Saturday, August 15, 2020

Hidden Figures Matter

Filed under: General — Tags: — m759 @ 3:00 PM

“Don’t forget the portcullis, Dutch Boy!”

Wednesday, August 12, 2020

Hidden Figure: Type Design at the East Village Other

Filed under: General — Tags: , — m759 @ 12:59 PM

    I.e.  . . . 

Monday, February 24, 2020

Hidden Figure

Filed under: General — m759 @ 2:55 PM

"There is  such a thing as  ▦  ."

— Saying adapted from a 1962 young-adult novel.

Friday, June 22, 2018

Hidden Figure

Filed under: General — m759 @ 11:29 AM

See as well Octavia Butler in this  journal.

Saturday, March 4, 2017

Hidden Figure

Filed under: General — m759 @ 3:17 AM

From this morning's New York Times  Wire

A cybersecurity-related image from Thursday evening

The diamond theorem correlation at the University of Bradford

Friday, August 17, 2012


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

Detail from last night's 1.3 MB image
"Search for the Lost Tesseract"—

The lost tesseract appears here on the cover of Wittgenstein's
Zettel  and, hidden in the form of a 4×4 array, as a subarray 
of the Miracle Octad Generator on the cover of Griess's
Twelve Sporadic Groups  and in a figure illustrating
the geometry of logic.

Another figure

IMAGE- Serbian chess innovator Svetozar Gligoric dies at 89

Gligoric died in Belgrade, Serbia, on Tuesday, August 14.

From this journal on that date

"Visual forms, he thought, were solutions to 
specific problems that come from specific needs."

— Michael Kimmelman in The New York Times
    obituary of E. H. Gombrich (November 7th, 2001)

Sunday, August 23, 2020

Jung on the Quaternity…

Filed under: General — m759 @ 3:12 PM

Or:  “1, 2, 3 , 4, who are we  for?”

Related material —

Wednesday, April 1, 2020

Adventures in the Matrix

Filed under: General — m759 @ 11:01 PM

Wednesday, March 11, 2020

Visualizing Mathieu Group Generators

Filed under: General — Tags: — m759 @ 6:23 AM

Marston Conder's M24 generators are illustrated by Cullinane's diamond-theorem (2x2 case) figures.

Update of March 17, 2020 —

The graphic images illustrate nicely Conder's six 4-cycles, but
their relationship, if any, to his eight 2-cycles is a mystery —

The Conder paper is at 


Monday, March 9, 2020

“Archimedes at Hiroshima” Continues.

Filed under: General — Tags: , , — m759 @ 7:34 PM

The title is from a post of January 10, 2019.

A figure from this journal on June 1, 2019

The following figure may help relate labelings of the
truncated octahedron (“permutahedron”) to labelings
of its fellow Archimedean solid, the cuboctahedron.

See as well other posts tagged Aitchison.

Wednesday, July 17, 2019

The Artsy Quantum Realm

Filed under: General — Tags: — m759 @ 6:38 PM

arXiv.org > quant-ph > arXiv:1905.06914 

Quantum Physics

Placing Kirkman's Schoolgirls and Quantum Spin Pairs on the Fano Plane: A Rainbow of Four Primary Colors, A Harmony of Fifteen Tones

J. P. Marceaux, A. R. P. Rau

(Submitted on 14 May 2019)

A recreational problem from nearly two centuries ago has featured prominently in recent times in the mathematics of designs, codes, and signal processing. The number 15 that is central to the problem coincidentally features in areas of physics, especially in today's field of quantum information, as the number of basic operators of two quantum spins ("qubits"). This affords a 1:1 correspondence that we exploit to use the well-known Pauli spin or Lie-Clifford algebra of those fifteen operators to provide specific constructions as posed in the recreational problem. An algorithm is set up that, working with four basic objects, generates alternative solutions or designs. The choice of four base colors or four basic chords can thus lead to color diagrams or acoustic patterns that correspond to realizations of each design. The Fano Plane of finite projective geometry involving seven points and lines and the tetrahedral three-dimensional simplex of 15 points are key objects that feature in this study.

Comments:16 pages, 10 figures

Subjects:Quantum Physics (quant-ph)

Cite as:arXiv:1905.06914 [quant-ph]

 (or arXiv:1905.06914v1 [quant-ph] for this version)

Submission history

From: A. R. P. Rau [view email] 
[v1] Tue, 14 May 2019 19:11:49 UTC (263 KB)

See also other posts tagged Tetrahedron vs. Square.

Monday, July 1, 2019

Inside the Exploded Cube

Filed under: General — m759 @ 10:15 PM

Metaphysical conceit | literature | Britannica.com


The metaphysical conceit, associated with the Metaphysical poets of the 17th century, is a more intricate and intellectual device. It usually sets up an analogy between one entity's spiritual qualities and an object in the physical world and sometimes controls the whole structure of the poem.…

This post's title refers to a metaphysical conceit 
in the previous post, Desperately Seeking Clarity.

Related material —

The source of the above mystical octahedron —

'Becoming Whole,' by Leslie Stein

      See also Jung's Imago Dei  in this journal.

Sunday, May 26, 2019

Sunday Shul: Connecting the Dots

Filed under: General — Tags: — m759 @ 9:00 AM

"It's very easy to say, 'Well, Jeff couldn't quite connect these dots,'"
director Jeff Nichols told BuzzFeed News. "Well, I wasn't actually
looking at the dots you were looking at."

— Posted on March 21, 2016, at 1:11 p.m,
Adam B. Vary, BuzzFeed News Reporter

"Magical arrays of numbers have been the talismans of mathematicians and mystics since the time of Pythagoras in the sixth century B.C. And in the 16th century, Rabbi Isaac ben Solomon Luria devised a cosmological world view that seems to have prefigured superstring theory, at least superficially. Rabbi Luria was a sage of the Jewish cabalist movement — a school of mystics that drew inspiration from the arcane oral tradition of the Torah.

According to Rabbi Luria's cosmology, the soul and inner life of the hidden God were expressed by 10 primordial numbers [sic ], known as the sefirot. [Link added.]"

— "Things Are Stranger Than We Can Imagine,"
by Malcolm W. BrowneNew York Times Book Review ,
Sunday, March 20, 1994

Wednesday, March 6, 2019

The Relativity Problem and Burkard Polster

Filed under: General,Geometry — Tags: — m759 @ 11:28 AM

From some 1949 remarks of Weyl—

"The relativity problem is one of central significance throughout geometry and algebra and has been recognized as such by the mathematicians at an early time."

