISAAC:
“Are Children the Ultimate Literary Critics?”
— Top of the News 29 (Nov. 1972): 32-36.
“Sets forth his own aims in writing for children
and laments ‘slice of life’ and chaos in
children’s literature. Maintains that children
like good plots, logic, and clarity,
and that they have a concern for
‘so-called eternal questions.'”

— An Annotated Listing of Criticism
by Linnea Hendrickson

“She returned the smile, then looked
across the room to her youngest brother,
Charles Wallace, and to their father,
who were deep in concentration, bent
over the model they were building
of a tesseract: the square squared,
and squared again: a construction
of the dimension of time.”

— A Swiftly Tilting Planet,
by Madeleine L’Engle

The image “http://www.log24.com/log/pix06B/061211-Swiftly2.gif” cannot be displayed, because it contains errors.

For “the dimension of time,”
see A Fold in Time,
Time Fold, and
Diamond Theory in 1937.

A Swiftly Tilting Planet is a fantasy for children set partly in Vespugia, a fictional country bordered by Chile and Argentina.

For a more adult audience —

In memory of General Augusto Pinochet, who died yesterday in Santiago, Chile, a quotation from Federico Garcia Lorca‘s lecture on “the Duende” (Buenos Aires, Argentina, 1933):

“… Philip of Austria… longing to discover the Muse and the Angel in theology, found himself imprisoned by the Duende of cold ardors in that masterwork of the Escorial, where geometry abuts with a dream and the Duende wears the mask of the Muse for the eternal chastisement of the great king.”

The image “http://www.log24.com/log/pix06B/061211-Escorial.jpg” cannot be displayed, because it contains errors.

The Escorial

Perhaps. Or perhaps Philip, “the lonely
hermit of the Escorial,” is less lonely now.

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Wednesday, December 6, 2006

Wednesday December 6, 2006

Filed under: General,Geometry — Tags: Waymark Prize — m759 @ 3:15 am

Mathematical Imagery

From the current
American Mathematical Society
“Mathematical Imagery” page:

From today’s New York Times:

“Rosie Lee Tompkins, a renowned African-American quiltmaker whose use of dazzling color and vivid geometric forms made her work internationally acclaimed despite her vehement efforts to remain completely unknown, was found dead on Friday at her home in Richmond, Calif. She was 70.” —Margalit Fox, NY Times 12/6/06

Tompkins was found dead
on December 1, 2006.

From Log24 on that date:

The image “http://www.log24.com/log/pix06B/061201-DayWithout.jpg” cannot be displayed, because it contains errors.

That entry contained an excerpt from
Tom Wolfe’s The Painted Word—

“What I saw before me was the critic-in-chief of The New York Times saying: In looking at a painting today, ‘to lack a persuasive theory is to lack something crucial.’ I read it again. It didn’t say ‘something helpful’ or ‘enriching’ or even ‘extremely valuable.’ No, the word was crucial….”

Related material:

Diamond Theory

and a politically correct
1995 feminist detective novel
about quilts,

A Piece of Justice.

From a summary of the novel:

The story deals with “one Gideon Summerfield, deceased.” Summerfield, a former tutor at (the fictional) St. Agatha’s College, Cambridge University, “is about to become the recipient of the Waymark prize. This prize is awarded in Mathematics and has the same prestige as the Nobel. Summerfield had a rather lackluster career at St. Agatha’s, with the exception of one remarkable result that he obtained. It is for this result that he is being awarded the prize, albeit posthumously.” Someone is apparently trying to prevent a biography of Summerfield from being published.

The following page contains
a critical part of the solution
to the mystery:

The image “http://www.log24.com/log/pix06B/PieceOfJustice138.gif” cannot be displayed, because it contains errors.

Meanwhile, back in real life…

It is said that the late Ms. Tompkins
liked to work while listening to the
soundtrack of “Saturday Night Fever.”

“It’s just your jive talkin’
you’re telling me lies, yeah
Jive talkin’
you wear a disguise
Jive talkin’
so misunderstood, yeah
Jive talkin’
You really no good”

These lyrics may also serve
to summarize reviews
of Diamond Theory written
in the summer of 2005.

For further details, see
Mathematics and Narrative.

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Monday, November 13, 2006

Monday November 13, 2006

Filed under: General — m759 @ 8:23 pm

Cognitive Blend:

Casino Royale
and
Time in the Rock

PA lottery Nov. 13, 2006: Mid-day 726, Evening 329

In today’s cognitive blend,
the role of Casino Royale
is played by the
Pennsylvania Lottery,
which points to 7/26,
Venus at St. Anne’s
(title of the closing chapter
of That Hideous Strength).

The role of
Time in the Rock
is played by a
Log24 entry of 3/29,
Diamond Theory in 1937.

“There is such a thing
as a tesseract.“

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Tuesday, October 31, 2006

Tuesday October 31, 2006

Filed under: General,Geometry — Tags: Galois's Space, Halloween 2006, Wrinkle Tesseract — m759 @ 11:00 pm

To Announce a Faith

From 7/07, an art review from The New York Times:

Endgame Art?
It's Borrow, Sample and Multiply
in an Exhibition at Bard College

"The show has an endgame, end-time mood….

I would call all these strategies fear of form…. the dismissal of originality is perhaps the oldest ploy in the postmodern playbook. To call yourself an artist at all is by definition to announce a faith, however unacknowledged, in some form of originality, first for yourself, second, perhaps, for the rest of us.

Fear of form above all means fear of compression– of an artistic focus that condenses experiences, ideas and feelings into something whole, committed and visually comprehensible."

— Roberta Smith

It is doubtful that Smith
would consider the
following "found" art an
example of originality.

It nevertheless does
"announce a faith."

The image “http://www.log24.com/log/pix06A/061031-PAlottery2.jpg” cannot be displayed, because it contains errors.

"First for yourself"

Today's mid-day
Pennsylvania number:
707

See Log24 on 7/07
and the above review.

"Second, perhaps,
for the rest of us"

Today's evening
Pennsylvania number:
384

This number is an
example of what the
reviewer calls "compression"–

"an artistic focus that condenses
experiences, ideas and feelings
into something
whole, committed
and visually comprehensible."

"Experiences"

See (for instance)

Joan Didion's writings
(1160 pages, 2.35 pounds)
on "the shifting phantasmagoria
which is our actual experience."

"Ideas"

See Plato.

"Feelings"

See A Wrinkle in Time.

"Whole"

The automorphisms
of the tesseract
form a group
of order 384.

"Committed"

See the discussions of
groups of degree 16 in
R. D. Carmichael's classic
Introduction to the Theory
of Groups of Finite Order.

"Visually comprehensible"

See "Diamond Theory in 1937,"
an excerpt from which
is shown below.

The image “http://www.log24.com/theory/images/Carmichael440abbrev.gif” cannot be displayed, because it contains errors.

The "faith" announced by
the above lottery numbers
on All Hallows' Eve is
perhaps that of the artist
Madeleine L'Engle:

"There is such a thing
as a tesseract."

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Tuesday, October 3, 2006

Tuesday October 3, 2006

Filed under: General,Geometry — Tags: Fields, Octad, Octad Generator, Sundown Ending — m759 @ 9:26 am

Serious

"I don't think the 'diamond theorem' is anything serious, so I started with blitzing that."

— Charles Matthews at Wikipedia, Oct. 2, 2006

"The 'seriousness' of a mathematical theorem lies, not in its practical consequences, which are usually negligible, but in the significance of the mathematical ideas which it connects. We may say, roughly, that a mathematical idea is 'significant' if it can be connected, in a natural and illuminating way, with a large complex of other mathematical ideas."

— G. H. Hardy, A Mathematician's Apology

Matthews yesterday deleted references to the diamond theorem and related material in the following Wikipedia articles:

Affine group‎
Reflection group‎
Symmetry in mathematics‎
Incidence structure‎
Invariant (mathematics)‎
Symmetry‎
Finite geometry‎
Group action‎
History of geometry‎

This would appear to be a fairly large complex of mathematical ideas.

See also the following "large complex" cited, following the above words of Hardy, in Diamond Theory:

Affine geometry, affine planes, affine spaces, automorphisms, binary codes, block designs, classical groups, codes, coding theory, collineations, combinatorial, combinatorics, conjugacy classes, the Conwell correspondence, correlations, design theory, duads, duality, error correcting codes, exceptional groups, finite fields, finite geometry, finite groups, finite rings, Galois fields, generalized quadrangles, generators, geometry, GF(2), GF(4), the (24,12) Golay code, group actions, group theory, Hadamard matrices, hypercube, hyperplanes, hyperspace, incidence structures, invariance, Karnaugh maps, Kirkman's schoolgirl problem, Latin squares, Leech lattice, linear groups, linear spaces, linear transformations, Mathieu groups, matrix theory, Meno, Miracle Octad Generator, MOG, multiply transitive groups, octads, the octahedral group, orthogonal arrays, outer automorphisms, parallelisms, partial geometries, permutation groups, PG(3,2), polarities, Polya-Burnside theorem, projective geometry, projective planes, projective spaces, projectivities, Reed-Muller codes, the relativity problem, Singer cycle, skew lines, sporadic simple groups, Steiner systems, symmetric, symmetry, symplectic, synthemes, synthematic, tesseract, transvections, Walsh functions, Witt designs.

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Sunday, July 9, 2006

Sunday July 9, 2006

Filed under: General,Geometry — m759 @ 11:00 am

Today’s birthday:
Tom Hanks, star of
“The Da Vinci Code”

Ben Nicholson
and the Holy Grail

Part I:
A Current Exhibit

“Kufi Blocks“*

by Ben Nicholson,
Illinois Institute of Technology

Part II:
Some Background

A. Diamond Theory, a 1976 preprint containing, in the original version, the designs on the faces of Nicholson’s “Kufi blocks,” as well as some simpler traditional designs, and

B. “Block Designs,” a web page illustrating design blocks based on the 1976 preprint.

Part III:
The Leonardo Connection

See Modern-Day Leonardos, part of an account of a Leonardo exhibit at Chicago’s Museum of Science and Industry that includes Ben Nicholson and his “Kufi Blocks.”

Part IV:
Nicholson’s Grail Quest

“I’m interested in locating the holy grail of the minimum means to express the most complex ideas.”

— Ben Nicholson in a 2005 interview

Nicholson’s quest has apparently lasted for some time. Promotional material for a 1996 Nicholson exhibit in Montreal says it “invites visitors of all ages to experience a contemporary architect’s search for order, meaning and logic in a world of art, science and mystery.” The title of that exhibit was “Uncovering Geometry.”

For web pages to which this same title might apply, see Quilt Geometry, Galois Geometry, and Finite Geometry of the Square and Cube.

* “Square Kufi” calligraphy is used in Islamic architectural ornament. I do not know what, if anything, is signified by Nicholson’s 6×12 example of “Kufi blocks” shown above.

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Friday, May 12, 2006

Friday May 12, 2006

Filed under: General,Geometry — Tags: Wrinkle Tesseract — m759 @ 3:00 am

Tesseract

"Does the word 'tesseract'
mean anything to you?"
— Robert A. Heinlein in
The Number of the Beast
(1980)

My reply–

Part I:

The image “http://www.log24.com/log/pix06A/WrinkleInTime1A.jpg” cannot be displayed, because it contains errors.

A Wrinkle in Time, by
Madeleine L'Engle
(first published in 1962)

Part II:

Diamond Theory in 1937
and
Geometry of the 4×4 Square

Part III:

Catholic Schools Sermon

Conclusion:

"Wells and trees were dedicated to saints. But the offerings at many wells and trees were to something other than the saint; had it not been so they would not have been, as we find they often were, forbidden. Within this double and intertwined life existed those other capacities, of which we know more now, but of which we still know little– clairvoyance, clairaudience, foresight, telepathy."

— Charles Williams, Witchcraft, Faber and Faber, London, 1941

Related material:

A New Yorker profile of Madeleine L'Engle from April 2004, which I found tonight online for the first time. For a related reflection on truth, stories, and values, see Saint's Day. For a wider context, see the Log24 entries of February 1-15, 2003 and February 1-15, 2006.

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Tuesday, April 25, 2006

Tuesday April 25, 2006

Filed under: General,Geometry — m759 @ 3:09 pm

Mathematics and Narrative, continued

“There is a pleasantly discursive treatment
of Pontius Pilate’s unanswered question
‘What is truth?'”

— H. S. M. Coxeter, 1987, introduction to
Richard J. Trudeau’s remarks on
the “Story Theory” of truth
as opposed to
the “Diamond Theory” of truth
in The Non-Euclidean Revolution

A Serious Position

“‘Teitelbaum,’ in German,
is ‘date palm.'”
— Generations, Jan. 2003

“In Hasidism, a mystical brand
of Orthodox Judaism, the grand rabbi
is revered as a kinglike link to God….”

— Today’s New York Times obituary
of Rabbi Moses Teitelbaum,
who died on April 24, 2006
(Easter Monday in the
Orthodox Church)

From Nextbook.org, “a gateway to Jewish literature, culture, and ideas”:

NEW BOOKS: 02.16.05
Proofs and Paradoxes
Alfred Teitelbaum changed his name to Tarski in the early 20s, the same time he changed religions, but when the Germans invaded his native Poland, the mathematician was in California, where he remained. His “great achievement was his audacious assault on the notion of truth,” says Martin Davis, focusing on the semantics and syntax of scientific language. Alfred Tarski: Life and Logic, co-written by a former student, Solomon Feferman, offers “remarkably intimate information,” such as abusive teaching and “extensive amorous involvements.”

From Wikipedia, an unsigned story:

“In 1923 Alfred Teitelbaum and his brother Wacław changed their surnames to Tarski, a name they invented because it sounded very Polish, was simple to spell and pronounce, and was unused. (Years later, he met another Alfred Tarski in northern California.) The Tarski brothers also converted to Roman Catholicism, the national religion of the Poles. Alfred did so, even though he was an avowed atheist, because he was about to finish his Ph.D. and correctly anticipated that it would be difficult for a Jew to obtain a serious position in the new Polish university system.”

The image “http://www.log24.com/log/pix06/060425-Tarski.jpg” cannot be displayed, because it contains errors.

Alfred Tarski

The image “http://www.log24.com/log/pix06/060424-Crimson2.jpg” cannot be displayed, because it contains errors.

See also

The Crimson Passion.

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Wednesday, March 29, 2006

Wednesday March 29, 2006

Filed under: General — Tags: March 26-29 2006 — m759 @ 12:00 pm

Darkness at Noon,
continued

It turns out that Medawar (see previous entry) also wrote a deeply hostile review of Koestler’s The Act of Creation. (See Pluto’s Republic.)

There are plenty more like Medawar, so it may be that a further effort at documentation of Diamond Theory is needed. See this evening’s entry, to follow.

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Monday, February 6, 2006

Monday February 6, 2006

Filed under: General,Geometry — m759 @ 10:00 am

The Diamond Theory of Truth

“Legend says that when the stones
are brought together the diamonds
inside of them will glow.”

— Harrison Ford in
“Indiana Jones and the
Temple of Doom”

In today’s online New York Times:

(1) A review of pop-archaeology TV,
   “Digging for the Truth,”
(2) a Sunday news story,
     “Looking for the Lie,”
(3) and a profile,
     “Storyteller in the Family.”

From (1):
“The season premiere ‘Digging for the Truth: The Real Temple of Doom,’ showed Mr. Bernstein in South America, exploring tunnels….”
From (2):
“… scientists are building a cognitive theory of deception to show what lying looks like….”
From (3):
“I did feel one had to get not just the facts, but the emotional underpinnings.”

Related material:

Log24 on
Harrison Ford’s birthday
last July–

The image “http://www.log24.com/log/pix05A/050713-Ford.jpg” cannot be displayed, because it contains errors.

— and Mathematics and Narrative.

See also Saturday’s entry,
Raiders of the Lost Matrix,
for logic as an aid in
detecting lies.

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Sunday, January 15, 2006

Sunday January 15, 2006

Filed under: General,Geometry — Tags: Geometry, The Nutshell — m759 @ 7:59 am

Inscape

My entry for New Year's Day links to a paper by Robert T. Curtis*
from The Arabian Journal for Science and Engineering
(King Fahd University, Dhahran, Saudi Arabia),
Volume 27, Number 1A, January 2002.