— Hermann Weyl, "Relativity Theory as a Stimulus in Mathematical Research," Proceedings of the American Philosophical Society , Vol. 93, No. 7, Theory of Relativity in Contemporary Science: Papers Read at the Celebration of the Seventieth Birthday of Professor Albert Einstein in Princeton, March 19, 1949  (Dec. 30, 1949), pp. 535-541

Weyl in 1946—:

"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

For some context, see Relativity Problem  in this journal.

In the case of PG(3,2), there is a choice of geometric models 
to be coordinatized: two such models are the traditional
tetrahedral model long promoted by Burkard Polster, and
the square model of Steven H. Cullinane.

The above Wikipedia section tacitly (and unfairly) assumes that
the model being coordinatized is the tetrahedral model. For
coordinatization of the square model, see (for instance) the webpage
Finite Relativity.

For comparison of the two models, see a figure posted here on
May 21, 2014 —

Labeling the Tetrahedral Model  (Click to enlarge) —

"Citation needed" —

The anonymous characters who often update the PG(3,2) Wikipedia article
probably would not consider my post of 2014, titled "The Tetrahedral
Model of PG(3,2)
," a "reliable source."

Sunday, October 21, 2018

For Connoisseurs of Bad Art

Filed under: General — Tags: , — m759 @ 7:00 PM

Yesterday afternoon's post "Study in Blue and Pink" featured 
an image related to the "Blade and Chalice" of Dan Brown 

Requiem for a comics character known as "The Blue Blade" —


"We all float down here."

About the corresponding "Pink Chalice," the less said the better.

Monday, October 13, 2014

Raiders of the Lost Theorem

Filed under: General,Geometry — Tags: — m759 @ 12:05 PM

(Continued from Nov. 16, 2013.)

The 48 actions of GL(2,3) on a 3×3 array include the 8-element
quaternion group as a subgroup. This was illustrated in a Log24 post,
Hamilton’s Whirligig, of Jan. 5, 2006, and in a webpage whose
earliest version in the Internet Archive is from June 14, 2006.

One of these quaternion actions is pictured, without any reference
to quaternions, in a 2013 book by a Netherlands author whose
background in pure mathematics is apparently minimal:

In context (click to enlarge):

Update of later the same day —

Lee Sallows, Sept. 2011 foreword to Geometric Magic Squares —

“I first hit on the idea of a geometric magic square* in October 2001,**
and I sensed at once that I had penetrated some previously hidden portal
and was now standing on the threshold of a great adventure. It was going
to be like exploring Aladdin’s Cave. That there were treasures in the cave,
I was convinced, but how they were to be found was far from clear. The
concept of a geometric magic square is so simple that a child will grasp it
in a single glance. Ask a mathematician to create an actual specimen and
you may have a long wait before getting a response; such are the formidable
difficulties confronting the would-be constructor.”

* Defined by Sallows later in the book:

“Geometric  or, less formally, geomagic  is the term I use for
a magic square in which higher dimensional geometrical shapes
(or tiles  or pieces ) may appear in the cells instead of numbers.”

** See some geometric  matrices by Cullinane in a March 2001 webpage.

Earlier actual specimens — see Diamond Theory  excerpts published in
February 1977 and a brief description of the original 1976 monograph:

“51 pp. on the symmetries & algebra of
matrices with geometric-figure entries.”

— Steven H. Cullinane, 1977 ad in
Notices of the American Mathematical Society

The recreational topic of “magic” squares is of little relevance
to my own interests— group actions on such matrices and the
matrices’ role as models of finite geometries.

Thursday, March 29, 2012


Filed under: General — m759 @ 9:09 PM
A Large & Startling Figure:
The Harry Crews Online Bibliography
Page updated: March 29, 2012, 08:16 PM
Copyright © 1998 – 2010

Tuesday, January 31, 2012


Filed under: General,Geometry — Tags: — m759 @ 12:21 PM

“… a finite set with  elements
is sometimes called an n-set ….”

Tesseract formed from a 4-set—

IMAGE- Tesseract.

The same 16 subsets or points can
be arranged in a 4×4 array that has,
when the array’s opposite edges are
joined together, the same adjacencies
as those of the above tesseract.

“There is  such a thing as a 4-set.”
— Saying adapted from a novel   

Update of August 12, 2012:

Figures like the above, with adjacent vertices differing in only one coordinate,
appear in a 1950 paper of H. S. M. Coxeter—

Saturday, June 4, 2011


Filed under: General — m759 @ 2:02 PM

"… a little mystery to figure out…." — Bonnie Raitt, 1991

Two writers walk into a bar…
(A phrase from Immoveable Feast on May 28, 2005. See also May 28 this  year.)

Yesterday's post First Class featured a picture of Mystique.

A related passage* from a book pictured here May 28

IMAGE- Alan McGlashan on 'the Hidden Treasure, which is the Center'

See also Wallace Stevens on the Center.

* on participation mystique — "The primitive mind
does not differentiate the supernatural from reality,
but rather uses 'mystical participation'
to manipulate the world."

Friday, October 15, 2010

Mathematics and Narrative, continued

Filed under: General — Tags: — m759 @ 6:29 AM

The Story of N


Roberta Smith in the New York Times  of July 7, 2006

Art Review

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

"… The show has an endgame, end-time mood, as if we are looking at the end of the end of the end of Pop, hyperrealism and appropriation art. The techniques of replication and copying have become so meticulous that they are beside the point. This is truly magic realism: the kind you can't see, that has to be explained. It is also a time when artists cultivate hybridism and multiplicity and disdain stylistic coherence, in keeping with the fashionable interest in collectivity, lack of ego, the fluidity of individual identity. But too often these avoidance tactics eliminate the thread of a personal sensibility or focus.

I would call all these strategies fear of form, which can be parsed as fear of materials, of working with the hands in an overt way and of originality. Most of all originality. Can we just say it? This far from Andy Warhol and Duchamp, 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. With a few exceptions, forms of collage and assemblage dominate this show: the putting together (or simply putting side by side) of existing images and objects prevails. The consistency of this technique in two and three dimensions should have been a red flag for the curators. Collage has driven much art since the late 1970's. Lately, and especially in this exhibition, it often seems to have become so distended and pulled apart that its components have become virtually autonomous and unrelated, which brings us back to square one. This is most obvious in the large installations of graphic works whose individual parts gain impact and meaning from juxtaposition but are in fact considered distinct artworks."

Margaret Atwood on art and the trickster

"The pleasures of fabulation, the charming and playful lie— this line of thought leads Hyde* to the last link in his subtitle, the connection of the trickster to art. Hyde reminds us that the wall between the artist and that American favourite son, the con-artist, can be a thin one indeed; that craft and crafty rub shoulders; and that the words artifice, artifact, articulation  and art  all come from the same ancient root, a word meaning 'to join,' 'to fit,' and 'to make.'  If it’s a seamless whole you want, pray to Apollo, who sets the limits within which such a work can exist.  Tricksters, however, stand where the door swings open on its hinges and the horizon expands: they operate where things are joined together, and thus can also come apart."

* Lewis Hyde, Trickster Makes This World: Mischief, Myth, and Art,  Farrar Straus & Giroux, January 1998

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

Atwood mentions "a seamless whole."

For some related remarks, see "A Study in Art Education" and the central figure pictured above. (There "N" can stand for "number," "nine," or "narrative.")

Thursday, July 1, 2010

Plato’s Code

Filed under: General,Geometry — m759 @ 12:00 PM

John Allen Paulos yesterday at Twitter

"Plato's code cracked? http://bit.ly/ad6k1S
Fascinating if not a hoax or hype."

The story that Paulos linked to is about a British
academic who claims to have found some
symbolism hidden in Plato's writings by
splitting each into 12 parts and correlating
the 12 parts with semitones of a musical scale.

I prefer a different approach to Plato that is
related to the following hoax and hype—


From Dan Brown's novel Angels & Demons  (2000)

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


Image-- From 'Alchemy,' by Holmyard, the diamond of Aristotle's 4 elements and 4 qualities

This  four-elements diamond summarizes the classical
four elements and four qualities neatly, but some scholars
might call the figure "hype" since it deals with an academically
disreputable subject, alchemy, and since its origin is unclear.

For the four elements' role in some literature more respectable
than Dan Brown's, see Poetry's Bones.

Although an author like Brown might spin the remarks
below into a narrative—  The Plato Code — they are
neither  hoax nor hype.


Image-- From the Diamond in Plato's Meno to Modern Finite Geometry



For related non-hoax, non-hype remarks, see
The Rational Enterprise: Logos in Plato's Theaetetus,
by Rosemary Desjardins.

Those who prefer  hoax and hype in their philosophy may consult
the writings of, say, Barbara Johnson, Rosalind Krauss, and—
in yesterday's NY Times's  "The Stone" columnNancy Bauer.

Image-- The Philosophers' Stone according to The New York Times

— The New York Times

Tuesday, February 16, 2010

Mysteries of Faith

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

From today's NY Times


Obituaries for mystery authors
Ralph McInerny and Dick Francis

From the date (Jan. 29) of McInerny's death–

"…although a work of art 'is formed around something missing,' this 'void is its vanishing point, not its essence.'"

Harvard University Press on Persons and Things (Walpurgisnacht, 2008), by Barbara Johnson

From the date (Feb. 14) of Francis's death–

2x2x2 cube

The EIghtfold Cube

The "something missing" in the above figure is an eighth cube, hidden behind the others pictured.

This eighth cube is not, as Johnson would have it, a void and "vanishing point," but is instead the "still point" of T.S. Eliot. (See the epigraph to the chapter on automorphism groups in Parallelisms of Complete Designs, by Peter J. Cameron. See also related material in this journal.) The automorphism group here is of course the order-168 simple group of Felix Christian Klein.

For a connection to horses, see
a March 31, 2004, post
commemorating the birth of Descartes
  and the death of Coxeter–

Putting Descartes Before Dehors

     Binary coordinates for a 4x2 array  Chess knight formed by a Singer 7-cycle

For a more Protestant meditation,
see The Cross of Descartes


Descartes's Cross

"I've been the front end of a horse
and the rear end. The front end is better."
— Old vaudeville joke

For further details, click on
the image below–

Quine and Derrida at Notre Dame Philosophical Reviews

Notre Dame Philosophical Reviews

Saturday, June 21, 2008

Saturday June 21, 2008

Filed under: General — Tags: — m759 @ 8:00 PM
For Mary Gaitskill

(See Eight is a Gate and
Faith, Doubt, Art, and
The New Yorker

A sructure from
today's previous entry:


From Notre-Dame de Paris:

"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."  

For some further details, see
the brief Log24 narrative
"Indiana Jones and
the Hidden Coffer
" as well as
Symmetry Framed and
the design of the doors
to Rick's Cafe Americain:

IMAGE- The perception of doors in 'Casablanca'

Everyone comes to Rick's.

Sunday, May 25, 2008

Sunday May 25, 2008

Filed under: General,Geometry — Tags: , — m759 @ 9:00 AM
Wechsler Cubes
“Confusion is nothing new.”
— Song lyric, Cyndi Lauper
Part I:
Magister Ludi

Hermann Hesse’s 1943 The Glass Bead Game (Picador paperback, Dec. 6, 2002, pp. 139-140)–

“For the present, the Master showed him a bulky memorandum, a proposal he had received from an organist– one of the innumerable proposals which the directorate of the Game regularly had to examine. Usually these were suggestions for the admission of new material to the Archives. One man, for example, had made a meticulous study of the history of the madrigal and discovered in the development of the style a curved that he had expressed both musically and mathematically, so that it could be included in the vocabulary of the Game. Another had examined the rhythmic structure of Julius Caesar’s Latin and discovered the most striking congruences with the results of well-known studies of the intervals in Byzantine hymns. Or again some fanatic had once more unearthed some new cabala hidden in the musical notation of the fifteenth century. Then there were the tempestuous letters from abstruse experimenters who could arrive at the most astounding conclusions from, say, a comparison of the horoscopes of Goethe and Spinoza; such letters often included pretty and seemingly enlightening geometric drawings in several colors.”