From that paper:

"Combinatorially, an outer automorphism [of S₆] can exist because the number of unordered pairs of 6 letters is equal to the number of ways in which 6 letters can be partitioned into three pairs. Which is to say that the two conjugacy classes of odd permutations of order 2 in S₆ contain the same number of elements, namely 15. Sylvester… refers to the unordered pairs as duads and the partitions as synthemes. Certain collections of five synthemes… he refers to as synthematic totals or simply totals; each total is stabilized within S₆ by a subgroup acting triply transitively on the 6 letters as PGL₂(5) acts on the projective line. If we draw a bipartite graph on (15+15) vertices by joining each syntheme to the three duads it contains, we obtain the famous 8-cage (a graph of valence 3 with minimal cycles of length 8)…."

Here is a way of picturing the 8-cage and a related configuration of points and lines:

The image “http://www.log24.com/theory/images/Cremona-Richmond.gif” cannot be displayed, because it contains errors.

Diamond Theory shows that this structure
can also be modeled by an "inscape"
made up of subsets of a
4×4 square array:

The image “http://www.log24.com/theory/images/Inscape.gif” cannot be displayed, because it contains errors.

The illustration below shows how the
points and lines of the inscape may
be identified with those of the
Cremona-Richmond configuration.

The image “http://www.log24.com/theory/images/Inscape2.gif” cannot be displayed, because it contains errors.

* "A fresh approach to the exceptional automorphism and covers of the symmetric groups"

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Sunday, November 20, 2005

Sunday November 20, 2005

Filed under: General,Geometry — Tags: Knight Move, Springer — m759 @ 4:04 pm

An Exercise
of Power

Johnny Cash:
“And behold,
a white horse.”

The image “http://www.log24.com/log/pix05B/051120-SpringerLogo9.gif” cannot be displayed, because it contains errors.
Adapted from
illustration below:

“There is a pleasantly discursive treatment of Pontius Pilate’s unanswered question ‘What is truth?'”

— H. S. M. Coxeter, 1987, introduction to Richard J. Trudeau’s remarks on the “Story Theory” of truth as opposed to the “Diamond Theory” of truth in The Non-Euclidean Revolution

— Richard J. Trudeau in
The Non-Euclidean Revolution

“‘Deniers’ of truth… insist that each of us is trapped in his own point of view; we make up stories about the world and, in an exercise of power, try to impose them on others.”

— Jim Holt in The New Yorker.

(Click on the box below.)

Exercise of Power:

Show that a white horse–

A Singer 7-Cycle

a figure not unlike the
symbol of the mathematics
publisher Springer–
is traced, within a naturally
arranged rectangular array of
polynomials, by the powers of x
modulo a polynomial
irreducible over a Galois field.

This horse, or chess knight–
“Springer,” in German–
plays a role in “Diamond Theory”
(a phrase used in finite geometry
in 1976, some years before its use
by Trudeau in the above book).

Related material

On this date:

In 1490, The White Knight
(Tirant lo Blanc The image “http://www.log24.com/images/asterisk8.gif” cannot be displayed, because it contains errors. )–
a major influence on Cervantes–
was published, and in 1910

The image “http://www.log24.com/log/pix05B/051120-Caballo1.jpg” cannot be displayed, because it contains errors.

the Mexican Revolution began.

Illustration:
Zapata by Diego Rivera,
Museum of Modern Art,
New York

The image “http://www.log24.com/images/asterisk8.gif” cannot be displayed, because it contains errors. Description from Amazon.com—

“First published in the Catalan language in Valencia in 1490…. Reviewing the first modern Spanish translation in 1969 (Franco had ruthlessly suppressed the Catalan language and literature), Mario Vargas Llosa hailed the epic’s author as ‘the first of that lineage of God-supplanters– Fielding, Balzac, Dickens, Flaubert, Tolstoy, Joyce, Faulkner– who try to create in their novels an all-encompassing reality.'”

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Friday, November 11, 2005

Friday November 11, 2005

Filed under: General,Geometry — Tags: Kabbalah, Sacerdotal Jargon — m759 @ 3:26 pm

720 in the Book
(continued)

From today's
New York Times:

The image “http://www.log24.com/log/pix05B/051111-BeeSeason.jpg” cannot be displayed, because it contains errors.

Phil Bray

Transcendence through spelling:
Richard Gere and Flora Cross
as father and daughter
in "Bee Season."

Words Made Flesh: Code, Culture, Imagination—

The earliest known foundation of the Kabbalah is the Sefer Yetzirah (Book of Creation) whose origin and history is unknown….

… letters create things by the virtue of an algorithm…

"From two letters or forms He composed two dwellings; from three, six; from four, twenty-four; from five, one hundred and twenty; from six, seven hundred and twenty…."
— Sefer Yetzirah

Foucault's Pendulum—

Mystic logic, letters whirling in infinite change, is the world of bliss, it is the music of thought, but see that you proceed slowly, and with caution, because your machine may bring you delirium instead of ecstasy. Many of Abulafia's disciples were unable to walk the fine line between contemplation of the names of God and the practice of magic.

Bee Season—

"The exercises we've been doing are Abulafia's. His methods are primarily a kind of Jewish yoga, a way to relax. For most, what Abulafia describes as shefa, the influx of the Divine, is a historical curiosity to be discussed and interpreted. Because, while anyone can follow Abulafia's instructions for permutation and chanting, very few can use them to achieve transcendence….

Spelling is a sign, Elly. When you win the national bee, we'll know that you are ready to follow in Abulafia's footsteps. Once you're able to let the letters guide you through any word you are given, you will be ready to receive shefa."

In the quiet of the room, the sound of Eliza and her father breathing is everything.

"Do you mean," Eliza whispers, "that I'll be able to talk to God?"

Related material:

Log24, Sept. 3, 2002,

Diamond Theory notes
of Feb. 4, 1986,
of April 26, 1986, and
of May 26, 1986,

Sacerdotal Jargon
(Log24, Dec. 5, 2002),

and 720 in the Book
(Log24, Epiphany 2004).

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Wednesday, September 28, 2005

Wednesday September 28, 2005

Filed under: General,Geometry — m759 @ 4:26 am

Mathematical Narrative,
continued:

There is a pleasantly discursive treatment
of Pontius Pilate’s unanswered question
“What is truth?”

— H. S. M. Coxeter, introduction to
Richard J. Trudeau’s
The Non-Euclidean Revolution

“People have always longed for truths about the world — not logical truths, for all their utility; or even probable truths, without which daily life would be impossible; but informative, certain truths, the only ‘truths’ strictly worthy of the name. Such truths I will call ‘diamonds’; they are highly desirable but hard to find….The happy metaphor is Morris Kline’s in Mathematics in Western Culture (Oxford, 1953), p. 430.”

— Richard J. Trudeau,
   The Non-Euclidean Revolution,
   Birkhauser Boston,
   1987, pages 114 and 117

“A new epistemology is emerging to replace the Diamond Theory of truth. I will call it the ‘Story Theory’ of truth: There are no diamonds. People make up stories about what they experience. Stories that catch on are called ‘true.’ The Story Theory of truth is itself a story that is catching on. It is being told and retold, with increasing frequency, by thinkers of many stripes…. My own viewpoint is the Story Theory…. I concluded long ago that each enterprise contains only stories (which the scientists call ‘models of reality’). I had started by hunting diamonds; I did find dazzlingly beautiful jewels, but always of human manufacture.”

— Richard J. Trudeau,
   The Non-Euclidean Revolution,
   Birkhauser Boston,
   1987, pages 256 and 259

An example of
the story theory of truth:

Actress Gwyneth Paltrow (“Proof”) was apparently born on either Sept. 27, 1972, or Sept. 28, 1972. Google searches yield “about 193” results for the 27th and “about 610” for the 28th.

Those who believe in the “story theory” of truth may therefore want to wish her a happy birthday today. Those who do not may prefer the contents of yesterday’s entry, from Paltrow’s other birthday.

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Friday, August 19, 2005

Friday August 19, 2005

Filed under: General,Geometry — Tags: Aug19 — m759 @ 2:00 pm

Mathematics and Narrative
continued

"There is a pleasantly discursive treatment of Pontius Pilate's unanswered question 'What is truth?'"

— H. S. M. Coxeter, 1987, introduction to Richard J. Trudeau's remarks on the "Story Theory" of truth as opposed to the "Diamond Theory" of truth " in The Non-Euclidean Revolution

"I had an epiphany: I thought 'Oh my God, this is it! People are talking about elliptic curves and of course they think they are talking mathematics. But are they really? Or are they talking about stories?'"

— An organizer of last month's "Mathematics and Narrative" conference

"A new epistemology is emerging to replace the Diamond Theory of truth. I will call it the 'Story Theory' of truth: There are no diamonds. People make up stories about what they experience. Stories that catch on are called 'true.' The Story Theory of truth is itself a story that is catching on. It is being told and retold, with increasing frequency, by thinkers of many stripes*…."

— Richard J. Trudeau in The Non-Euclidean Revolution

"'Deniers' of truth… insist that each of us is trapped in his own point of view; we make up stories about the world and, in an exercise of power, try to impose them on others."

— Jim Holt in this week's New Yorker magazine. Click on the box below.

The image “http://www.log24.com/log/pix05B/050819-Critic4.jpg” cannot be displayed, because it contains errors.

* Many stripes —

"What disciplines were represented at the meeting?"
"Apart from historians, you mean? Oh, many: writers, artists, philosophers, semioticians, cognitive psychologists – you name it."

— An organizer of last month's "Mathematics and Narrative" conference

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Thursday, August 18, 2005

Thursday August 18, 2005

Filed under: General — m759 @ 12:48 am

Sermon for
World Youth Day

(Cologne, Aug. 16-21, 2005)

“And the light shineth in darkness;
and the darkness comprehended it not.”
— The Gospel according to St. John,
Chapter 1, Verse 5

Part I: The Light

The Shining of May 29
and
Diamond Theory

Part II: The Darkness

Mathematics and Narrative
and
Reply to My Fan Mail

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Tuesday, August 16, 2005

Tuesday August 16, 2005

Filed under: General — m759 @ 12:07 pm

Narrative and Latin Squares

From The Independent, 15 August 2005:

“Millions of people now enjoy Sudoku puzzles. Forget the pseudo-Japanese baloney: sudoku grids are a version of the Latin Square created by the great Swiss mathematician Leonhard Euler in the late 18th century.”

The Independent was discussing the conference on “Mathematics and Narrative” at Mykonos in July.

From the Wikipedia article on Latin squares:

“The popular Sudoku puzzles are a special case of Latin squares; any solution to a Sudoku puzzle is a Latin square. Sudoku imposes the additional restriction that 3×3 subgroups must also contain the digits 1–9 (in the standard version).

The Diamond 16 Puzzle illustrates a generalized concept of Latin-square orthogonality: that of “orthogonal squares” (Diamond Theory, 1976) or “orthogonal matrices”– orthogonal, that is, in a combinatorial, not a linear-algebra sense (A. E. Brouwer, 1991).”

This last paragraph, added to Wikipedia on Aug. 14, may or may not survive the critics there.

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Saturday, August 6, 2005

Saturday August 6, 2005

Filed under: General,Geometry — Tags: The Empty Quarter — m759 @ 1:25 pm

The Fugue

   "True joy is a profound remembering, and true grief is the same.
    Thus it was, when the dust storm that had snatched Cal up finally died, and he opened his eyes to see the Fugue spread out before him, he felt as though the few fragile moments of epiphany he'd tasted in his twenty-six years– tasted but always lost– were here redeemed and wed. He'd grasped fragments of this delight before. Heard rumour of it in the womb-dream and the dream of love; known it in lullabies. But never, until now, the whole, the thing entire.
    It would be, he idly thought, a fine time to die.
    And a finer time still to live, with so much laid out before him."

— Clive Barker,
Weaveworld,
Book Two:
The Fugue

From Monday:

Weaveworld,
Book Three:
Out of the
Empty Quarter

"The wheels of its body rolled,
the visible mathematics
of its essence turning on itself…."

From Friday:

For the meaning
of this picture, see
Geometry of the
4×4 Square.

For graphic designs
based on this geometry,
see Theme and Variations
and Diamond Theory.

For these designs in the
context of a Bach fugue,
see Timothy A. Smith's
essay (pdf) on

Fugue No. 21 in B-Flat Major
from Book II of
The Well-Tempered Clavier
by Johann Sebastian Bach.

Smith also offers a
Shockwave movie
that uses diamond theory
to illustrate this fugue.

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Wednesday, July 20, 2005

Wednesday July 20, 2005

Filed under: General — m759 @ 7:20 pm

Moon Day

Words that may or may not have been said on July 20, 1969:

“That’s one small step for a man; one giant leap for mankind.”

Another rhetorical contrast,
from a different date —

One small step for me:

Sunday, November 03, 2002

Music to Read By

In honor of Roger Cooke’s review of Helson’s Harmonic Analysis, 2nd Edition, today’s site music is “Moonlight in Vermont.”

12:00 AM – 5 comments

One giant leap for mankind:

Date Posted: 11/03/02 Sun

“The ‘Diamond Theory’ website of Steven Cullinane shows a man who is incapable of telling the truth: a pathological liar who hates and despises the mathematical community; a sociopath caught between the conflicting desires to earn the admiration of mathematicians, and his desire to insult those who ignore him and refuse him his self-perceived due measure of honor and reverie. As such, Steven Cullinane is constantly trying to purchase recognition when he has the funds to advertise on google.com, or steal that recognition by lying and deceiving dmoz.org when money isn’t enough. As you can see from the correspondence below, Jed Pack has clearly pointed out serious errors in Steven Cullinane’s calculations. Now, instead of admitting that he has been caught with his pants down, Steven Cullinane is questioning Jed Pack’s education! Surely, Jed Pack is a more competent mathematician than Steven Cullinane.”

For further details, see Crankbuster.

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Wednesday, July 13, 2005

Wednesday July 13, 2005

Filed under: General — Tags: Cliffhanger — m759 @ 1:00 pm

Today's birthday: Harrison Ford

Location, Location, Location

Wikipedia on Temple of Doom:

The image “http://www.log24.com/log/pix05A/050713-Ford.jpg” cannot be displayed, because it contains errors.

"Most of the filming was done
on location in Sri Lanka."

Math Forum User Profile for: crankbuster
UserID:	226278
Name:
Email:
Registered:	7/5/05
Occupation:	Teacher
Location:	Srilanka
Homepage:	http://www.math16.com/
Biography:	Mathematics teacher in Srilanka.
Total Posts:	10

Recent Messages:

1. Re: Steven Cullinane is a Crank
   Jul 12, 2005
2. Re: Steven Cullinane is a Crank
   Jul 12, 2005
3. Re: Steven Cullinane is a Crank
   Jul 12, 2005
4. Re: Steven Cullinane's "Diamond Theory"
   Jul 8, 2005
5. Steven Cullinane's "Diamond Theory"
   Jul 7, 2005
6. Re: Steven Cullinane is a Crank
   Jul 7, 2005
7. Re: Steven Cullinane is a Crank
   Jul 5, 2005
8. Re: Steven Cullinane is a Crank
   Jul 5, 2005
9. Steven Cullinane is a Crank
   Jul 5, 2005
10. Steven Cullinane is a Crank
   Jul 5, 2005

       Google Groups view of
   the main thread (at sci.math)
       to which crankbuster has posted

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Monday, July 11, 2005

Monday July 11, 2005

Filed under: General — m759 @ 12:00 am

Logos
for St. Benedict’s Day

Click on either of the logos below for religious meditations — on the left, a Jewish meditation from the Conference of Catholic Bishops; on the right, an Aryan meditation from Stormfront.org.

Both logos represent different embodiments of the “story theory” of truth, as opposed to the “diamond theory” of truth. Both logos claim, in their own ways, to represent the eternal Logos of the Christian religion. I personally prefer the “diamond theory” of truth, represented by the logo below.

See also the previous entry
and the entries of 7/11, 2003.

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Thursday, May 26, 2005

Thursday May 26, 2005

Filed under: General,Geometry — Tags: Logos and Branding — m759 @ 4:23 pm

Drama of the Diagonal

"The beautiful in mathematics
resides in contradiction.
Incommensurability, logoi alogoi, was
the first splendor in mathematics."
— Simone Weil, Oeuvres Choisies,
éd. Quarto , Gallimard, 1999, p. 100

Logos Alogos
by S. H. Cullinane

"To a mathematician, mathematical entities have their own existence, they habitate spaces created by their intention. They do things, things happen to them, they relate to one another. We can imagine on their behalf all sorts of stories, providing they don't contradict what we know of them. The drama of the diagonal, of the square…"

— Dennis Guedj, abstract of "The Drama of Mathematics," a talk to be given this July at the Mykonos conference on mathematics and narrative.