Part II:
A Bulky Memorandum

From Siri Hustvedt, author of Mysteries of the Rectangle: Essays on Painting (Princeton Architectural Press, 2005)– What I Loved: A Novel (Picador paperback, March 1, 2004, page 168)–

A description of the work of Bill Wechsler, a fictional artist:

“Bill worked long hours on a series of autonomous pieces about numbers. Like O’s Journey, the works took place inside glass cubes, but these were twice as large– about two feet square. He drew his inspiration from sources as varied as the Cabbala, physics, baseball box scores, and stock market reports. He painted, cut, sculpted, distorted, and broke the numerical signs in each work until they became unrecognizable. He included figures, objects, books, windows, and always the written word for the number. It was rambunctious art, thick with allusion– to voids, blanks, holes, to monotheism and the individual, the the dialectic and yin-yang, to the Trinity, the three fates, and three wishes, to the golden rectangle, to seven heavens, the seven lower orders of the sephiroth, the nine Muses, the nine circles of Hell, the nine worlds of Norse mythology, but also to popular references like A Better Marriage in Five Easy Lessons and Thinner Thighs in Seven Days. Twelve-step programs were referred to in both cube one and cube two. A miniature copy of a book called The Six Mistakes Parents Make Most Often lay at the bottom of cube six. Puns appeared, usually well disguised– one, won; two, too, and Tuesday; four, for, forth; ate, eight. Bill was partial to rhymes as well, both in images and words. In cube nine, the geometric figure for a line had been painted on one glass wall. In cube three, a tiny man wearing the black-and-white prison garb of cartoons and dragging a leg iron has

— End of page 168 —

opened the door to his cell. The hidden rhyme is “free.” Looking closely through the walls of the cube, one can see the parallel rhyme in another language: the German word drei is scratched into one glass wall. Lying at the bottom of the same box is a tiny black-and-white photograph cut from a book that shows the entrance to Auschwitz: ARBEIT MACHT FREI. With every number, the arbitrary dance of associations worked togethere to create a tiny mental landscape that ranged in tone from wish-fulfillment dream to nightmare. Although dense, the effect of the cubes wasn’t visually disorienting. Each object, painting, drawing, bit of text, or sculpted figure found its rightful place under the glass according to the necessary, if mad, logic of numerical, pictorial, and verbal connection– and the colors of each were startling. Every number had been given a thematic hue. Bill had been interested in Goethe’s color wheel and in Alfred Jensen’s use of it in his thick, hallucinatory paintings of numbers. He had assigned each number a color. Like Goethe, he included black and white, although he didn’t bother with the poet’s meanings. Zero and one were white. Two was blue. Three was red, four was yellow, and he mixed colors: pale blue for five, purples in six, oranges in seven, greens in eight, and blacks and grays in nine. Although other colors and omnipresent newsprint always intruded on the basic scheme, the myriad shades of a single color dominated each cube.

The number pieces were the work of a man at the top of his form. An organic extension of everything Bill had done before, these knots of symbols had an explosive effect. The longer I looked at them, the more the miniature constructions seemed on the brink of bursting from internal pressure. They were tightly orchestrated semantic bombs through which Bill laid bare the arbitrary roots of meaning itself– that peculiar social contract generated by little squiggles, dashes, lines, and loops on a page.”

Part III:
Wechsler Cubes(named not for
Bill Wechsler, the
fictional artist above,
but for the non-fictional
David Wechsler) —

From 2002:

Above: Dr. Harrison Pope, Harvard professor of psychiatry, demonstrates the use of the Wechsler Adult Intelligence Scale “block design” subtest.

Part IV:
A Magic Gallery
Log24, March 4, 2004


Figures from the
Kaleidoscope Puzzle
of Steven H. Cullinane:

Poem by Eugen Jost:
Zahlen und Zeichen,
Wörter und Worte

Mit Zeichen und Zahlen
vermessen wir Himmel und Erde
auf weiss
schaffen wir neue Welten
oder gar Universen

 Numbers and Names,
Wording and Words

With numbers and names
we measure heaven and earth
on white
we create new worlds
and universes

English translation
by Catherine Schelbert
A related poem:

by Hermann Hesse

From time to time
we take our pen in hand
and scribble symbols
on a blank white sheet
Their meaning is
at everyone’s command;
it is a game whose rules
are nice and neat.

But if a savage
or a moon-man came
and found a page,
a furrowed runic field,
and curiously studied
lines and frame:
How strange would be
the world that they revealed.
a magic gallery of oddities.
He would see A and B
as man and beast,
as moving tongues or
arms or legs or eyes,
now slow, now rushing,
all constraint released,
like prints of ravens’
feet upon the snow.
He’d hop about with them,
fly to and fro,
and see a thousand worlds
of might-have-been
hidden within the black
and frozen symbols,
beneath the ornate strokes,
the thick and thin.
He’d see the way love burns
and anguish trembles,
He’d wonder, laugh,
shake with fear and weep
because beyond this cipher’s
cross-barred keep
he’d see the world
in all its aimless passion,
diminished, dwarfed, and
spellbound in the symbols,
and rigorously marching
He’d think: each sign
all others so resembles
that love of life and death,
or lust and anguish,
are simply twins whom
no one can distinguish…
until at last the savage
with a sound
of mortal terror
lights and stirs a fire,
chants and beats his brow
against the ground
and consecrates the writing
to his pyre.
Perhaps before his
consciousness is drowned
in slumber there will come
to him some sense
of how this world
of magic fraudulence,
this horror utterly
behind endurance,
has vanished as if
it had never been.
He’ll sigh, and smile,
and feel all right again.

— Hermann Hesse (1943),
Buchstaben,” from
Das Glasperlenspiel,
translated by
Richard and Clara Winston

Saturday, February 23, 2008

Saturday February 23, 2008

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

"An acute study of the links
between word and fact"
Nina daVinci Nichols

Thanks to a Virginia reader for a reminder:
Virginia /391062427/item.html? 2/22/2008 7:37 PM
The link is to a Log24 entry
that begins as follows…

An Exercise

of Power

Johnny Cash:
"And behold,
a white horse."

Springer logo - A chess knight
Chess Knight
(in German, Springer)

This, along with the "jumper" theme in the previous two entries, suggests a search on springer jumper.That search yields a German sports phrase, "Springer kommt!"  A search on that phrase yields the following:
"Liebe Frau vBayern,
mich würde interessieren wie man
mit diesem Hintergrund
zu Springer kommt?"

Background of "Frau vBayern" from thePeerage.com:

Anna-Natascha Prinzessin zu Sayn-Wittgenstein-Berleburg 

F, #64640, b. 15 March 1978Last Edited=20 Oct 2005

     Anna-Natascha Prinzessin zu Sayn-Wittgenstein-Berleburg was born on 15 March 1978. She is the daughter of Ludwig Ferdinand Prinz zu Sayn-Wittgenstein-Berleburg and Countess Yvonne Wachtmeister af Johannishus. She married Manuel Maria Alexander Leopold Jerg Prinz von Bayern, son of Leopold Prinz von Bayern and Ursula Mohlenkamp, on 6 August 2005 at Nykøping, Södermanland, Sweden.


The date of the above "Liebe Frau vBayern" inquiry, Feb. 1, 2007, suggests the following:

From Log24 on
St. Bridget's Day, 2007:

The quotation
"Science is a Faustian bargain"
and the following figure


The 63 yang-containing hexagrams of the I Ching as a Singer 63-cycle

From a short story by
the above Princess:

"'I don't even think she would have wanted to change you. But she for sure did not want to change herself. And her values were simply a part of her.' It was true, too. I would even go so far as to say that they were her basis, if you think about her as a geometrical body. That's what they couldn't understand, because in this age of the full understanding for stretches of values in favor of self-realization of any kind, it was a completely foreign concept."

To make this excellent metaphor mathematically correct,
change "geometrical body" to "space"… as in

"For Princeton's Class of 2007"

Review of a 2004 production of a 1972 Tom Stoppard play, "Jumpers"–

John Lahr on Tom Stoppard's play Jumpers

Related material:

Knight Moves (Log24, Jan. 16),
Kindergarten Theology (St. Bridget's Day, 2008),

The image “My space -(the affine space of six dimensions over the two-element field
(Click on image for details.)

Wednesday, October 24, 2007

Wednesday October 24, 2007

Filed under: General,Geometry — Tags: , , — m759 @ 11:11 PM
Descartes’s Twelfth Step

An earlier entry today (“Hollywood Midrash continued“) on a father and son suggests we might look for an appropriate holy ghost. In that context…


A search for further background on Emmanuel Levinas, a favorite philosopher of the late R. B. Kitaj (previous two entries), led (somewhat indirectly) to the following figures of Descartes:

The color-analogy figures of Descartes
This trinity of figures is taken from Descartes’ Rule Twelve in Rules for the Direction of the Mind. It seems to be meant to suggest an analogy between superposition of colors and superposition of shapes.Note that the first figure is made up of vertical lines, the second of vertical and horizontal lines, and the third of vertical, horizontal, and diagonal lines. Leon R. Kass recently suggested that the Descartes figures might be replaced by a more modern concept– colors as wavelengths. (Commentary, April 2007). This in turn suggests an analogy to Fourier series decomposition of a waveform in harmonic analysis. See the Kass essay for a discussion of the Descartes figures in the context of (pdf) Science, Religion, and the Human Future (not to be confused with Life, the Universe, and Everything).

Compare and contrast:

The harmonic-analysis analogy suggests a review of an earlier entry’s
link today to 4/30–  Structure and Logic— as well as
re-examination of Symmetry and a Trinity

(Dec. 4, 2002).

See also —

A Four-Color Theorem,
The Diamond Theorem, and
The Most Violent Poem,

Emma Thompson in 'Wit'

from Mike Nichols’s birthday, 2003.

Saturday, June 30, 2007

Saturday June 30, 2007

Filed under: General,Geometry — Tags: — m759 @ 10:04 PM
is God

Frank Rich in
The New York Times

 November 2004–

Desperate Housewives ad on Monday Night Football

"Desperate Housewives"
ad on "Monday
Night Football"

"Desperate Housewives"… ranks No. 5 among all prime-time shows for ages 12-17. ("Monday Night Football" is No. 18.) This may explain in part why its current advertisers include products like Fisher-Price toys, the DVD of "Elf" and the forthcoming Tim Allen holiday vehicle, "Christmas With the Kranks."

Those who cherish the First Amendment can only hope that the Traditional Values Coalition, OneMillionMoms.com, OneMillionDads.com and all the rest send every e-mail they can to the F.C.C. demanding punitive action against the stations that broadcast "Desperate Housewives." A "moral values" crusade that stands between a TV show this popular and its audience will quickly learn the limits of its power in a country where entertainment is god.

— "The Great Indecency Hoax," a New York Times column by Frank Rich quoted in Log24 on Nov. 26, 2004

The entertainment continues.  A rabbi's obituary in today's New York Times (see previous entry) served as ad-bait for "Joshua," a Fox Searchlight film opening July 6.

A search for a less sacrilegious memorial to the rabbi yields the following:

Project MUSE link on Rabbi Abraham Klausner

The "Project MUSE" link above
works only at
subscribing libraries.

  It seems that here, too,
the rabbi is being
used as bait.

  For a perhaps preferable
 reference to bait, in the
context of St. Peter as
a "fisher of men," see
the Christian "mandorla"
or "vesica piscis,"
a figure hidden within
the geometry of Rome's
St. Peter's Square–
which, despite its name,
is an oval:

Mandorla and ovator tondo in St. Peter's Square” cannot be displayed, because it contains errors.

For the geometric
construction of the
 Roman oval, see
"ovato tondo" in
Rudolf Arnheim's
The Power of the Center.

For a less theoretical account
of the religious significance
of the mandorla, see
the 2001 film
The Center of the World.

Friday, February 16, 2007

Friday February 16, 2007

Filed under: General — Tags: — m759 @ 6:16 AM

The Judas Seat

Janet Maslin in today’s New York Times:

“The much-borrowed Brown formula involves some very specific things. The name of a great artist, artifact or historical figure must be in the book’s story, not to mention on its cover. The narrative must start in the present day with a bizarre killing, then use that killing as a reason to investigate the past. And the past must yield a secret so big, so stunning, so saber-rattling that all of civilization may be changed by it. Probably not for the better.

This formula is neatly summarized….”