For the drama of the diagonal of the square, see

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Thursday May 26, 2005

Filed under: General,Geometry — m759 @ 4:00 am

The Changing

The previous entry dealt with a transformation
of the diamond figure from Plato’s Meno
into a visual proof of the Pythagorean theorem:

The image “http://www.log24.com/theory/images/DiamondTurning.jpg” cannot be displayed, because it contains errors.

Here is a transformation of Plato’s diamond
into the “gyronny” pattern of heraldry:

The image “http://www.log24.com/theory/images/Gyronny.gif” cannot be displayed, because it contains errors.

Viking Heraldry

The image “http://www.log24.com/theory/images/DiamondChanging2.jpg” cannot be displayed, because it contains errors.

For the mathematics dealing with
this sort of transformation, see
The Diamond 16 Puzzle and Diamond Theory.

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Saturday, March 12, 2005

Saturday March 12, 2005

Filed under: General,Geometry — m759 @ 5:09 am

Three Eleanors

Continued from March 10:

For some children…

The image “http://www.log24.com/log/pix05/050310-Burton.jpg” cannot be displayed, because it contains errors.

It takes three Eleanors.

1 2 3

For Alice, a beautiful child
who died in London
on Tuesday
at 72:

Today’s New York Times says that
Alice, the author of Fairy Tale,
was a
“passionately traditional Catholic.”

For related material, see
Immortal Diamond:
O’Hara, Hopkins, and Joyce.

See also the conflict between Trudeau’s
“diamond theory” and
“story theory”
of truth,

and Suzanne Keen‘s article from the
Catholic publication Commonweal:

Getting to Truth by Lying.

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Thursday, January 27, 2005

Thursday January 27, 2005

Filed under: General,Geometry — Tags: on050127 — m759 @ 2:29 am

Crystal Night

From artbook.com:

Mies van der Rohe:
Mies in Berlin

The image “http://www.log24.com/log/pix05/050127-Mies.jpg” cannot be displayed, because it contains errors.

Winner of
The Society of Architectural Historians
2002 Philip Johnson Award
for Excellence

Exhibition Catalog

"Published to accompany
a groundbreaking 2001 exhibition at
The Museum of Modern Art, New York."

From Mies and the Mastodon,
by Martin Filler, The New Republic,
issue dated Aug. 6, 2001:

"It would have been wiser for the new MoMA catalog… to have addressed the issue of his politics…. By ignoring such a central subject… the show gives off a mild stench of cover-up…. Only the German-born Rosemarie Haag Bletter (full disclosure: she is my wife) alludes to the verboten topic in her [catalog] essay on Mies's flirtation with crystal imagery, drawing a sharp parallel between the architect's extensive use of Kristallglas (plate glass) and the ensuing devastation of Kristallnacht, which erupted just three months after he left for the States."

Also from Filler's essay:

"Mies's rigorously simplified structures, typified by grids of steel and glass and an absence of applied ornament, represented the Platonic ideal of modernism for many people."

For more on history, politics, and
Mies's disciple Philip Johnson,
who died Tuesday evening, see

"We Cannot Not Know History."

For more on aesthetics, see the
Log24.net entry of Tuesday noon,

Diamonds Are Forever.

For more on a Platonic ideal of sorts,
see the following figure in two versions:

Version A, from Plato's Meno and
Diamond Theory,

The image “http://www.log24.com/log/pix05/050127-MenoDiamond.gif” cannot be displayed, because it contains errors.

and Version B,

The image “http://www.log24.com/log/pix05/050125-Forever.gif” cannot be displayed, because it contains errors.

from the date of Johnson's death
at his "famous crystalline box."

Was less more?

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Thursday, November 11, 2004

Thursday November 11, 2004

Filed under: General — m759 @ 4:04 pm

Blasphemy?

From the profile of
psifenix at livejournal.com:

"Perhaps I shall embrace Islam.
Its standards for poetry
seem very high."

— Susan Voight in the novel
Freedom & Necessity,
by Emma Bull and Steven Brust

From psifenix
at livejournal.com
on Oct. 3, 2004:

"S₄ is the one true God,
and C₂×C₂ is
His [normal] prophet."

— psifenix

"Group theory is so beautiful. If I ever get the chance to go back in time or mess with an agrarian civilization, I'm going to try to get them to worship finite groups of small order. Why? Symmetry is nice, and you can turn groups into pretty pictures, too… Each group could represent an attribute of god, and the nonabelian attributes would be the scary and wrathful ones. The prime cyclic groups could symbolize the cycles of life and death. And the trivial group would be the origin of all things!

I'm on crack, but that's okay!"

— psifenix

"You can turn groups
into pretty pictures" —

IMAGE- 'Study of O' and 'Portrait of O' (the octahedral group)

Shown above are two ways of
picturing the octahedral group O,
also known as
the symmetric group S₄.

For further details, see
Diamond Theory.

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Sunday, September 5, 2004

Sunday September 5, 2004

Filed under: General — m759 @ 4:00 pm

Symmetry and Change
in the Dreamtime

Notes from the Journal
of Steven H. Cullinane

Summary:

Aug 31 2004 07:31:01 PM	Early Evening, Shining Star
Sep 01 2004 09:00:35 AM	Words and Images
Sep 01 2004 12:07:28 PM	Whale Rider
Sep 02 2004 11:11:42 AM	Heaven and Earth
Sep 02 2004 07:00:23 PM	Whale Road
Sep 03 2004 12:00:54 AM	Cinderella’s Slipper
Sep 03 2004 10:01:56 AM	Another September Morn
Sep 03 2004 12:00:25 PM	Noon
Sep 03 2004 01:13:49 PM	De Nada
Sep 03 2004 03:17:13 PM	Ite, Missa Est

Tuesday, August 31, 2004

Symmetry and Change, Part 1…

Early Evening,
Shining Star

7:31:01 PM ET

Hexagram 01
The Creative:

The Image

Heaven

The movement of heaven
is full of power.

Click on picture
for details.

The Clare Lawler Prize
for Literature goes to…

Under the Volcano,
Chapter VI:

“What have I got out of my life? Contacts with famous men… The occasion Einstein asked me the time, for instance. That summer evening…. smiles when I say I don’t know. And yet asked me. Yes: the great Jew, who has upset the whole world’s notions of time and space, once leaned down… to ask me… ragged freshman… at the first approach of the evening star, the time. And smiled again when I pointed out the clock neither of us had noticed.”

For the thoughts on time
of another famous man,
from Mexico, see the
Nobel Prize acceptance speech
of Octavio Paz,
“In Search of the Present.”

Wednesday, September 1, 2004

Symmetry and Change, Part 2…

Words and Images

9:00:35 AM ET

Hexagram 35
Progress:

The Image

Fire

Earth

The sun rises over the earth.

From Aug. 18, 2004:

“Oh, my Lolita. I have only words
to play with!” (Nabokov, Lolita)

“This is the best toy train set
a boy ever had!”
(Orson Welles, after first touring
RKO Studios, quoted in Halliwell)

“As the quotes above by Nabokov and Welles suggest, we need to be able to account for the specific functions available to narrative in each medium, for the specific elements that empirical creators will ‘play with’ in crafting their narratives.”

— Donald F. Larsson

For
James Whale
and
William French Anderson —

Words
In the Spirit of
Dave Barry’s Book of Bad Songs:

Stay for just a while…
Stay, and let me look at you.
It’s been so long, I hardly knew you.
Standing in the door…
Stay with me a while.
I only want to talk to you.
We’ve traveled halfway ’round the world
To find ourselves again.

September morn…
We danced until the night
      became a brand new day,
Two lovers playing scenes
      from some romantic play.
September morning still can
      make me feel this way.

Look at what you’ve done…
Why, you’ve become a grown-up girl…

— Neil Diamond

Images
In the Spirit of
September Morn:

The Last Day of Summer:
Photographs by Jock Sturges

“In 1990, the FBI entered Sturges’s studio and seized his work, claiming violation of child pornography laws.”

Related material:

Bill’s Diamond Theory

and

Log24 entries of
Aug. 15, 2004.

Those interested in the political implications of Diamond’s songs may enjoy Neil Performs at Kerry Fundraiser.

I personally enjoyed this site’s description of Billy Crystal’s remarks, which included “a joke about former President Clinton’s forthcoming children’s book — ‘It’s called The Little Engine That Could Because It Could.'”

“Puff, puff, woo, woo, off we go!”

Wednesday, September 1, 2004

Symmetry and Change, Part 3…

Whale Rider

12:07:28 PM

Hexagram 28
Preponderance of
the Great:

The Image

Lake

Wind

The lake rises
above the trees.

Cullinane College News:

“Congratulations to Clare Lawler, who participated very successfully in the recently held Secondary Schools Judo Championships in Wellington.”

For an explanation of this entry’s title, see the previous two entries and

Oxford Word
(Log24, July 10, 2004)

Thursday, September 2, 2004

Symmetry and Change, Part 4…

Heaven and Earth

11:11:42 AM ET

Hexagram 42
Increase:

The Image

Wind

Thunder

Wind and thunder:
the image of Increase.

“This time resembles that of
the marriage of heaven and earth”

Kylie

Finney

Well if you want to ride
you gotta ride it like you find it.
Get your ticket at the station
of the Rock Island Line. — Lonnie Donegan (d. Nov. 3) and others

The Rock Island Line’s namesake depot
in Rock Island, Illinois

“What it all boiled down to really was everybody giving everybody else a hard time for no good reason whatever… You just couldn’t march to your own music. Nowadays, you couldn’t even hear it… It was lost, the music which each person had inside himself, and which put him in step with things as they should be.”

— The Grifters, Ch. 10, 1963, by
James Myers Thompson

“The Old Man’s still an artist
with a Thompson.”
— Terry in “Miller’s Crossing”

For some of “the music which
each person had inside,”
click on the picture
with the Thompson.

It may be that Kylie is,
in her own way, an artist…
with a 357:

(Hits counter at
The Quality of Diamond
as of 11:05 AM Sept. 2, 2004)

For more on
“the marriage of heaven and earth,”
see
Plato, Pegasus, and the Evening Star.

Thursday, September 2, 2004

Symmetry and Change, Part 5…

Whale Road

7:00:23 PM

Hexagram 23
Splitting Apart:

The Image

Mountain

Earth

The mountain rests
on the earth.

“… the plot is different but the monsters, names, and manner of speaking will ring a bell.”

— Frank Pinto, Jr., review of Seamus Heaney’s translation of Beowulf

Other recommended reading, found during a search for the implications of today’s previous entry, “Hexagram 42”:

Water Wings.

This excellent meditation
on symmetry and change
comes from a site whose
home page
has the following image:

Friday, September 3, 2004

Symmetry and Change, Part 6…

Cinderella’s Slipper

12:00:54 AM ET

Hexagram 54
The Marrying Maiden:

The Image

Thunder

Lake

See

The hundredletter
thunderwords of
Finnegans Wake

“… a Thoreau-like retreat
by a nearby lake….
Both men have
a ‘touch of the poet’….
The symmetry is perfect.”

Friday, September 3, 2004

Symmetry and Change, Part 7…

Another September Morn

10:01:56 AM ET

Hexagram 56:
The Wanderer

The Image

Fire

Mountain

Fire on the mountain,
Run boys run…
Devil’s in the House of
The Rising Sun!

Friday, September 3, 2004

Symmetry and Change, Part 8…

Noon

12:00:25 PM ET

Hexagram 25
Innocence:

The Image

Heaven

Thunder

Under heaven
thunder rolls.

Friday, September 3, 2004

Symmetry and Change, Part 9…

De Nada

Helen Lane

1:13:49 PM ET

Hexagram 49
Revolution:

The Image

Lake

Fire Fire in the lake:
the image of Revolution.

“I sit now in a little room off the bar at four-thirty in the morning drinking ochas and then mescal and writing this on some Bella Vista notepaper I filched the other night…. But this is worst of all, to feel your soul dying. I wonder if it is because to-night my soul has really died that I feel at the moment something like peace. Or is it because right through hell there is a path, as Blake well knew, and though I may not take it, sometimes lately in dreams I have been able to see it? …And this is how I sometimes think of myself, as a great explorer who has discovered some extraordinary land from which he can never return to give his knowledge to the world: but the name of this land is hell. It is not Mexico of course but in the heart.”

— Malcolm Lowry, Under the Volcano

Friday, September 3, 2004

Symmetry and Change, conclusion…

Ite, Missa Est

3:17:13 PM ET

Hexagram 13
Fellowship With Men:

The Image

Heaven

Fire

Heaven together with fire.

“A pretty girl —
is like a melody —- !”

For details, see
A Mass for Lucero.

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Wednesday, September 1, 2004

Wednesday September 1, 2004

Filed under: General — m759 @ 9:00 am

Symmetry and Change, Part 2…

Words and Images

9:00:35 AM ET

Hexagram 35
Progress:

The Image

Fire

Earth

The sun rises over the earth.

From Aug. 18, 2004:

“Oh, my Lolita. I have only words
to play with!” (Nabokov, Lolita)

“This is the best toy train set
a boy ever had!”
(Orson Welles, after first touring
RKO Studios, quoted in Halliwell)

— Donald F. Larsson

For
James Whale
and
William French Anderson —

Words
In the Spirit of
Dave Barry’s Book of Bad Songs:

September morn…
We danced until the night
      became a brand new day,
Two lovers playing scenes
      from some romantic play.
September morning still can
      make me feel this way.

Look at what you’ve done…
Why, you’ve become a grown-up girl…

— Neil Diamond

Images
In the Spirit of
September Morn:

The Last Day of Summer:
Photographs by Jock Sturges

“In 1990, the FBI entered Sturges’s studio and seized his work, claiming violation of child pornography laws.”

Related material:

Bill’s Diamond Theory

and

Log24 entries of
Aug. 15, 2004.

Those interested in the political implications of Diamond’s songs may enjoy Neil Performs at Kerry Fundraiser.

“Puff, puff, woo, woo, off we go!”

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Wednesday, May 19, 2004

Wednesday May 19, 2004

Filed under: General,Geometry — Tags: May 19 Gestalt, Parable — m759 @ 2:00 pm

Style

In memory of Lynn H. Loomis:

The above diagram is from a
(paper) journal note of October 21, 1999.

It pictures the relationship of my own discovery, diamond theory (at center), to the field, harmonic analysis, of Professor Loomis, a writer whose style I have long admired.

A quotation from the 1999 note:

"…it is not impossible to draw a fairly sharp dividing line between our mental disposition in the case of esthetic response and that of the responses of ordinary life. A far more difficult question arises if we try to distinguish it from the responses made by us to certain abstract mental constructions such as those of pure mathematics…. Perhaps the distinction lies in this, that in the case of works of art the whole end and purpose is found in the exact quality of the emotional state, whereas in the case of mathematics the purpose is the constatation of the universal validity of the relations without regard to the quality of the emotion accompanying apprehension. Still, it would be impossible to deny the close similarity of the orientation of faculties and attention in the two cases."
— Roger Fry, Transformations (1926), Doubleday Anchor paperback, 1956, p. 8

In other words, appreciating mathematics is much like appreciating art.

(Digitized diagram courtesy of Violet.)

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Monday, April 5, 2004

Monday April 5, 2004

Filed under: General,Geometry — Tags: Change Logos, Gods and Giants, Strange Change — m759 @ 4:03 am

Ideas and Art

— Motto of
Plato's Academy

From Minimalist Fantasies,
by Roger Kimball, May 2003:

All I want anyone to get out of my paintings, and all I ever get out of them, is the fact that you can see the whole idea without any confusion. … What you see is what you see.
—Frank Stella, 1966

Minimal Art remains too much a feat of ideation, and not enough anything else. Its idea remains an idea, something deduced instead of felt and discovered.
— Clement Greenberg, 1967

The artists even questioned whether art needed to be a tangible object. Minimalism … Conceptualism — suddenly art could be nothing more than an idea, a thought on a piece of paper….
— Michael Kimmelman, 2003

There was a period, a decade or two ago, when you could hardly open an art journal without encountering the quotation from Frank Stella I used as an epigraph. The bit about “what you see is what you see” was reproduced ad nauseam. It was thought by some to be very deep. In fact, Stella’s remarks—from a joint interview with him and Donald Judd—serve chiefly to underscore the artistic emptiness of the whole project of minimalism. No one can argue with the proposition that “what you see is what you see,” but there’s a lot to argue with in what he calls “the fact that you can see the whole idea without any confusion.” We do not, of course, see ideas. Stella’s assertion to the contrary might be an instance of verbal carelessness, but it is not merely verbal carelessness. At the center of minimalism, as Clement Greenberg noted, is the triumph of ideation over feeling and perception, over aesthetics.
— Roger Kimball, 2003

From How Not Much Is a Whole World,
by Michael Kimmelman, April 2, 2004

Decades on, it's curious how much Minimalism, the last great high modern movement, still troubles people who just can't see why … a plain white canvas with a line painted across it

"William Clark,"
by Patricia Johanson, 1967

should be considered art. That line might as well be in the sand: on this side is art, it implies. Go ahead. Cross it.

….

The tug of an art that unapologetically sees itself as on a par with science and religion is not to be underestimated, either. Philosophical ambition and formal modesty still constitute Minimalism's bottom line.

If what results can sometimes be more fodder for the brain than exciting to look at, it can also have a serene and exalted eloquence….

That line in the sand doesn't separate good art from bad, or art from nonart, but a wide world from an even wider one.

I maintain that of course
we can see ideas.

Example: the idea of
invariant structure.

"What modern painters
are trying to do,
if they only knew it,
is paint invariants."

— James J. Gibson, Leonardo,
Vol. 11, pp. 227-235.
Pergamon Press Ltd., 1978

For a discussion
of how this works, see
Block Designs,
4×4 Geometry, and
Diamond Theory.

Incidentally, structures like the one shown above are invariant under an important subgroup of the affine group AGL(4,2)… That is to say, they are not lost in translation. (See previous entry.)

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Friday, February 20, 2004

Friday February 20, 2004

Filed under: General,Geometry — Tags: Finite Relativity — m759 @ 3:24 pm

Finite Relativity

Today is the 18th birthday of my note

“The Relativity Problem in Finite Geometry.”

That note begins with a quotation from Weyl:

“This is the relativity problem: to fix objectively a class of equivalent coordinatizations and to ascertain the group of transformations S mediating between them.”

— Hermann Weyl, The Classical Groups, Princeton University Press, 1946, p. 16

Here is another quotation from Weyl, on the profound branch of mathematics known as Galois theory, which he says

“… is nothing else but the relativity theory for the set Sigma, a set which, by its discrete and finite character, is conceptually so much simpler than the infinite set of points in space or space-time dealt with by ordinary relativity theory.”

— Weyl, Symmetry, Princeton University Press, 1952, p. 138

This second quotation applies equally well to the much less profound, but more accessible, part of mathematics described in Diamond Theory and in my note of Feb. 20, 1986.

Comments (1)

Thursday, January 29, 2004

Thursday January 29, 2004

Filed under: General,Geometry — Tags: Against Gombrich, Gombrich-Douat, Truchet Tiles — m759 @ 2:30 pm

Misunderstanding
in the Theory of Design

"Whether or not we can follow the theorist in his demonstrations, there is one misunderstanding we must avoid at all cost. We must not confuse the analyses of geometrical symmetries with the mathematics of combination and permutation….

The earliest (and perhaps the rarest) treatise on the theory of design drives home this insight with marvellous precision."

— E. H. Gombrich, 1979, in
The Sense of Order

This is perhaps the stupidest remark I have ever read. The "treatise on the theory of design" that Gombrich refers to is

Dominique Douat, Méthode Pour Faire une Infinité de Desseins Differents…. Paris, 1722.

For some background, see

Truchet Tiling,

Truchet & Types:
Tiling Systems and Ornaments, and

Douat Designs.

Certain of the Truchet/Douat patterns have rather intriguing mathematical properties, sketched in my website Diamond Theory. These properties become clear if and only if we we do what Gombrich declares that we must not do: "confuse the analyses of geometrical symmetries with the mathematics of combination and permutation." (The verb "confuse" should, of course, be replaced by the verb "combine.")

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Saturday, December 20, 2003

Saturday December 20, 2003

Filed under: General,Geometry — Tags: Permutahedron — m759 @ 5:00 pm

White, Geometric, and Eternal

This afternoon's surfing:

Prompted by Edward Rothstein's own Fides et Ratio encyclical in today's NY Times, I googled him.

At the New York Review of Books, I came across the following by Rothstein:

"… statements about TNT can be represented within TNT: the formal system can, in a precise way, 'talk' about itself."

This naturally prompted me to check what is on TNT on this, the feast day of St. Emil Artin. At 5 PM this afternoon, we have Al Pacino in "The Devil's Advocate" — a perfect choice for the festival of an alleged saint.

Preparing for Al, I meditated on the mystical significance of the number 373, as explained in Zen and Language Games: the page number 373 in Robert Stone's theological classic A Flag for Sunrise conveys the metaphysical significance of the phrase "diamonds are forever" — "the eternal in the temporal," according to Stone's Catholic priest. This suggests a check of another theological classic, Pynchon's Gravity's Rainbow. Page 373 there begins with the following description of prewar Berlin:

"white and geometric."

This suggests the following illustration of a white and geometric object related to yesterday's entry on Helmut Wielandt:

From antiquark.com

Figure 1

(This object, which illustrates the phrase "makin' the changes," also occurs in this morning's entry on the death of a jazz musician.)

A further search for books containing "white" and "geometric" at Amazon.com yields the following:

Figure 2

From Mosaics, by
Fassett, Bahouth, and Patterson:

"A risco fountain in Mexico city, begun circa 1740 and made up of Mexican pottery and Chinese porcelain, including Ming.

The delicate oriental patterns on so many different-sized plates and saucers [are] underlined by the bold blue and white geometric tiles at the base."

Note that the tiles are those of Diamond Theory; the geometric object in figure 1 above illustrates a group that plays a central role in that theory.

Finally, the word "risco" (from Casa del Risco) associated with figure 2 above leads us to a rather significant theological site associated with the holy city of Santiago de Compostela:

Figure 3

Vicente Risco's
Dedalus in Compostela.

Figure 3 shows James Joyce (alias Dedalus), whose daughter Lucia inspired the recent entry Jazz on St. Lucia's Day — which in turn is related, by last night's 2:45 entry and by Figure 1, to the mathematics of group theory so well expounded by the putative saint Emil Artin.

"His lectures are best described as
polished diamonds."
— Fine Hall in its Golden Age,
by Gian-Carlo Rota

If Pynchon plays the role of devil's advocate suggested by his creation, in Gravity's Rainbow, of the character Emil Bummer, we may hope that Rota, no longer in time but now in eternity, can be persuaded to play the important role of saint's advocate for his Emil.

Update of 6:30 PM 12/20/03:

Riddled:

The Absolutist Faith
of The New York Times

White and Geometric, but not Eternal.

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Saturday, November 1, 2003

Saturday November 1, 2003

Filed under: General,Geometry — Tags: All Saints — m759 @ 1:05 pm

Symmetry in Diamond Theory:
Robbing Peter to Pay Paul

"Groups arise in most areas of pure and applied mathematics, usually as a set of operators or transformations of some structure. The appearance of a group generally reflects some kind of symmetry in the object under study, and such symmetry may be considered one of the fundamental notions of mathematics."

— Peter Webb

"Counter-change is sometimes known as Robbing Peter to Pay Paul."

— Helen Kelley Patchwork


Paul Robeson in King Solomon's Mines	Counterchange symmetry

For a look at the Soviet approach
to counterchange symmetry, see

The Kishinev School of Discrete Geometry.

The larger cultural context:

See War of Ideas (Oct. 24),
The Hunt for Red October (Oct. 25),
On the Left (Oct. 25), and
ART WARS for Trotsky's Birthday (Oct. 26).

Comments (2)

Friday, October 17, 2003

Friday October 17, 2003

Filed under: General,Geometry — m759 @ 4:15 pm

Happy Birthday, Arthur Miller

Miller, the author of “The Crucible,” is what Russell Baker has called a “tribal storyteller.”

From an essay by Baker in The New York Review of Books, issue dated November 6, 2003 (Fortieth Anniversary Issue):

“Among the privileges enjoyed by rich, fat, superpower America is the power to invent public reality. Politicians and the mass media do much of the inventing for us by telling us stories which purport to unfold a relatively simple reality. As our tribal storytellers, they shape our knowledge and ignorance of the world, not only producing ideas and emotions which influence the way we live our lives, but also leaving us dangerously unaware of the difference between stories and reality.”

— Russell Baker, “The Awful Truth,” NYRB 11/6/03, page 8

Here is a rather similar view of the media:

“Who Rules America?”.

The attentive student of this second essay will have no difficulty finding a single four-letter word to replace both of Baker’s phrases “rich, fat, superpower America” and “politicians and the mass media.”

Baker’s concern for “the difference between stories and reality” is reflected in my own website The Diamond Theory of Truth. In summary:

“Is it safe?” — Sir Laurence Olivier

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Friday, October 10, 2003

Friday October 10, 2003

Filed under: General — m759 @ 1:35 am

The West Wing’s
Story Line

From USATODAY.com, Oct. 9, 2003:

News analysis by Judy Keen, USA TODAY

Posted 10/9/2003 9:40 PM
Updated 10/9/2003 9:42 PM

WASHINGTON — President Bush’s fierce defense Thursday of the war with Iraq was part of an effort to regain control of the debate over the wisdom of the conflict….

Bush’s insistence that the United States “won’t run from a challenge” in Iraq was a sign that he and his top aides are doing what they always do when they’re in trouble. They attempt to recapture equilibrium by confronting critics and trying to control the story line.

See also the “story theory of truth”
versus the “diamond theory of truth.”

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Thursday, September 4, 2003

Thursday September 4, 2003

Filed under: General — Tags: Wind over Water — m759 @ 4:23 pm

Epitaphs

The late philosopher Donald Davidson (see previous entry) had a gift for titles. For example:

“The Folly of Trying to Define Truth”
(Journal of Philosophy June 1996, pp. 263-278) and

“A Nice Derangement of Epitaphs”
(In R. Grandy and R. Warner (eds.), Philosophical Grounds of Rationality, Oxford: Oxford University Press, 1986).

For my thoughts on the former, see

Pilate, Truth, and Friday the Thirteenth,

The Diamond Theory of Truth, and

Sept. 2, 2002 (Laurindo Almeida’s Birthday).

For my thoughts on the latter, see

Happy Birthday, Mary Shelley (2003),

For Mary Shelley’s Birthday (2002),

and, in honor of J. R. R. Tolkien, who died on the date September 2,

The Article on Epitaphs

at Wikipedia Encyclopedia, which contains the following:

“J. R. R. Tolkien is buried next to his wife, and on their tombstone the names ‘Beren’ and ‘Luthien’ are engraved, a fact that sheds light on the love story of Beren and Luthien which is recorded in several versions in his works.”

A nice derangement, indeed.

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Wednesday, September 3, 2003

Wednesday September 3, 2003

Filed under: General,Geometry — Tags: Cubehenge, Kummerhenge, Six-Set, Symmetric Generation, The Nutshell — m759 @ 3:00 pm

Reciprocity

From my entry of Sept. 1, 2003:

"…the principle of taking and giving, of learning and teaching, of listening and storytelling, in a word: of reciprocity….

… E. M. Forster famously advised his readers, 'Only connect.' 'Reciprocity' would be Michael Kruger's succinct philosophy, with all that the word implies."

— William Boyd, review of Himmelfarb, New York Times Book Review, October 30, 1994

Last year's entry on this date:

Today's birthday:
James Joseph Sylvester

"Mathematics is the music of reason."
— J. J. Sylvester

The picture above is of the complete graph K₆… Six points with an edge connecting every pair of points… Fifteen edges in all.

If diamond theory illustrates any general philosophical principle, it is probably the interplay of opposites…. "Reciprocity" in the sense of Lao Tzu. See

Reciprocity and Reversal in Lao Tzu.

For a sense of "reciprocity" more closely related to Michael Kruger's alleged philosophy, see the Confucian concept of Shu (Analects 15:23 or 24) described in

Shu: Reciprocity.

Click on the design for details.

Those who prefer a Jewish approach to physics can find the star of David, in the form of K₆, applied to the sixteen 4×4 Dirac matrices, in

A Graphical Representation
of the Dirac Algebra.

The star of David also appears, if only as a heuristic arrangement, in a note that shows generating partitions of the affine group on 64 points arranged in two opposing triplets.

The Jewel of Arithmetic and

The Golden Theorem.

Comments (1)

Monday, August 18, 2003

Monday August 18, 2003

Filed under: General,Geometry — Tags: Gopnik, Sorkin — m759 @ 3:09 pm

Entries since Xanga’s
August 10 Failure:

Sunday, August 17, 2003 2:00 PM

A Thorny Crown of…

West Wing's Toby Ziegler

From the first episode of
the television series
“The West Wing“:

Original airdate: Sept. 22, 1999
Written by Aaron Sorkin

MARY MARSH
That New York sense of humor. It always–

CALDWELL
Mary, there’s absolutely no need…

MARY MARSH
Please, Reverend, they think they’re so much smarter. They think it’s smart talk. But nobody else does.

JOSH
I’m actually from Connecticut, but that’s neither here nor there. The point is that I hope…

TOBY
She meant Jewish.

[A stunned silence. Everyone stares at Toby.]

TOBY (CONT.)
When she said “New York sense of humor,” she was talking about you and me.

JOSH
You know what, Toby, let’s just not even go there.

Going There, Part I

Crown of Ideas

Kirk Varnedoe, 57, art historian and former curator of the Museum of Modern Art, died Thursday, August 14, 2003.

From his New York Times obituary:

” ‘He loved life in its most tangible forms, and so for him art was as physical and pleasurable as being knocked down by a wave,’ said Adam Gopnik, the writer and a former student of his who collaborated on Mr. Varnedoe’s first big show at the Modern, ‘High & Low.’ ‘Art was always material first — it was never, ever bound by a thorny crown of ideas.’ ”

For a mini-exhibit of ideas in honor of Varnedoe, see

Fahne Hoch.

Verlyn Klinkenborg on Varnedoe:

“I was always struck by the tangibility of the words he used…. It was as if he were laying words down on the table one by one as he used them, like brushes in an artist’s studio. That was why students crowded into his classes and why the National Gallery of Art had overflow audiences for his Mellon Lectures earlier this year. Something synaptic happened when you listened to Kirk Varnedoe, and, remarkably, something synaptic happened when he listened to you. You never knew what you might discover together.”

Perhaps even a “thorny crown of ideas“?

“Crown of Thorns”
Cathedral, Brasilia

Varnedoe’s death coincided with
the Great Blackout of 2003.

“To what extent does this idea of a civic life produced by sense of adversity correspond to actual life in Brasília? I wonder if it is something which the city actually cultivates. Consider, for example the cathedral, on the monumental axis, a circular, concrete framed building whose sixteen ribs are both structural and symbolic, making a structure that reads unambiguously as a crown of thorns; other symbolic elements include the subterranean entrance, the visitor passing through a subterranean passage before emerging in the light of the body of the cathedral. And it is light, shockingly so….”

— Modernist Civic Space: The Case of Brasilia, by Richard J. Williams, Department of History of Art, University of Edinburgh, Scotland

Going There, Part II

Simple, Bold, Clear

Art historian Kirk Varnedoe was, of course, not the only one to die on the day of the Great Blackout.

Claude Martel, 34, a senior art director of The New York Times Magazine, also died on Thursday, August 14, 2003.

Janet Froelich, the magazine’s art director, describes below a sample of work that she and Martel did together:

“A new world of ideas”

Froelich notes that “the elements are simple, bold, and clear.”

For another example of elements with these qualities, see my journal entry

Fahne Hoch.

The flag design in that entry
might appeal to Aaron Sorkin’s
Christian antisemite:

Fahne,
S. H. Cullinane,
Aug. 15, 2003

Dr. Mengele,
according to
Hollywood

Note that the elements of the flag design have the qualities described so aptly by Froelich– simplicity, boldness, clarity:

They share these qualities with the Elements of Euclid, a treatise on geometrical ideas.

For the manner in which such concepts might serve as, in Gopnik’s memorable phrase, a “thorny crown of ideas,” see

“Geometry for Jews” in

ART WARS: Geometry as Conceptual Art.

See also the discussion of ideas in my journal entry on theology and art titled

Understanding: On Death and Truth

and the discussion of the word “idea” (as well as the word, and the concept, “Aryan”) in the following classic (introduced by poet W. H. Auden):

Saturday, August 16, 2003 6:00 AM

Varnedoe’s Crown

Kirk Varnedoe, 57, art historian and former curator of the Museum of Modern Art, died Thursday, August 14, 2003.