Cover illustration
The Judas Seat:
The Narrative:

The Secret:

Part I

“Little ‘Jack’ Horner was actually Thomas Horner, steward to the Abbot of Glastonbury during the reign of King Henry VIII…. Always keen to raise fresh funds, Henry had shown a interest in Glastonbury (and other abbeys). Hoping to appease the royal appetite, the nervous Abbot, Richard Whiting, allegedly sent Thomas Horner to the King with a special gift. This was a pie containing the title deeds to twelve manor houses in the hope that these would deflect the King from acquiring Glastonbury Abbey. On his way to London, the not so loyal courier Horner apparently stuck his thumb into the pie and extracted the deeds for Mells Manor, a plum piece of real estate. The attempted bribe failed and the dissolution of the monasteries (including Glastonbury) went ahead from 1536 to 1540. Richard Whiting was subsequently executed, but the Horner family kept the house, so the moral of this one is: treachery and greed pay off, but bribery is a bad idea.” –Chris Roberts, Heavy Words Lightly Thrown: The Reason Behind the Rhyme

Part II

“The Grail Table has thirteen seats, one of which is kept vacant in memory of Judas Iscariot who betrayed Christ.” —Symbolism of King Arthur’s Round Table

“In medieval romance, the grail was said to have been brought to Glastonbury in Britain by Joseph of Arimathea and his followers. In the time of Arthur, the quest for the Grail was the highest spiritual pursuit.” —The Camelot Project

Part III

The Log24 entry
for the date–
February 13, 2007–
of the above Bible scholar’s death,

and the three entries preceding it:

“And what the dead had no speech for, when living,
they can tell you, being dead:
the communication of the dead is tongued with fire
beyond the language of the living.”

— T. S. Eliot, Four Quartets

Tuesday, January 9, 2007

Tuesday January 9, 2007

Filed under: General,Geometry — m759 @ 9:00 PM
Logos and Logic
(private, cut from prev. entry)

The diamond is used in modal logic to symbolize possibility.

  The 3×3 grid may also be used
to illustrate “possibility.”  It leads,
as noted at finitegeometry.org, to
the famed “24-cell,” which may be
pictured either as the diamond
figure from Plato’s Meno

The image “http://www.log24.com/theory/images/poly-24cell-sm.jpg” cannot be displayed, because it contains errors.

Click for details.

  — or as a figure
with 24 vertices:

The image “http://www.log24.com/theory/images/poly-24cell-02sm.jpg” cannot be displayed, because it contains errors.

Click for details.

The “diamond” version of the
24-cell seems unrelated to the
second version that shows all
vertices and edges, yet the
second version is implicit,
or hidden, in the first.
Hence “possibility.”

Neither version of the 24-cell
seems related in any obvious
way to the 3×3 grid, yet both
versions are implicit,
or hidden, in the grid.
Hence “possibility.”

Saturday, September 2, 2006

Saturday September 2, 2006

Filed under: General — Tags: — m759 @ 7:31 AM
Today's birthdays:

Salma Hayek

Salma Hayek and Julie Taymor

"Shinin' like a diamond
 she had tombstones
in her eyes.

The image “http://www.log24.com/log/pix06A/060902-StarAndDiamond2.gif” cannot be displayed, because it contains errors.

(For the above figures,
see Log24, 5/17/06,
"Tombstone," and
Log24, 9/13/03,
"For the Man in Black.")

and Keanu Reeves


The image “http://www.log24.com/log/pix05/050218-Highwater.jpg” cannot be displayed, because it contains errors.

(For the above figure,
see Log24, 2/18/05,
"In Hoc Signo.")

Related material:

From Notre-Dame de Paris:

"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."  

For some further details, see
the brief Log24 narrative
"Indiana Jones and
the Hidden Coffer
" as well as
Symmetry Framed and
the design of the doors
to Rick's Cafe Americain:

The image “http://www.log24.com/log/pix06A/060902-Ricks.jpg” cannot be displayed, because it contains errors.

Friday, May 19, 2006

Friday May 19, 2006

Filed under: General — m759 @ 1:00 AM
Star and Diamond

” ‘I know what it is you last saw,’ she said; ‘for that is also in my mind. Do not be afraid! But do not think that only by singing amid the trees, nor even by the slender arrows of elvenbows, is this land of Lothlórien maintained and defended against the Enemy. I say to you, Frodo, that even as I speak to you, I perceive the Dark Lord and know his mind, or all his mind that concerns the Elves. And he gropes ever to see me and my thought. But still the door is closed!’
      She lifted up her white arms, and spread out her hands towards the East in a gesture of rejection and denial. Eärendil, the Evening Star, most beloved of the Elves, shone clear above. So bright was it that the figure of the Elven-lady cast a dim shadow on the ground. Its ray glanced upon a ring about her finger; it glittered like polished gold overlaid with silver light, and a white stone in it twinkled as if the Even-star had come to rest upon her hand. Frodo gazed at the ring with awe; for suddenly it seemed to him that he understood.
      ‘Yes,’ she said, divining his thought, ‘it is not permitted to speak of it, and Elrond could not do so. But it cannot be hidden from the Ring-Bearer, and one who has seen the Eye. Verily it is in the land of Lórien upon the finger of Galadriel that one of the Three remains. This is Nenya, the Ring of Adamant, and I am its keeper.’ ”

— J. R. R. Tolkien, The Lord of the Rings

Related material:

The last 3 entries,
as well as
Mathematics and Narrative

“How much story
do you want?”
— George Balanchine  

Wednesday, January 4, 2006

Wednesday January 4, 2006

Filed under: General — m759 @ 2:00 AM
The Shining

The Shining according to
the Catholic Church:

“The Transfiguration of Christ is the culminating point of His public life…. Jesus took with him Peter and James and John and led them to a high mountain apart, where He was transfigured before their ravished eyes.  St. Matthew and St. Mark express this phenomenon by the word metemorphothe, which the Vulgate renders transfiguratus est.   The Synoptics explain the true meaning of the word by adding ‘his face did shine as the sun: and his garments became white as snow,’ according to the Vulgate, or ‘as light,’  according to the Greek text.  This dazzling brightness which emanated from His whole Body was produced by an interior shining of His Divinity.”

— The Catholic Encyclopedia of 1912

The Shining according to
Paul Preuss:

From Broken Symmetries, 1983, Chapter 16:

“He’d toyed with ‘psi’ himself…. The reason he and so many other theoretical physicists were suckers for the stuff was easy to understand– for two-thirds of a century an enigma had rested at the heart of theoretical physics, a contradiction, a hard kernel of paradox….   

Peter [Slater] had never thirsted after ‘hidden variables’ to explain what could not be pictured.  Mathematical relationships were enough to satisfy him, mere formal relationships which existed at all times, everywhere, at once.  It was a thin nectar, but he was convinced it was the nectar of the gods….

Those so-called crazy psychics were too sane, that was their problem– they were too stubborn to admit that the universe was already more bizarre than anything they could imagine in their wildest dreams of wizardry.”