From his New York Times obituary:

For a mini-exhibit of ideas in honor of Varnedoe, see

Fahne Hoch.

Verlyn Klinkenborg on Varnedoe:

Perhaps even a “thorny crown of ideas”?

“Crown of Thorns”
Cathedral, Brasilia

Varnedoe’s death coincided with
the Great Blackout of 2003.

“To what extent does this idea of a civic life produced by sense of adversity correspond to actual life in Brasília? I wonder if it is something which the city actually cultivates. Consider, for example the cathedral, on the monumental axis, a circular, concrete framed building whose sixteen ribs are both structural and symbolic, making a structure that reads unambiguously as a crown of thorns; other symbolic elements include the subterranean entrance, the visitor passing through a subterranean passage before emerging in the light of the body of the cathedral. And it is light, shockingly so….”

— Modernist Civic Space: The Case of Brasilia, by Richard J. Williams, Department of History of Art, University of Edinburgh, Scotland

Friday, August 15, 2003 3:30 PM

ART WARS:

The Boys from Brazil

It turns out that the elementary half-square designs used in Diamond Theory

also appear in the work of artist Nicole Sigaud.

Sigaud’s website The ANACOM Project has a page that leads to the artist Athos Bulcão, famous for his work in Brasilia.

From the document

Conceptual Art in an
Authoritarian Political Context:
Brasilia, Brazil,

by Angélica Madeira:

“Athos created unique visual plans, tiles of high poetic significance, icons inseparable from the city.”

As Sigaud notes, two-color diagonally-divided squares play a large part in the art of Bulcão.

The title of Madeira’s article, and the remarks of Anna Chave on the relationship of conceptual/minimalist art to fascist rhetoric (see my May 9, 2003, entries), suggest possible illustrations for a more politicized version of Diamond Theory:

Fahne,
S. H. Cullinane,
Aug. 15, 2003

Dr. Mengele,
according to
Hollywood

Is it safe?

These illustrations were suggested in part by the fact that today is the anniversary of the death of Macbeth, King of Scotland, and in part by the following illustrations from my journal entries of July 13, 2003 comparing a MOMA curator to Lady Macbeth:

Die Fahne Hoch,
Frank Stella,
1959

Dorothy Miller,
MOMA curator,
died at 99 on
July 11, 2003.

Thursday, August 14, 2003 3:45 AM

Famous Last Words

The ending of an Aug. 14 Salon.com article on Mel Gibson’s new film, “The Passion”:

” ‘The Passion’ will most likely offer up the familiar puerile, stereotypical view of the evil Jew calling for Jesus’ blood and the clueless Pilate begging him to reconsider. It is a view guaranteed to stir anew the passions of the rabid Christian, and one that will send the Jews scurrying back to the dark corners of history.”

— Christopher Orlet

“Scurrying”?! The ghost of Joseph Goebbels, who famously portrayed Jews as sewer rats doing just that, must be laughing — perhaps along with the ghost of Lady Diana Mosley (née Mitford), who died Monday.

This goes well with a story that Orlet tells at his website:

“… to me, the most genuine last words are those that arise naturally from the moment, such as

Joseph Goebbels

Voltaire’s response to a request that he foreswear Satan: ‘This is no time to make new enemies.’ ”

For a view of Satan as an old, familiar, acquaintance, see the link to Prince Ombra in my entry last October 29 for Goebbels’s birthday.

Wednesday, August 13, 2003 3:00 PM

Best Picture

For some reflections inspired in part by

Today’s anniversary of a
1727 Moravian Pentecost,
An April 7, 2003,
lecture at Moravian College, and
St. Bonaventure’s 1259
remarks on the Trinity,

click here.

Tuesday, August 12, 2003 4:44 PM

Atonement:

A sequel to my entry “Catholic Tastes” of July 27, 2003.

Some remarks of Wallace Stevens that seem appropriate on this date:

“It may be that one life is a punishment
For another, as the son’s life for the father’s.”

— Esthétique du Mal, Wallace Stevens

Joseph Patrick Kennedy, Jr.

“Unless we believe in the hero, what is there
To believe? ….
Devise, devise, and make him of winter’s
Iciest core, a north star, central
In our oblivion, of summer’s
Imagination, the golden rescue:
The bread and wine of the mind….”

— Examination of the Hero in a Time of War, Wallace Stevens

Etymology of “Atonement”:

“Middle English atonen, to be reconciled, from at one, in agreement“

At One

“… We found,
If we found the central evil, the central good….
… we and the diamond globe at last were one.”

— Asides on the Oboe, Wallace Stevens

Tuesday, August 12, 2003 1:52 PM

Franken & ‘Stein,
Attorneys at Law

“Tue August 12, 2003 04:10 AM ET
NEW YORK (Reuters) – Fox News Network is suing humor writer Al Franken for trademark infringement over the phrase ‘fair and balanced’ on the cover of his upcoming book, saying it has been ‘a signature slogan’ of the network since 1996.”

Franken:
Fair?

‘Stein:
Balanced?

For answers, click on the pictures
of Franken and ‘Stein.

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Sunday, August 17, 2003

Sunday August 17, 2003

Filed under: General,Geometry — Tags: Cube Coloring, Cube School — m759 @ 6:21 pm

Diamond theory is the theory of affine groups over GF(2) acting on small square and cubic arrays. In the simplest case, the symmetric group of degree 4 acts on a two-colored diamond figure like that in Plato's Meno dialogue, yielding 24 distinct patterns, each of which has some ordinary or color-interchange symmetry .

This symmetry invariance can be generalized to (at least) a group of order approximately 1.3 trillion acting on a 4x4x4 array of cubes.

The theory has applications to finite geometry and to the construction of the large Witt design underlying the Mathieu group of degree 24.

Saturday, August 16, 2003

Saturday August 16, 2003

Filed under: General,Geometry — Tags: Against Gombrich, Gopnik, Truchet Tiles — m759 @ 2:16 am

My Personal Thorny Crown

Kirk Varnedoe, 57, art historian and former curator of the Museum of Modern Art, died Thursday, August 14, 2003.

From his New York Times obituary:

" 'He loved life in its most tangible forms, and so for him art was as physical and pleasurable as being knocked down by a wave,' said Adam Gopnik, the writer and a former student of his who collaborated on Mr. Varnedoe's first big show at the Modern, 'High & Low.' 'Art was always material first — it was never, ever bound by a thorny crown of ideas.' "

For some background on the phrase "thorny crown of ideas," see the web page

Understanding.

The phrase "thorny crown of ideas" is also of interest in the light of recent controversy over Mel Gibson's new film, "The Passion."

For details of the controversy, see Christopher Orlet's Aug. 14 essay at Salon.com,

Mel Gibson vs. "The Jews"

For a real "thorny crown of ideas," consider the following remarks by another art historian:

The earliest (and perhaps the rarest) treatise on the theory of design drives home this insight with marvellous precision."

— E. H. Gombrich, 1979, in
The Sense of Order

This is perhaps the most stupid remark I have ever read. The "treatise on the theory of design" that Gombrich refers to is

Dominique Douat, Methode pour faire une infinité de desseins differents avec des carreaux mipartis de deux couleurs par une ligne diagonale : ou observations du Pere Dominique Douat Religieux Carmes de la Province de Toulouse sur un memoire inséré dans l'Histoire de l'Académie Royale des Sciences de Paris l'année 1704, présenté par le Reverend Sebastien Truchet religieux du même ordre, Academicien honoraire, imprimé chez Jacques Quillau, Imprimeur Juré de l'Université, Paris 1722.

This is the title given at the web page

Truchet & Types:
Tiling Systems and Ornaments,

which gives some background.

Certain of the Truchet/Douat patterns have rather intriguing mathematical properties, sketched in my website Diamond Theory. These properties become clear if and only we we do what Gombrich moronically declares that we must not do: "confuse the analyses of geometrical symmetries with the mathematics of combination and permutation." (The verb "confuse" should, of course, be replaced by the verb "combine.")

What does all this have to do with

Mel Gibson vs. "The Jews" ?

As jesting Pilate seems to have realized, whenever Jews (or, for that matter, Christians) tell stories, issues of truth may arise. Such issues, as shown by current events in that damned Semitic Hell-on-Earth that used to be referred to as "the Holy Land," can be of life-and-death importance.

Scene from
The Passion

The Roman soldiers may have fashioned a physical crown of thorns, but the Jews are quite capable of fashioning a very uncomfortable crown of, as Gopnik says, "ideas."

Here is an example.

"Ernst Hans Josef Gombrich, who as an author went by the name E. H. Gombrich, was born in Vienna in 1909….

The Gombrich family was Jewish, but his parents felt this had no particular relevance. In later years Mr. Gombrich said that whether someone was Jewish or not was a preoccupation for the Gestapo."

— Michael Kimmelman's obituary for Gombrich in the New York Times. Kimmelman is chief art critic for the New York Times and author of the Times's Aug. 15 Varnedoe obituary.

The web page Understanding cited above contains a link to

Pilate, Truth, and Friday the Thirteenth,

a page combining some religious remarks with a quotation of an extremely patronizing and superficial reference to my own work (and, in passing, to Truchet/Douat patterns).

This reference, and the above-quoted remark by Gombrich, constitute my own modest claim to what the Jew Gopnik jokingly calls a "thorny crown of ideas."

To me it is no joke.

This partly accounts for the rather strained quality of the attempt at humor in a web page I put together yesterday in response to Varnedoe's obituary:

Fahne Hoch, Macbeth!

Another reason for the strained quality is my being struck by the synchronicity of reading Varnedoe's obituary shortly after I had done a journal entry related to the death in July of an earlier Museum of Modern Art curator. Like Robert A. Heinlein, I think the God of the Jews is a lousy deity and an even worse father figure. I do, however, believe in synchronicity.

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Saturday, August 9, 2003

Saturday August 9, 2003

Filed under: General — m759 @ 12:07 pm

Beware of…
Jews Peddling Stories:

An episode in the ongoing saga of the conflict between the "story theory of truth" and the "diamond theory of truth."

The following set of pictures summarizes some reflections on truth and reality suggested by the August 9, 2003, New York Times obituary of writer William Woolfolk, who died on July 20, 2003.

Woolfolk was the author of The Sex Goddess and was involved in the production of the comic book series The Spirit (see below).

The central strategy of the three Semitic religions — Judaism, Christianity, and Islam — is to pretend that we are all characters in a story whose author is God. This strategy suggests the following Trinity, based on the work of William Woolfolk (The Sex Goddess and The Spirit) and Steven Spielberg ("Catch Me If You Can"). Like other Semitic tales, the story of this Trinity should not be taken too seriously.

Woolfolk as a Jewish God	Woolfolk's Story
The Father as a Lutheran God	The Father's Story
The Son	The Son's Story
The Holy Spirit	The Holy Spirit's Story

A Confession of Faith:

Theology Based On the Film
"Catch Me If You Can":

The Son to God the Lutheran Father:

"I'm nothing really, just a kid in love with your daughter."

This is taken from a review of "Catch Me If You Can" by Thomas S. Hibbs.

For some philosophical background to this confession, see Hibbs's book

Shows About Nothing:
Nihilism in Popular Culture
from The Exorcist to Seinfeld.

By the way, today is the anniversary of the dropping on Nagasaki
of a made-in-USA Weapon of Mass Destruction, a plutonium bomb
affectionately named Fat Man.

Fat Man was a sequel to an earlier Jewish story,

Trinity.

Comments (4)

Monday, August 4, 2003

Monday August 4, 2003

Filed under: General — Tags: Milioti's Razr — m759 @ 1:00 pm

Venn's Trinity

Today is the birthday of logician John Venn.

From the St. Andrews History of Mathematics site:

"Venn considered three discs R, S, and T as typical subsets of a set U. The intersections of these discs and their complements divide U into 8 non-overlapping regions, the unions of which give 256 different Boolean combinations of the original sets R, S, T."

Last night's entry, "A Queer Religion," gave a Catholic view of the Trinity. Here are some less interesting but more fruitful thoughts inspired by Venn's diagram of the Trinity (or, indeed, of any three entities):

"To really know a subject you've got to learn a bit of its history…."
— John Baez, August 4, 2002

"We both know what memories can bring;
They bring diamonds and rust."
— Joan Baez, April 1975

For the "diamonds" brought by memories of the 2⁸combinations described above, consider how the symmetric group S₈ is related to the symmetries of the finite projective space PG(3,2). (See Diamond Theory.)

For the "rust," consider the following:

"Lay not up for yourselves treasures upon earth, where moth and rust doth corrupt…."
— Matthew 6:19

The letters R, U, S, T in the Venn diagram above are perhaps relevant here, symbolizing, if you will, the earthly confusion of language, as opposed to the heavenly clarity of mathematics.

As for MOTH, see the article Hometown Zeroes (which brings us yet again to the Viper Room, scene of River Phoenix's death) and the very skillfully designed website MOTHEMATICS.

Comments (1)

Friday, July 11, 2003

Friday July 11, 2003

Filed under: General,Geometry — m759 @ 6:00 am

Links for St. Benedict

Today is the feast of St. Benedict.

Here is a link from the left:

The Trial of Depleted Uranium,
by Saint Philip Berrigan

Here is a link from the right:

On a Preview of “The Passion,”
a film by Saint Mel Gibson

Both Berrigan and Gibson are devout Catholics. (I use the present tense for Berrigan, though he is dead, since, as a saint, he is not very dead.) Both are worthy of respect, and should be listened to carefully, even though the religion they espouse is that of Hitler and Torquemada.

Logos

For more details, see sites related to the above links…. Click on either of the logos below — on the left, a Jewish meditation from the Conference of Catholic Bishops; on the right, an Aryan meditation from Stormfront.org.

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Saturday, July 5, 2003

Saturday July 5, 2003

Filed under: General,Geometry — m759 @ 7:21 pm

Elementary,
My Dear Gropius

“What is space, how can it be understood and given a form?”
— Walter Gropius

Stoicheia:

Stoicheia,” Elements, is the title of
Euclid’s treatise on geometry.

Stoicheia is apparently also related to a Greek verb meaning “march” or “walk.”

According to a website on St. Paul’s phrase “ta stoicheia tou kosmou,” which might be translated

The Elements of the Cosmos,

“… the verbal form of the root stoicheo was used to mean, ‘to be in a line,’ ‘to march in rank and file.’ … The general meaning of the noun form (stoicheion) was ‘what belongs to a series.’ “

As noted in my previous entry, St. Paul used a form of stoicheo to say “let us also walk (stoichomen) by the Spirit.” (Galatians 5:25) The lunatic ravings* of Saul of Tarsus aside, the concepts of walking, of a spirit, and of elements may be combined if we imagine the ghost of Gropius strolling with the ghosts of Plato, Aristotle, and Euclid, and posing his question about space. Their reply might be along the following lines:

Combining stoicheia with a peripatetic peripateia (i.e., Aristotelian plot twist), we have the following diagram of Aristotle’s four stoicheia (elements),

which in turn is related, by the “Plato’s diamond” figure in the monograph Diamond Theory, to the Stoicheia, or Elements, of Euclid.

Quod erat demonstrandum.

* A phrase in memory of the Paulist Norman J. O’Connor, the “jazz priest” who died on St. Peter’s day, Sunday, June 29, 2003. Paulists are not, of course, entirely mad; the classic The Other Side of Silence: A Guide to Christian Meditation, by the Episcopal priest Morton Kelsey, was published by the Paulist Press.

Its cover (above), a different version of the four-elements theme, emphasizes the important Jungian concept of quaternity. Jung is perhaps the best guide to the bizarre world of Christian symbolism. It is perhaps ironic, although just, that the Paulist Fathers should distribute a picture of “ta stoicheia tou kosmou,” the concept that St. Paul himself railed against.

The above book by Kelsey should not be confused with another The Other Side of Silence, a work on gay history, although confusion would be understandable in light of recent ecclesiastical revelations.

Let us pray that if there is a heaven, Father O’Connor encounters there his fellow music enthusiast Cole Porter rather than the obnoxious Saul of Tarsus.

Comments (2)

Saturday July 5, 2003

Filed under: General,Geometry — m759 @ 4:17 am

Elements

In memory of Walter Gropius, founder of the Bauhaus and head of the Harvard Graduate School of Design. Gropius died on this date in 1969. He said that

"The objective of all creative effort in the visual arts is to give form to space. … But what is space, how can it be understood and given a form?"

"Alle bildnerische Arbeit will Raum gestalten. … Was ist Raum, wie können wir ihn erfassen und gestalten?"