Saturday, November 12, 2005

Saturday November 12, 2005

Filed under: General,Geometry — Tags: , — m759 @ 1:28 PM
Glory Season

"…his eyes ranged the Consul's books disposed quite neatly… on high shelves around the walls: Dogme et Ritual de la Haute Magie, Serpent and Siva Worship in Central America, there were two long shelves of this, together with the rusty leather bindings and frayed edges of the numerous cabbalistic and alchemical books, though some of them looked fairly new, like the Goetia of the Lemegaton of Solomon the King, probably they were treasures, but the rest were a heterogeneous collection…."

Malcolm Lowry, Under the Volcano, Chapter VI

"… when Saul does reach for a slim leather-bound volume Eliza cannot help but feel that something momentous is about to happen.  There is care in the way he carries the book on the short journey from its shelf, as if it were constructed not of leather and parchment but of flesh and blood….
    "Otzar Eden HaGanuz," Saul says.  "The Hidden Eden.  In this book, Abulafia describes the process of permutation…. Once you have mastered it, you will have mastered words, and once you have mastered words, you will be ready to receive shefa."

Bee Season: A Novel

"In the Inner Game, we call the Game Dhum Welur, the Mind of God."

The Gameplayers of Zan, a novel featuring games based on cellular automata

"Regarding cellular automata, I'm trying to think in what SF books I've seen them mentioned. Off the top of my head, only three come to mind:

The Gameplayers of Zan M.A. Foster
Permutation City Greg Egan
Glory Season David Brin"

— Jonathan L. Cunningham, Usenet

    "If all that 'matters' are fundamentally mathematical relationships, then there ceases to be any important difference between the actual and the possible. (Even if you aren't a mathematical Platonist, you can always find some collection of particles of dust to fit any required pattern. In Permutation City this is called the 'logic of the dust' theory.)….
    … Paul Durham is convinced by the 'logic of the dust' theory mentioned above, and plans to run, just for a few minutes, a complex cellular automaton (Permutation City) started in a 'Garden of Eden' configuration — one which isn't reachable from any other, and which therefore must have been the starting point of a simulation….  I didn't understand the need for this elaborate set-up, but I guess it makes for a better story than 'well, all possible worlds exist, and I'm going to tell you about one of them.'"

— Danny Yee, review of Permutation City

"Y'know, I never imagined the competition version involved so many tricky permutations."

— David Brin, Glory Season, 1994 Spectra paperback, p. 408

Related material:
The image “http://www.log24.com/log/pix05B/051112-EdenFigs.jpg” cannot be displayed, because it contains errors.

Figure 2



"… matter is consciousness expressed in the intermixing of force and form, but so heavily structured and constrained by form that its behaviour becomes describable using the regular and simple laws of  physics. This is shown in Figure 2.
    The glyph in Figure 2 is the basis for a kabbalistic diagram called the Etz Chaiim, or Tree of Life. The first principle of being or consciousness is called Keter, which means Crown. The raw energy of consciousness is called Chokhmah or Wisdom, and the capacity to give form to the energy of consciousness is called Binah, which is sometimes translated as Understanding, and sometimes as Intelligence. The outcome of the interaction of force and form, the physical world, is called Malkhut or Kingdom.  This is shown… in Figure 3."

Figure 3

"This quaternary is a Kabbalistic representation of God-the-Knowable, in the sense that it the most abstract representation of God we are capable of comprehending….
    God-the-Knowable has four aspects, two male and two female: Keter and Chokhmah are both represented as male, and Binah and Malkhut are represented as female. One of the titles of Chokhmah is Abba, which means Father, and one of the titles of Binah is Imma, which means Mother, so you can think of Chokhmah as God-the-Father, and Binah as God-the-Mother. Malkhut is the daughter, the female spirit of God-as-Matter, and it would not be wildly wrong to think of her as Mother Earth. And what of God-the-Son? Is there also a God-the-Son in Kabbalah? There is…."

A Depth of Beginning: Notes on Kabbalah by Colin Low (pdf)

See also
Cognitive Blending and the Two Cultures,
Mathematics and Narrative,
Deep Game,
and the previous entry.

Tuesday, July 5, 2005

Tuesday July 5, 2005

Filed under: General — m759 @ 5:14 AM

For Christopher Fry
and the White Goddess:

The Edge of Eternity

Christian humanist playwright Christopher Fry, author of The Lady’s Not for Burning, died at 97 on June 30, 2005.

From Log24 on June 30:

Robert Graves, author of
The White Goddess:
A Historical Grammar of Poetic Myth

How may the King hold back?
Royally then he barters life for love.

Or of the undying snake from chaos hatched,
Whose coils contain the ocean,
Into whose chops with naked sword he springs,
Then in black water, tangled by the reeds,
Battles three days and nights…

From Cold Mountain:

“He sat awhile on a rock, and then got up and walked all morning through the dim woods. The track was ill used, so coiled and knotted he could not say what its general tendency was. It aimed nowhere certain but up. The brush and bracken grew thick in the footway, and the ground seemed to be healing over, so that in some near future the way would not even remain as scar. For several miles it mostly wound its way through a forest of immense hemlocks, and the fog lay among them so thick that their green boughs were hidden. Only the black trunks were visible, rising into the low sky like old menhirs stood up by a forgotten race to memorialize the darkest events of their history….

They climbed to a bend and from there they walked on great slabs of rock. It seemed to Inman that they were at the lip of a cliff, for the smell of the thin air spoke of considerable height, though the fog closed off all visual check of loftiness….

Then he looked back down and felt a rush of vertigo as the lower world was suddenly revealed between his boot toes. He was indeed at the lip of a cliff, and he took one step back…. The country around was high, broken. Inman looked about and was startled to see a great knobby mountain forming up out of the fog to the west, looming into the sky.  The sun broke through a slot in the clouds, and a great band of Jacob’s ladder suddenly hung in the air like a gauze curtain between Inman and the blue mountain….

Inman looked at the big grandfather mountain and then he looked beyond it to the lesser mountains as they faded off into the southwest horizon, bathed in faint smoky haze. Waves of mountains. For all the evidence the eye told, they were endless. The grey overlapping humps of the farthest peaks distinguished themselves only as slightly darker values of the pale grey air. The shapes and their ghostly appearance spoke to Inman in a way he could not clearly interpret. They graded off like the tapering of pain from the neck wound as it healed.”

See also the entries of July 3.

The crone figure in this section of Cold Mountain is not entirely unrelated to the girl accused of being a witch in Fry’s play and to Graves’s White Goddess.