Gropius

— "The Theory and Organization
of the Bauhaus" (1923)

I designed the following logo for my Diamond Theory site early this morning before reading in a calendar that today is the date of Gropius's death. Hence the above quote.

"And still those voices are calling
from far away…"

— The Eagles

Stoicheia:

("Stoicheia," Elements, is the title of
Euclid's treatise on geometry.)

Comments (3)

Sunday, June 22, 2003

Sunday June 22, 2003

Filed under: General,Geometry — m759 @ 2:28 am

The Real Hogwarts

is at no single geographical location; it is distributed throughout the planet, and it is perhaps best known (apart from its disguises in the fiction of J. K. Rowling, C. S. Lewis, Charles Williams, and other Inklings) as Christ Church. Some relevant links:

Christ Church College, Oxford

Harry Potter at Christ Church, Oxford

Christchurch, New Zealand

University of Canterbury
Physical Sciences Library:
Keeping Current with the Web:
Maths & Statistics, June 2002

Diamond Theory:
Symmetry in Binary Spaces
http://m759.freeservers.com/
The author of this site is Steven Cullinane, who has also written booklets on the subject. The web site provides detailed discussions of Diamond Theory, and is intended for college math students or mathematicians. According to Cullinane, Diamond Theory is best classified in the subject of “finite automorphism groups of algebraic, geometric, or combinatorial structures.” The site also includes links to other resources. From the NSDL Scout Report for Math, Engineering and Technology, Volume 1, No. 9, 7 June 2002, Copyright Internet Scout Project 1994-2002. http://scout.cs.wisc.edu

Christ Church, Christchurch Road,
Virginia Water, England

The Vicar of Christ Church has written a scathing attack on white-trash religion, “Christian Zionism,” that should be required reading for all prospective U. S. demagogues.

Finally, on this Sunday in June, with The New York Review of Books of July 3, 2003, headlining the religion of Scientism (Freeman Dyson reviewing Gleick’s new book on Newton), it seems fitting to provide a link to an oasis of civilisation in the home town of mathematician John Nash — Bluefield, West Virginia.

Christ Church,
Bluefield, West Virginia

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Saturday, June 14, 2003

Saturday June 14, 2003

Filed under: General,Geometry — m759 @ 5:00 pm

Indiana Jones
and the Hidden Coffer

In memory of Bernard Williams,

Oxford philosopher, who died Tuesday, June 10, 2003.

“…in… Truth and Truthfulness [September, 2002], he sought to speak plainly, and took on the post-modern, politically correct notion that truth is merely relative…”

— Christopher Lehmann-Haupt

A new epistemology is emerging to replace the Diamond Theory of truth. I will call it the ‘Story Theory’ of truth: There are no diamonds. People make up stories about what they experience. Stories that catch on are called ‘true.’ The Story Theory of truth is itself a story that is catching on. It is being told and retold, with increasing frequency, by thinkers of many stripes…. My own viewpoint is the Story Theory….”

— Richard J. Trudeau, The Non-Euclidean Revolution, Birkhauser Boston, 1987

Today is the feast day of Saint Jorge Luis Borges (b. Buenos Aires, August 24, 1899 – d. Geneva, June 14, 1986).

From Borges’s “The Aleph“:

“The Faithful who gather at the mosque of Amr, in Cairo, are acquainted with the fact that the entire universe lies inside one of the stone pillars that ring its central court…. The mosque dates from the seventh century; the pillars come from other temples of pre-Islamic religions…. Does this Aleph exist in the heart of a stone?”

(“Los fieles que concurren a la mezquita de Amr, en el Cairo, saben muy bien que el universo está en el interior de una de las columnas de piedra que rodean el patio central…. la mezquita data del siglo VII; las columnas proceden de otros templos de religiones anteislámicas…. ¿Existe ese Aleph en lo íntimo de una piedra?”)

From The Hunchback of Notre Dame:

Un cofre de gran riqueza
Hallaron dentro un pilar,
Dentro del, nuevas banderas
Con figuras de espantar.*

* A coffer of great richness
In a pillar’s heart they found,
Within it lay new banners,
With figures to astound.

See also the figures obtained by coloring and permuting parts of the above religious symbol.

Lena Olin and Harrison Ford
in “Hollywood Homicide“

Comments (2)

Monday, May 26, 2003

Monday May 26, 2003

Filed under: General,Geometry — m759 @ 4:25 am

Mental Health Month, Day 26:

Many Dimensions,
Part II — The Blue Matrix

But seriously…

John Baez in July 1999:

"…it's really the fact that the Leech lattice is 24-dimensional that lets us compactify 26-dimensional spacetime in such a way as to get a bosonic string theory with the Monster group as symmetries."

Well, maybe. I certainly hope so. If the Leech lattice and the Monster group turn out to have some significance in theoretical physics, then my own work, which deals with symmetries of substructures of the Leech lattice and the Monster, might be viewed in a different light. Meanwhile, I take (cold) comfort from some writers who pursue the "story" theory of truth, as opposed to the "diamond" theory. See the following from my journal:

Evariste Galois and the Rock that Changed Things, and

A Time to Gather Stones Together: Readings for Yom Kippur.

See, too, this web page on Marion Zimmer Bradley's fictional

Matrices, or Blue Star-Stones, and

the purely mathematical site Diamond Theory, which deals with properties of the above "blue matrix" and its larger relatives.

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Thursday, May 22, 2003

Thursday May 22, 2003

Filed under: General,Geometry — m759 @ 12:25 am

Mental Health Month:
Springtime for Wagner

“And now what you’ve all been waiting for…

Wagner!“

— Colin Hay as Zac in the film “Cosi“

“When I sought those who would sympathize with my plans, I had only you, the friends of my particular art, my most personal work and creation, to turn to.”

— Wagner’s address at the ceremony for the laying of the foundation stone of the Festival Theater in Bayreuth, May 22 (Wagner’s birthday), 1872

“The new computer package DISCRETA which was created in Bayreuth is in the process of permanent development.”

— “A Computer Approach to the Enumeration of Block Designs Which Are Invariant With Respect to a Prescribed Permutation Group”

The above is a preprint from Dresden.

See, too, the work of Bierbrauer, who received his doctorate at Mainz in 1977 and taught at Heidelberg from 1977 to 1994. Bierbrauer’s lecture notes give a particularly good background for the concepts involved in my Diamond Theory, in the tradition of Witt and Artin. See

Introduction to Group Theory
and Applications,

by Jürgen Bierbrauer, 138 pp., PostScript

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Monday, May 12, 2003

Monday May 12, 2003

Filed under: General — m759 @ 3:33 am

Death and Truth

Material related to my May 9, 2003, notes:

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Friday, May 9, 2003

Friday May 9, 2003

Filed under: General,Geometry — Tags: Transparent Things — m759 @ 7:20 pm

ART WARS:
The Religion of Cubism

In the dome of the Capitol at Washington, DC, a painting depicts The Apotheosis of Washington . Personally, I prefer the following pair of pictures, which might be titled Apotheosis of the Cube.

logo

Die

A New York Times article says Tony Smith's instructions for fabricating Die were as follows:

"a six-foot cube of quarter-inch hot-rolled steel with diagonal internal bracing."

The transparent cube in the upper picture above shows the internal diagonals. The fact that there are four of these may be used to demonstrate the isomorphism of the group of rotations of the cube with the group of permutations on an arbitrary set of four elements. For deeper results, see Diamond Theory.

For an explanation of why our current president might feel that the cube deserves an apotheosis, see the previous entry, "The Rhetoric of Power."

See, too, Nabokov's Transparent Things :

"Its ultimate vision was the incandescence of a book or a box grown completely transparent and hollow. This is, I believe, it: not the crude anguish of physical death but the incomparable pangs of the mysterious mental maneuver needed to pass from one state of being to another. Easy, you know, does it, son."

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Friday May 9, 2003

Filed under: General,Geometry — m759 @ 6:30 pm

ART WARS

The Rhetoric of Power:
A meditation for Mental Health Month

From “Secondary Structures,” by Tom Moody, Sculpture Magazine, June 2000:

“By the early ’90s, the perception of Minimalism as a ‘pure’ art untouched by history lay in tatters. The coup de grâce against the movement came not from an artwork, however, but from a text. Shortly after the removal of Richard Serra’s Tilted Arc from New York City’s Federal Plaza, Harvard art historian Anna Chave published ‘Minimalism and the Rhetoric of Power’ (Arts Magazine, January 1990), a rousing attack on the boys’ club that stops just short of a full-blown ad hominem rant. Analyzing artworks (Walter de Maria’s aluminum swastika, Morris’s ‘carceral images,’ Flavin’s phallic ‘hot rods’), critical vocabulary (Morris’s use of ‘intimacy’ as a negative, Judd’s incantatory use of the word ‘powerful’), even titles (Frank Stella’s National Socialist-tinged Arbeit Macht Frei and Reichstag), Chave highlights the disturbing undercurrents of hypermasculinity and social control beneath Minimalism’s bland exterior. Seeing it through the eyes of the ordinary viewer, she concludes that ‘what [most] disturbs [the public at large] about Minimalist art may be what disturbs them about their own lives and times, as the face it projects is society’s blankest, steeliest face; the impersonal face of technology, industry and commerce; the unyielding face of the father: a face that is usually far more attractively masked.’ ”

From Maureen Dowd’s New York Times column of June 9, 2002:

“The shape of the government is not as important as the policy of the government. If he makes the policy aggressive and pre-emptive, the president can conduct the war on terror from the National Gallery of Art.”

From the New York Times,
Friday, May 2, 2003:

The National Gallery of Art in Washington has just acquired Tony Smith’s first steel sculpture: “Die,” created in 1962 and fabricated in 1968.

“It’s a seminal icon of postwar American art,” said Earl A. Powell III, director of the National Gallery.

	Die (Tony Smith)
	Bishop Moore

From a New York Times obituary,
Friday, May 2, 2003:

Bishop Dies

by Ari L. Goldman

Paul Moore Jr., the retired Episcopal bishop of New York who for more than a decade was the most formidable liberal Christian voice in the city, died yesterday at home in Greenwich Village. He was 83….

Bishop Moore argued for his agenda in the most Christian of terms, refusing to cede Biblical language to the Christian right. Although he retired as bishop in 1989, he continued to speak out, taking to the pulpit of his former church as recently as March 24, even as illness overtook him, to protest the war in Iraq.

“It appears we have two types of religion here,” the bishop said, aiming his sharpest barbs at President Bush. “One is a solitary Texas politician who says, `I talk to Jesus, and I am right.’ The other involves millions of people of all faiths who disagree.”

He added: “I think it is terrifying. I believe it will lead to a terrible crack in the whole culture as we have come to know it.”….

[In reference to another question] Bishop Moore later acknowledged that his rhetoric was strong, but added, “In this city you have to speak strongly to be heard.”

Paul Moore’s early life does not immediately suggest an affinity for the kinds of social issues that he would later champion…. His grandfather was one of the founders of Bankers Trust. His father was a good friend of Senator Prescott Bush, whose son, George H. W. Bush, and grandson, George W. Bush, would become United States presidents.

Related material (update of May 12, 2003):

Question:

Which of the two theories of truth in reading (2) above is exemplified by Moore’s March 24 remarks?

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Monday, April 28, 2003

Monday April 28, 2003

Filed under: General,Geometry — Tags: 4403, Octad, Octad Generator — m759 @ 12:07 am

ART WARS:

Toward Eternity

April is Poetry Month, according to the Academy of American Poets. It is also Mathematics Awareness Month, funded by the National Security Agency; this year's theme is "Mathematics and Art."

Some previous journal entries for this month seem to be summarized by Emily Dickinson's remarks:

"Because I could not stop for Death–
He kindly stopped for me–
The Carriage held but just Ourselves–
And Immortality.

………………………
Since then–'tis Centuries–and yet
Feels shorter than the Day
I first surmised the Horses' Heads
Were toward Eternity– "

Consider the following journal entries from April 7, 2003:

Math Awareness Month

April is Math Awareness Month.
This year's theme is "mathematics and art."

An Offer He Couldn't Refuse

Today's birthday: Francis Ford Coppola is 64.

"There is a pleasantly discursive treatment
of Pontius Pilate's unanswered question
'What is truth?'."

— H. S. M. Coxeter, 1987, introduction to Richard J. Trudeau's remarks on the "Story Theory" of truth as opposed to the "Diamond Theory" of truth in The Non-Euclidean Revolution

From a website titled simply Sinatra:

"Then came From Here to Eternity. Sinatra lobbied hard for the role, practically getting on his knees to secure the role of the street smart punk G.I. Maggio. He sensed this was a role that could revive his career, and his instincts were right. There are lots of stories about how Columbia Studio head Harry Cohn was convinced to give the role to Sinatra, the most famous of which is expanded upon in the horse's head sequence in The Godfather. Maybe no one will know the truth about that. The one truth we do know is that the feisty New Jersey actor won the Academy Award as Best Supporting Actor for his work in From Here to Eternity. It was no looking back from then on."

From a note on geometry of April 28, 1985:

The "horse's head" figure above is from a note I wrote on this date 18 years ago. The following journal entry from April 4, 2003, gives some details:

The Eight

Today, the fourth day of the fourth month, plays an important part in Katherine Neville's The Eight. Let us honor this work, perhaps the greatest bad novel of the twentieth century, by reflecting on some properties of the number eight. Consider eight rectangular cells arranged in an array of four rows and two columns. Let us label these cells with coordinates, then apply a permutation.

Decimal
labeling

Binary
labeling

Algebraic
labeling

Permutation
labeling

The resulting set of arrows that indicate the movement of cells in a permutation (known as a Singer 7-cycle) outlines rather neatly, in view of the chess theme of The Eight, a knight. This makes as much sense as anything in Neville's fiction, and has the merit of being based on fact. It also, albeit rather crudely, illustrates the "Mathematics and Art" theme of this year's Mathematics Awareness Month.

The visual appearance of the "knight" permutation is less important than the fact that it leads to a construction (due to R. T. Curtis) of the Mathieu group M₂₄ (via the Curtis Miracle Octad Generator), which in turn leads logically to the Monster group and to related "moonshine" investigations in the theory of modular functions. See also "Pieces of Eight," by Robert L. Griess.

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Friday, April 25, 2003

Friday April 25, 2003

Filed under: General,Geometry — Tags: Columns, Temple, Valéry — m759 @ 7:59 pm

Mark

Today is the feast of Saint Mark. It seems an appropriate day to thank Dr. Gerald McDaniel for his online cultural calendar, which is invaluable for suggesting blog topics.

Yesterday's entry "Cross-Referenced" referred to a bizarre meditation of mine titled "The Matthias Defense," which combines some thoughts of Nabokov on lunacy with some of my own thoughts on the Judeo-Christian tradition (i.e., also on lunacy). In this connection, the following is of interest:

From a site titled Meaning of the Twentieth Century —

"Freeman Dyson has expressed some thoughts on craziness. In a Scientific American article called 'Innovation in Physics,' he began by quoting Niels Bohr. Bohr had been in attendance at a lecture in which Wolfgang Pauli proposed a new theory of elementary particles. Pauli came under heavy criticism, which Bohr summed up for him: 'We are all agreed that your theory is crazy. The question which divides us is whether it is crazy enough to have a chance of being correct. My own feeling is that is not crazy enough.' To that Freeman added: 'When a great innovation appears, it will almost certainly be in a muddled, incomplete and confusing form. To the discoverer, himself, it will be only half understood; to everyone else, it will be a mystery. For any speculation which does not at first glance look crazy, there is no hope!' "

— Kenneth Brower, The Starship and the Canoe, 1979, pp. 146, 147

It is my hope that the speculation, implied in The Matthias Defense, that the number 162 has astonishing mystical properties (as a page number, article number, etc.) is sufficiently crazy to satisfy Pauli and his friend Jung as well as the more conventional thinkers Bohr and Dyson. It is no less crazy than Christianity, and has a certain mad simplicity that perhaps improves on some of that religion's lunatic doctrines.

Some fruits of the "162 theory" —

Searching on Google for muses 162, we find the following Orphic Hymn to Apollo and a footnote of interest:

27 Tis thine all Nature's music to inspire,
28 With various-sounding, harmonising lyre;
29 Now the last string thou tun'ft to sweet accord,
30 Divinely warbling now the highest chord….

"Page 162 Verse 29…. Now the last string…. Gesner well observes, in his notes to this Hymn, that the comparison and conjunction of the musical and astronomical elements are most ancient; being derived from Orpheus and Pythagoras, to Plato. Now, according to the Orphic and Pythagoric doctrine, the lyre of Apollo is an image of the celestial harmony…."

For the "highest chord" in a metaphorical sense, see selection 162 of the 1919 edition of The Oxford Book of English Verse (whose editor apparently had a strong religious belief in the Muses (led by Apollo)). This selection contains the phrase "an ever-fixèd mark" — appropriately enough for this saint's day. The word "mark," in turn, suggests a Google search for the phrase "runes to grave" Hardy, after a poem quoted in G. H. Hardy's A Mathematician's Apology.

Such a search yields a website that quotes Housman as the source of the "runes" phrase, and a further search yields what is apparently the entire poem:

Smooth Between Sea and Land

by A. E. Housman

Smooth between sea and land
Is laid the yellow sand,
And here through summer days
The seed of Adam plays.

Here the child comes to found
His unremaining mound,
And the grown lad to score
Two names upon the shore.

Here, on the level sand,
Between the sea and land,
What shall I build or write
Against the fall of night?

Tell me of runes to grave
That hold the bursting wave,
Or bastions to design
For longer date than mine.

Shall it be Troy or Rome
I fence against the foam
Or my own name, to stay
When I depart for aye?

Nothing: too near at hand
Planing the figured sand,
Effacing clean and fast
Cities not built to last
And charms devised in vain,
Pours the confounding main.

(Said to be from More Poems (Knopf, 1936), p. 64)

Housman asks the reader to tell him of runes to grave or bastions to design. Here, as examples, are one rune and one bastion.

The rune known as
"Dagaz"

Represents
the balance point or "still point."

The Nike Bastion

Dagaz: (Pronounced thaw-gauze, but with the "th" voiced as in "the," not unvoiced as in "thick") (Day or dawn.)

From Rune Meanings:

Dagaz means "breakthrough, awakening, awareness. Daylight clarity as opposed to nighttime uncertainty. A time to plan or embark upon an enterprise. The power of change directed by your own will, transformation. Hope/happiness, the ideal. Security and certainty. Growth and release. Balance point, the place where opposites meet."

Also known as "the rune of transformation."

For the Dagaz rune in another context, see Geometry of the I Ching. The geometry discussed there does, in a sense, "hold the bursting wave," through its connection with Walsh functions, hence with harmonic analysis.

Temple of Athena Nike on the Nike Bastion, the Acropolis, Athens. Here is a relevant passage from Paul Valéry's Eupalinos ou L'Architecte about another temple of four columns:

Et puis… Écoute, Phèdre (me disait-il encore), ce petit temple que j'ai bâti pour Hermès, à quelques pas d'ici, si tu savais ce qu'il est pour moi ! — Où le passant ne voit qu'une élégante chapelle, — c'est peu de chose: quatre colonnes, un style très simple, — j'ai mis le souvenir d'un clair jour de ma vie. Ô douce métamorphose ! Ce temple délicat, nul ne le sait, est l'image mathématique d'une fille de Corinthe que j'ai heureusement aimée. Il en reproduit fidèlement les proportions particulières. Il vit pour moi !

Four columns, in a sense more suited to Hardy's interests, are also a recurrent theme in The Diamond 16 Puzzle and Diamond Theory.

Apart from the word "mark" in The Oxford Book of English Verse, as noted above, neither the rune nor the bastion discussed has any apparent connection with the number 162… but seek and ye shall find.

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Monday, April 7, 2003

Monday April 7, 2003

Filed under: General,Geometry — Tags: Octad, on030407 — m759 @ 1:17 pm

An Offer He Couldn't Refuse

Today's birthday: Francis Ford Coppola is 64.

"There is a pleasantly discursive treatment
of Pontius Pilate's unanswered question
'What is truth?'."

— H. S. M. Coxeter, 1987, introduction to Richard J. Trudeau's remarks on the "Story Theory" of truth as opposed to the "Diamond Theory" of truth in The Non-Euclidean Revolution

From a website titled simply Sinatra:

"Then came From Here to Eternity. Sinatra lobbied hard for the role, practically getting on his knees to secure the role of the street smart punk G.I. Maggio. He sensed this was a role that could revive his career, and his instincts were right. There are lots of stories about how Columbia Studio head Harry Cohn was convinced to give the role to Sinatra, the most famous of which is expanded upon in the horse's head sequence in The Godfather. Maybe no one will know the truth about that. The one truth we do know is that the feisty New Jersey actor won the Academy Award as Best Supporting Actor for his work in From Here to Eternity. It was no looking back from then on."

From a note on geometry of April 28, 1985:

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Monday, March 24, 2003

Monday March 24, 2003

Filed under: General — m759 @ 12:52 pm

Orwell’s question, according to
an admirer of leftist Noam Chomsky:

“When so much of the BS is right out in the open,
why is it that we know so little about it?
Why don’t we see what’s right in front of our eyes?”

Oscar

Deep Chomsky:
Lying, Truth-Telling,
and the Social Order

Michael
Moore

“First of all, I’d like to thank the Academy….”
— Quotation attributed to Plato

The New Yorker of March 31, 2003, discusses leftist academic Noam Chomsky. The online edition provides a web page listing pro-Chomsky links.

Chomsky’s influence is based in part on the popularity of his half-baked theories on linguistics, starting in the 1950’s with “deep structure” and “transformational,” or “generative,” grammar.

Chomsky has abandoned many of his previous ideas and currently touts what he calls The Minimalist Program.

For some background on Chomsky’s recent linguistic notions, see the expository essay “Syntactic Theory,” by Elly van Gelderen of the Arizona State University English Department. Van Gelderen lists her leftist political agenda on her “Other Interests” page. Her department may serve as an example of how leftists have converted many English departments in American universities to propaganda factories.

Some attacks on Chomsky’s scholarship:

The Emperor’s New Linguistics

The New Grammarians’ Funeral

Beyond Chomsky

Could Chomsky Be Wrong?

Forty-four Reasons Why the Chomskians Are Mistaken

Call for Papers, Chomsky 2003

Chomsky’s (Mis)Understanding of Human Thinking

Anatomy of a Revolution… Chomsky in 1962

…Linguistic Theory: The Rationality of Noam Chomsky

A Bibliography

Some attacks on Chomsky’s propaganda:

LeftWatch.com Chomsky page

Destructive Generation excerpt

The Sick Mind of Noam Chomsky

Partners in Hate: Noam Chomsky and the Holocaust Deniers

Chomsky and Plato’s Diamond

Like another purveyor of leftist nonsense, Jacques Derrida, Chomsky is fond of citing Plato as a precedent. In particular, what Chomsky calls “Plato’s problem” is discussed in Plato’s Meno. For a look at the diamond figure that plays a central role in that dialogue, see Diamond Theory. For an excellent overview of related material in Plato, see Theory of Forms.

Comments (2)

Thursday, March 13, 2003

Thursday March 13, 2003

Filed under: General,Geometry — m759 @ 2:45 am

Birthday Song

Today is the birthday of the late Jewish media magnate and art collector Walter H. Annenberg, whose name appears on a website that includes the following text:

Shape and Space in Geometry

“Making quilt blocks is an excellent way to explore symmetry. A quilt block is made of 16 smaller squares. Each small square consists of two triangles. Study this example of a quilt block:

This block has a certain symmetry. The right half is a mirror image of the left, and the top half is a mirror of the bottom.”

Symmetries of patterns such as the above are the subject of my 1976 monograph “ Diamond Theory,” which also deals with “shape and space in geometry,” but in a much more sophisticated way. For more on Annenberg, see my previous entry, “Daimon Theory.” For more on the historical significance of March 13, see Neil Sedaka, who also has a birthday today, in “ Jews in the News.”

Sedaka is, of course, noted for the hit tune “Happy Birthday, Sweet Sixteen,” our site music for today.

See also Geometry for Jews and related entries.

For the phrase “diamond theory” in a religious and philosophical context, see

Pilate, Truth, and Friday the Thirteenth.

“It’s quarter to three….” — Frank Sinatra

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Wednesday, March 12, 2003

Wednesday March 12, 2003

Filed under: General,Geometry — m759 @ 2:03 am

Daimon Theory

Today is allegedly the anniversary of the canonization, in 1622, of two rather important members of the Society of Jesus (Jesuits):

Ignatius Loyola…
Click here for Loyola’s legacy of strategic intelligence.

Francis Xavier…
Click here for Xavier’s legacy of strategic stupidity.

We can thank (or blame) a Jesuit (Gerard Manley Hopkins) for the poetic phrase “immortal diamond.” He may have been influenced by Plato, who has Socrates using a diamond figure in an argument for the immortality of the soul. Confusingly, Socrates also talked about his “daimon” (pronounced dye-moan). Combining these similar-sounding concepts, we have Doctor Stephen A. Diamond writing about daimons — a choice of author and topic that neatly combines the strategic intelligence of Loyola with the strategic stupidity of Xavier.

The cover illustration is perhaps not of Dr. Diamond himself.

A link between diamond theory and daimon theory is furnished by the charitable legacy of the non-practicing Jew Walter Annenberg.

For Annenberg and diamond theory, see this site on the elementary geometry of quilt blocks, which credits the Annenberg Foundation for support.

For Annenberg and daimon theory, see this site on Socrates, which has a similar Annenberg support credit.

Advanced disciples of Annenberg can learn much from the Perseus site about daimon theory. Let us pray that Abrahamic religious bigotry does not stand in their way. Less advanced disciples of Annenberg may find fulfillment in teaching children the beauty of elementary 4×4 quilt-block symmetry. Let us pray that academic bigotry does not prevent these same children, when they have grown older, from learning the deeper, and more difficult, beauties of diamond theory.

Daimon Theory

Diamond Theory

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Friday, February 21, 2003

Friday February 21, 2003

Filed under: General — m759 @ 7:08 pm

Shabbos Kodesh

Sabbath readings, music, video, etc.:

ThePhoenix.com club listing for
The Station, West Warwick, Rhode Island,
night of February 21-22, 2003:
“Sabbath Bloody Sabbath, Awake, Gov’t Cheese”
Sabbath Bloody Sabbath home page
Black Sabbath lyrics archive
Queen Shabbos Kodesh Holy Jewish Shabbat
Bill’s Diamond Theory

“Friday night and the lights are low…” — ABBA

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Sunday, January 5, 2003

Sunday January 5, 2003

Filed under: General,Geometry — Tags: Yankee Puzzle — m759 @ 12:12 am

Whirligig

Thus the whirligig of time brings in his revenges.
Twelfth Night. Act v. Sc. 1.

Twelfth night is the night of January 5-6.

Tonight is twelfth night in Australia; 4 AM Jan. 5
in New York City is 8 PM Jan. 5 in Sydney.

An October 6 entry:

Twenty-first Century Fox

On Sunday, October 6, 1889, the Moulin Rouge music hall opened in Paris, an event that to some extent foreshadowed the opening of Fox Studios Australia in Sydney on November 7, 1999. The Fox ceremonies included, notably, Kylie Minogue singing "Diamonds are a Girl's Best Friend."

Red Windmill

Kylie Minogue

For the mathematical properties of the red windmill (moulin rouge) figure at left, see Diamond Theory.

An October 5 entry:

The Message from Vega

"Mercilessly tasteful"
— Andrew Mueller,
review of Suzanne Vega's
"Songs in Red and Gray"

In accordance with the twelfth-night
"whirligig of time" theme,
here are two enigmatic quilt blocks:

Devil's Claws, or
Hourglass Var. 3

Yankee Puzzle, or
Hourglass Var. 5

One can approach these symbols in either a literary or a mathematical fashion. For a purely mathematical discussion of the differences in the two symbols' structure, see Diamond Theory. Those who prefer literary discussions may make up their own stories.

"Plato is wary of all forms of rapture other than reason's. He is most deeply leery of, because himself so susceptible to, the literary imagination. He speaks of it as a kind of holy madness or intoxication and goes on to link it to Eros, another derangement that joins us, but very dangerously, with the gods."

— Rebecca Goldstein in The New York Times,
December 16, 2002

"It's all in Plato, all in Plato; bless me,
what do they teach them at these schools?"

— C. S. Lewis in the Narnia Chronicles

Comments (1)

Monday, December 16, 2002

Monday December 16, 2002

Filed under: General,Geometry — m759 @ 10:00 pm

Rebecca Goldstein
at Heaven’s Gate

This entry is in gratitude for Rebecca Goldstein’s
excellent essay
in The New York Times of December 16, 2002.

She talks about the perennial conflict between two theories of truth that Richard Trudeau called the “story theory” and the “diamond theory.” My entry of December 13, 2002, “Rhyme Scheme,” links the word “real” to an article in the Stanford Encyclopedia of Philosophy that contains the following:

“According to a platonist about arithmetic, the truth of the sentence ‘7 is prime’ entails the existence of an abstract object, the number 7. This object is abstract because it has no spatial or temporal location, and is causally inert. A platonic realist about arithmetic will say that the number 7 exists and instantiates the property of being prime independently of anyone’s beliefs, linguistic practices, conceptual schemes, and so on. A certain kind of nominalist rejects the existence claim which the platonic realist makes: there are no abstract objects, so sentences such as ‘7 is prime’ are false…”

This discussion of “sevenness,” along with the discussion of “eightness” in my December 14, 2002, note on Bach, suggest that I supply a transcription of a note in my paper journal from 2001 that deals with these matters.

From a paper journal note of October 5, 2001:

The 2001 Silver Cup Award
for Realism in Mathematics
goes to…
Glynis Johns, star of
The Sword and the Rose,
Shake Hands with the Devil, and
No Highway in the Sky.

Glynis Johns is 78 today.

“Seven is heaven,
Eight is a gate.”
— from Dealing with Memory Changes
as You Grow Older,
by Kathleen Gose and Gloria Levi

“There is no highway in the sky.”
— Quotation attributed to Albert Einstein.
(See Gotthard Günther’s website
“Achilles and the Tortoise, Part 2”.)

“Don’t give up until you
Drink from the silver cup
And ride that highway in the sky.”
—America, 1974

See also page 78 of
Realism in Mathematics
(on Gödel’s Platonism)
by Penelope Maddy,
Clarendon Press, Oxford, 1990
(reprinted, 2000).

Added 12/17/02: See also
the portrait of Rebecca Goldstein in
Hadassah Magazine
Volume
78
Number 10
(June/July 1997).

For more on the Jewish propensity to
assign mystical significance to numbers, see
Rabbi Zwerin’s Kol Nidre Sermon.

For the significance of “seven” in Judaism, see
Zayin: The Woman of Valor.
For the significance of “eight” in Judaism, see
Chet: The Life Dynamic.

For the cabalistic significance of
“Seven is heaven, Eight is a gate,”
note that Zayin, Seven, signifies
“seven chambers of Paradise”
and that Chet, Eight, signifies
the “gateway to infinity.”

For the significance of the date 12.17, see
Tet: The Concealed Good.

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Friday, November 29, 2002

Friday November 29, 2002

Filed under: General,Geometry — Tags: Groups Act, Pascal's Diamond — m759 @ 1:06 pm

A Logocentric Archetype

Today we examine the relativist, nominalist, leftist, nihilist, despairing, depressing, absurd, and abominable work of Samuel Beckett, darling of the postmodernists.

One lens through which to view Beckett is an essay by Jennifer Martin, "Beckettian Drama as Protest: A Postmodern Examination of the 'Delogocentering' of Language." Martin begins her essay with two quotations: one from the contemptible French twerp Jacques Derrida, and one from Beckett's masterpiece of stupidity, Molloy. For a logocentric deconstruction of Derrida, see my note, "The Shining of May 29," which demonstrates how Derrida attempts to convert a rather important mathematical result to his brand of nauseating and pretentious nonsense, and of course gets it wrong. For a logocentric deconstruction of Molloy, consider the following passage:

"I took advantage of being at the seaside to lay in a store of sucking-stones. They were pebbles but I call them stones…. I distributed them equally among my four pockets, and sucked them turn and turn about. This raised a problem which I first solved in the following way. I had say sixteen stones, four in each of my four pockets these being the two pockets of my trousers and the two pockets of my greatcoat. Taking a stone from the right pocket of my greatcoat, and putting it in my mouth, I replaced it in the right pocket of my greatcoat by a stone from the right pocket of my trousers, which I replaced by a stone from the left pocket of my trousers, which I replaced by a stone from the left pocket of my greatcoat, which I replaced by the stone which was in my mouth, as soon as I had finished sucking it. Thus there were still four stones in each of my four pockets, but not quite the same stones….But this solution did not satisfy me fully. For it did not escape me that, by an extraordinary hazard, the four stones circulating thus might always be the same four."

Beckett is describing, in great detail, how a damned moron might approach the extraordinarily beautiful mathematical discipline known as group theory, founded by the French anticleric and leftist Evariste Galois. Disciples of Derrida may play at mimicking the politics of Galois, but will never come close to imitating his genius. For a worthwhile discussion of permutation groups acting on a set of 16 elements, see R. D. Carmichael's masterly work, Introduction to the Theory of Groups of Finite Order, Ginn, Boston, 1937, reprinted by Dover, New York, 1956.

There are at least two ways of approaching permutations on 16 elements in what Pascal calls "l'esprit géométrique." My website Diamond Theory discusses the action of the affine group in a four-dimensional finite geometry of 16 points. For a four-dimensional euclidean hypercube, or tesseract, with 16 vertices, see the highly logocentric movable illustration by Harry J. Smith. The concept of a tesseract was made famous, though seen through a glass darkly, by the Christian writer Madeleine L'Engle in her novel for children and young adults, A Wrinkle in Tme.

This tesseract may serve as an archetype for what Pascal, Simone Weil (see my earlier notes), Harry J. Smith, and Madeleine L'Engle might, borrowing their enemies' language, call their "logocentric" philosophy.

For a more literary antidote to postmodernist nihilism, see Archetypal Theory and Criticism, by Glen R. Gill.

For a discussion of the full range of meaning of the word "logos," which has rational as well as religious connotations, click here.

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Tuesday, November 26, 2002

Tuesday November 26, 2002

Filed under: General,Geometry — Tags: Logos and Branding — m759 @ 10:00 pm

Notes toward a Supreme Fact

In "Notes toward a Supreme Fiction," Wallace Stevens lists criteria for a work of the imagination:

It Must Be Abstract
It Must Change
It Must Give Pleasure.

For a work that seems to satisfy these criteria, see the movable images at my diamond theory website. Central to these images is the interplay of rational sides and irrational diagonals in square subimages.

"Logos and logic, crystal hypothesis,
Incipit and a form to speak the word
And every latent double in the word…."

— "Notes toward a Supreme Fiction," Section 1, Canto VIII

Recall that "logos" in Greek means "ratio," as well as (human or divine) "word." Thus when I read the following words of Simone Weil today, I thought of Stevens.

"The beautiful in mathematics resides in contradiction. Incommensurability, logoi alogoi , was the first splendor in mathematics."

— Simone Weil, Oeuvres Choisies , éd. Quarto, Gallimard, 1999, p. 100

In the conclusion of Section 3, Canto X, of "Notes," Stevens says

"They will get it straight one day at the Sorbonne.
We shall return at twilight from the lecture
Pleased that the irrational is rational…."

This is the logoi alogoi of Simone Weil.

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Monday, November 25, 2002

Monday November 25, 2002

Filed under: General,Geometry — m759 @ 1:00 pm

Swashbucklers and Misfits

There are two theories of truth, according to a a book on the history of geometry —

The “Story Theory” and the “Diamond Theory.”

For those who prefer the story theory…

From a review by Brian Hayes of A Beautiful Mind:

“Mathematical genius is rare enough. Cloaked in madness, or wrapped in serious eccentricity, it’s the stuff legends are made of.

There are brilliant and productive mathematicians who go to the office from nine to five, play tennis on the weekend, and worry about fixing the gearbox in the Volvo. Not many of them become the subjects of popular biographies. Instead we read about the great swashbucklers and misfits of mathematics, whose stories combine genius with high romance or eccentricity.”

Russell Crowe,
swashbuckler

Marilyn
Monroe,
misfit

Hollywood has recently given us a mathematical Russell Crowe. For a somewhat tougher sell, Marilyn Monroe as a mathematician, see “Insignificance,” 1985: “Marilyn Monroe on her hands and knees explains the theory of relativity to Albert Einstein.”

For a combination of misfit and swashbuckler in one Holy Name, see today’s earlier note, “The Artist’s Signature.”

See also my note of October 4, 2002, on Michelangelo, and the description of “the face of God” in this review.

Comments (1)

Thursday, November 21, 2002

Thursday November 21, 2002

Filed under: General — m759 @ 11:20 am

Back Again

Sorry for the hiatus in weblog entries since November 9. There were two reasons for this…

The five entries ending Nov. 9 formed a sort of story, taken as a whole, and I didn’t want to break up the set. But now I have archived this set of five entries. See my Diamond 16 Puzzle notes.
A very nasty entry in my Diamond Theory Forum site shook me up, and I haven’t felt like blogging until now.

Comments (1)

Saturday, November 9, 2002

Saturday November 9, 2002

Filed under: General,Geometry — m759 @ 4:44 am

Birthdate of Hermann Weyl

Weyl

Plato’s Diamond

Result of a Google search.

Category: Science > Math > Algebra > Group Theory

Weyl, H.: Symmetry.
Description of the book Symmetry by Weyl, H., published by Princeton University Press. … pup.princeton.edu/titles/
865.html – 7k – Nov. 8, 2002

Sunday, October 6, 2002

Sunday October 6, 2002

Filed under: General — m759 @ 4:40 am

Twenty-first Century Fox

Red Windmill

Kylie Minogue

For the mathematical properties of the red windmill (moulin rouge) figure at left, see Diamond Theory.

Comments (2)

Wednesday, October 2, 2002

Wednesday October 2, 2002

Filed under: General — Tags: Four Color — m759 @ 9:52 am

A Crackpot with Power

The following is an greatly abbreviated version of a sci.math group thread on an attempted proof of the four-color theorem.

Chip Eastham 2000-10-13 :
There is a nicely presented approach to proving the Four Color Theorem… at the following… site:

http://www.geocities.com/dharwadker/index.html
“Default” 2000-10-13:
Where in the proof is the hypothesis of “requiring N colors” (not colorable with N-1 colors) used?
Michael Varney 2000-10-14:
(Following some banter) Go play elsewhere if you buy into 4CT crackpot proofs.
“Default” 2000-10-14:
The proposed 4CT proof is hardly crackpot, and may contain some new ideas (or reformulations of old ones).
Michael Varney 2000-10-14:
That’s what all crackpots say. Join the club.
David Eppstein 2000-10-14:
My first-glance reaction is that it’s an amazing collection of undigested chunks of heavy equipment. It seems more designed to confuse any expert (by making sure it contains something the expert doesn’t understand) than to convince anyone of the truth of the 4CT.
“Default” 2000-10-15:
Skimming the proof I did not see any place where the minimality of the chromatic number N was used, nor any explanation of why a 12-fold covering is introduced (other than it fits the numerology needed to rule out a Steiner system). This makes me skeptical about the proof, but it’s hardly crackpot.

The author of this attempted proof, Ashay Dharwadker, is now an editor of the following Open Directory Project categories:

Science: Math: Combinatorics and
Science: Math: Combinatorics: Graph Theory.

I agree with “Default,” Eppstein, and Varney.

As “Default” notes, the proof is invalid, since it does not even use the hypotheses of the theorem. I pointed this out in November 2000 in a sub-page of a website in the Open Directory combinatorics category,

I also agree with Eppstein that Dharwadker’s writing seems “designed to confuse.”

Finally, I strongly agree with Varney that Dharwadker is a crackpot. I reluctantly arrived at this conclusion only last night, after learning that

Dharwadker, who formerly had edited only the graph theory Open Directory category, now is a co-editor of its parent category, combinatorics, and that
My website containing a criticism of Dharwadker’s work has been deleted from the Open Directory combinatics listings. This site, “Diamond Theory,” is only incidentally related to Dharwadker’s attempted proof, and has been in the Open Directory combinatorics listing for about two years.

Crackpots are annoying, but crackpots with power are both contemptible and infuriating. I am currently trying to rectify the appalling mistake made by whoever appointed Dharwadker to a position of responsibility.

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Tuesday, September 3, 2002

Tuesday September 3, 2002

Filed under: G-Notes,General,Geometry — Tags: Symmetric Generation, The Nutshell — m759 @ 6:00 pm

Today's birthday: James Joseph Sylvester

"Mathematics is the music of reason." — J. J. Sylvester

Sylvester, a nineteenth-century mathematician, coined the phrase "synthematic totals" to describe some structures based on 6-element sets that R. T. Curtis has called "rather unwieldy objects." See Curtis's abstract, Symmetric Generation of Finite Groups, John Baez's essay, Some Thoughts on the Number 6, and my website, Diamond Theory. See also the abstract of a December 7, 2000, talk, Mathematics and the Art of M. C. Escher, in which Curtis notes that graphic designs can "often convey a mathematical idea more eloquently than pages of symbolism."

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Saturday, August 31, 2002

Saturday August 31, 2002

Filed under: General,Geometry — m759 @ 3:36 am

Today’s birthday: Dr. Maria Montessori

THE MONTESSORI METHOD: CHAPTER VI

HOW LESSONS SHOULD BE GIVEN

“Let all thy words be counted.”
Dante, Inf., canto X.

CONCISENESS, SIMPLICITY, OBJECTIVITY.

…Dante gives excellent advice to teachers when he says, “Let thy words be counted.” The more carefully we cut away useless words, the more perfect will become the lesson….

Another characteristic quality of the lesson… is its simplicity. It must be stripped of all that is not absolute truth…. The carefully chosen words must be the most simple it is possible to find, and must refer to the truth.

The third quality of the lesson is its objectivity. The lesson must be presented in such a way that the personality of the teacher shall disappear. There shall remain in evidence only the object to which she wishes to call the attention of the child….

Above: Dr. Harrison Pope, Harvard professor of psychiatry, demonstrates the use of the Wechsler Adult Intelligence Scale “block design” subtest.

Mathematicians mean something different by the phrase “block design.”

A University of London site on mathematical design theory includes a link to my diamond theory site, which discusses the mathematics of the sorts of visual designs that Professor Pope is demonstrating. For an introduction to the subject that is, I hope, concise, simple, and objective, see my diamond 16 puzzle.

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Sunday, August 4, 2002

Sunday August 4, 2002

Filed under: General,Geometry — Tags: Cubehenge — m759 @ 2:52 pm

The Story Theory of Truth

versus

The Diamond Theory of Truth

One year ago today, Lorenzo Music, the voice of Carlton the doorman on Rhoda, died. His eulogy from Valerie Harper:

"Valerie's heart is breaking, but Rhoda is certain that Carlton the doorman is giving St. Peter at the gate a run for his money."

Today's birthday: Logician John Venn.

Appearing for the story theory…

Flannery O'Connor:

"In the long run, a people is known, not by its statements or statistics, but by the stories it tells. Fiction is the most impure and the most modest and the most human of the arts."

Appearing for the diamond theory…

Mary McCarthy and G. H. Hardy:

From the Hollywood Investigator:

On October 18, 1979, Mary McCarthy said on PBS's Dick Cavett Show: "Every word she writes is a lie, including 'and' and 'the.'"

Don't forget "a," as in "a people is known" —

"Greek mathematics is permanent, more permanent even than Greek literature. Archimedes will be remembered when Aeschylus is forgotten, because languages die and mathematical ideas do not."

— G. H. Hardy in A Mathematician's Apology

And a closing rebuttal from the story theory…

Martin Heidegger and Dean Martin:

Words of wisdom from Martin Heidegger, Catholic Nazi:

"The nature of art is poetry. The nature of poetry, in turn, is the founding of truth…. In the work, truth is thrown toward… an historical group of men."

— Poetry, Language, Thought, page 75, translated by Albert Hofstadter, Harper & Row paperback, 1975

And from Dean Martin, avatar of anti-art :

That's Amore:

– Artist: Dean Martin as sung on "Dean Martin's Greatest Hits"
– Capitol 4XL-9389
– peak Billboard position # 2 in 1953
– from the movie "the Caddy" starring Dean, Jerry Lewis, and Donna Reed
– Words and Music by Harry Warren and Jack Brooks

(In Napoli where love is King, when boy meets girl, here's what they say)

When the moon hits your eye like a big-a pizza pie,
That's amore!
When the world seems to shine like you've had too much wine,
That's amore!

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Saturday, July 20, 2002

Saturday July 20, 2002

Filed under: General,Geometry — Tags: Change Logos, Eightfold Cube, four-set, Map Systems, Octad, Octad Generator, Strange Change — m759 @ 10:13 pm

ABSTRACT: Finite projective geometry explains the surprising symmetry properties of some simple graphic designs– found, for instance, in quilts. Links are provided for applications to sporadic simple groups (via the "Miracle Octad Generator" of R. T. Curtis), to the connection between orthogonal Latin squares and projective spreads, and to symmetry of Walsh functions.

We regard the four-diamond figure D above as a 4×4 array of two-color diagonally-divided square tiles.

Let G be the group of 322,560 permutations of these 16 tiles generated by arbitrarily mixing random permutations of rows and of columns with random permutations of the four 2×2 quadrants.

THEOREM: Every G-image of D (as at right, below) has some ordinary or color-interchange symmetry.

Example:

For an animated version, click here.

Remarks:

Some of the patterns resulting from the action of G on D have been known for thousands of years. (See Jablan, Symmetry and Ornament, Ch. 2.6.) It is perhaps surprising that the patterns' interrelationships and symmetries can be explained fully only by using mathematics discovered just recently (relative to the patterns' age)– in particular, the theory of automorphism groups of finite geometries.

Using this theory, we can summarize the patterns' properties by saying that G is isomorphic to the affine group A on the linear 4-space over GF(2) and that the 35 structures of the 840 = 35 x 24 G-images of D are isomorphic to the 35 lines in the 3-dimensional projective space over GF(2).

This can be seen by viewing the 35 structures as three-sets of line diagrams, based on the three partitions of the four-set of square two-color tiles into two two-sets, and indicating the locations of these two-sets of tiles within the 4×4 patterns. The lines of the line diagrams may be added in a binary fashion (i.e., 1+1=0). Each three-set of line diagrams sums to zero– i.e., each diagram in a three-set is the binary sum of the other two diagrams in the set. Thus, the 35 three-sets of line diagrams correspond to the 35 three-point lines of the finite projective 3-space PG(3,2).

For example, here are the line diagrams for the figures above:

Shown below are the 15 possible line diagrams resulting from row/column/quadrant permutations. These 15 diagrams may, as noted above, be regarded as the 15 points of the projective 3-space PG(3,2).

The symmetry of the line diagrams accounts for the symmetry of the two-color patterns. (A proof shows that a 2nx2n two-color triangular half-squares pattern with such line diagrams must have a 2×2 center with a symmetry, and that this symmetry must be shared by the entire pattern.)

Among the 35 structures of the 840 4×4 arrays of tiles, orthogonality (in the sense of Latin-square orthogonality) corresponds to skewness of lines in the finite projective space PG(3,2). This was stated by the author in a 1978 note. (The note apparently had little effect. A quarter-century later, P. Govaerts, D. Jungnickel, L. Storme, and J. A. Thas wrote that skew (i.e., nonintersecting) lines in a projective space seem "at first sight not at all related" to orthogonal Latin squares.)

We can define sums and products so that the G-images of D generate an ideal (1024 patterns characterized by all horizontal or vertical "cuts" being uninterrupted) of a ring of 4096 symmetric patterns. There is an infinite family of such "diamond" rings, isomorphic to rings of matrices over GF(4).

The proof uses a decomposition technique for functions into a finite field that might be of more general use.

The underlying geometry of the 4×4 patterns is closely related to the Miracle Octad Generator of R. T. Curtis– used in the construction of the Steiner system S(5,8,24)– and hence is also related to the Leech lattice, which, as Walter Feit has remarked, "is a blown up version of S(5,8,24)."

For a movable JavaScript version of these 4×4 patterns, see The Diamond 16 Puzzle.

The above is an expanded version of Abstract 79T-A37, "Symmetry invariance in a diamond ring," by Steven H. Cullinane, Notices of the American Mathematical Society, February 1979, pages A-193, 194.

For a discussion of other cases of the theorem, click here.

Related pages:

The Diamond 16 Puzzle

Diamond Theory in 1937:
A Brief Historical Note

Notes on Finite Geometry

Geometry of the 4×4 Square

Binary Coordinate Systems

The 35 Lines of PG(3,2)

Map Systems:
Function Decomposition over a Finite Field

The Diamond Theorem–
The 2×2, the 2x2x2, the 4×4, and the 4x4x4 Cases

Diamond Theory

Latin-Square Geometry