From Fry’s obituary in The Guardian:

“Though less of a public theorist than Eliot, Fry still believed passionately in the validity of poetic drama. As he wrote in the magazine Adam: ‘In prose, we convey the eccentricity of things, in poetry their concentricity, the sense of relationship between them: a belief that all things express the same identity and are all contained in one discipline of revelation.'”

From Fry’s obituary in today’s New York Times:

“His plays radiated an optimistic faith in God and humanity, evoking, in his words, ‘a world in which we are poised on the edge of eternity, a world which has deeps and shadows of mystery, and God is anything but a sleeping partner.’ He said he wrote his plays in poetry because that was ‘the language in which man expresses his own amazement’ at the complexity both of himself and of a reality which, beneath the surface, was ‘wildly, perilously, inexplicably fantastic.'”

Saturday, May 22, 2004

Saturday May 22, 2004

Filed under: General — m759 @ 6:14 AM

Star Wars

In memory of Melvin J. Lasky, editor, 1958-1990, of the CIA-funded journal Encounter:

“Once called as lively, and as bitchy, as a literary cocktail party, Encounter published articles of unrivalled authority on politics, history and literature.”

— Obituary in the Telegraph 

Lasky died on Wednesday, May 19, 2004.  From a journal entry of my own on that date:

This newly-digitized diagram is from a
paper journal note of October 21, 1999.

Note that the diagram’s overall form is that of an eight-point star.  Here is an excerpt from a Fritz Leiber story dealing with such a star, the symbol of a fictional organization:

Time traveling, which is not quite the good clean boyish fun it’s cracked up to be, started for me when this woman with the sigil on her forehead looked in on me from the open doorway of the hotel bedroom where I’d hidden myself and the bottles and asked me, “Look, Buster, do you want to live?”

Her right arm was raised and bent, the elbow touching the door frame, the hand brushing back the very dark bangs from her forehead to show me the sigil, as if that had a bearing on her question.

The sigil was an eight-limbed asterisk made of fine dark lines and about as big as a silver dollar.  An X superimposed on a plus sign.  It looked permanent.

… “Here is how it stacks up:  You’ve bought your way with something other than money into an organization of which I am an agent….”

“It’s a very big organization,” she went on, as if warning me.  “Call it an empire or a power if you like.  So far as you are concerned, it has always existed and always will exist.  It has agents everywhere, literally.  Space and time are no barriers to it.  Its purpose, so far as you will ever be able to know it, is to change, for its own aggrandizement, not only the present and the future, but also the past.  It is a ruthlessly competitive organization and is merciless to its employees.”

“I. G. Farben?” I asked grabbing nervously and clumsily at humor.

She didn’t rebuke my flippancy, but said, “And it isn’t the Communist Party or the Ku Klux Klan, or the Avenging Angels or the Black Hand, either, though its enemies give it a nastier name.”

“Which is?” I asked.

“The Spiders,” she said.

That word gave me the shudders, coming so suddenly.  I expected the sigil to step off her forehead and scuttle down her face and leap at me—something like that.

She watched me.  “You might call it the Double Cross,” she suggested, “if that seems better.”

— Fritz Leiber,
   “Damnation Morning,” 1959

From last year’s entry,
Indiana Jones and the Hidden Coffer,
of 6/14:

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

Finally, from an excellent site
on the Knights Templar,
a quotation from Umberto Eco:

When all the archetypes burst out shamelessly, we plumb the depths of Homeric profundity. Two cliches make us laugh but a hundred cliches move us because we sense dimly that the cliches are talking among themselves, celebrating a reunion . . . Just as the extreme of pain meets sensual pleasure, and the extreme of perversion borders on mystical energy, so too the extreme of banality allows us to catch a glimpse of the Sublime.

— “Casablanca: Cult Movies and Intertextual Collage” (1984) from Travels in Hyperreality

Friday, April 30, 2004

Friday April 30, 2004

Filed under: General — Tags: — m759 @ 6:24 AM


These are the folios of April,
All the library of spring

Christopher Morley

The above quotation is dedicated to Quay A. McCune, M.D., whose copy of Bartlett's Familiar Quotations I purchased for two dollars at a Friends of the Library sale on July 2, 1999.  Dr. McCune's copy of Bartlett was the twelfth edition, of November 1948, in a February 1952 reprint.  It was edited by Christopher Morley.

Incidentally, Morley's father Frank, a professor of mathematics, is the discoverer of Morley's theorem, which says that the angle trisectors of any triangle, of whatever shape, determine an equilateral "Morley triangle" hidden within the original triangle.


Those familiar with Dorothy Sayers's explication of the Trinity, The Mind of the Maker, will recognize that this figure represents a triumph over the heresies she so skillfully describes in the chapter "Scalene Trinities."  From another chapter:

"… this is the Idea that is put forward for our response. There is nothing mythological about Christian Trinitarian doctrine: it is analogical. It offers itself freely for meditation and discussion; but it is desirable that we should avoid the bewildered frame of mind of the apocryphal Japanese gentleman who complained:

'Honourable Father, very good;
 Honourable Son, very good; but
 Honourable Bird
     I do not understand at all.'

'Honourable Bird,' however, has certain advantages as a pictorial symbol, since, besides reminding us of those realities which it does symbolise, it also reminds us that the whole picture is a symbol and no more."

In the Morley family trinity, if Frank is the Father and Christopher is the Son, we must conclude that the Holy Spirit is Christopher's mother — whose maiden name was, appropriately, Bird.

Friday, April 16, 2004

Friday April 16, 2004

Filed under: General — m759 @ 10:00 AM

Mistakes Were Made

By Al Kamen, Washington Post 
Friday, April 16, 2004

“… Bush, in his news conference Tuesday…. found a way to make not one, not two, but three factual errors in a single 15-word sentence, which must be something of a world indoor record. Bush said it is still possible that inspectors will find weapons of mass destruction in Iraq.

‘They could still be there. They could be hidden, like the 50 tons of mustard gas in a turkey farm,’ he said, referring to Libya’s WMD disclosures last month.

The White House, according to Reuters, said the accurate figure was 23.6 metric tons or 26 tons, not 50. The stuff was found at various locations, not at a turkey farm. And there was no mustard gas on the farm at all, but unfilled chemical munitions.

Other than that, the sentence was spot on.”

Other mistakes …

“It’s not at all like CIA Director George J. Tenet to forget not one, but two, conversations with President Bush in the critical month [August] before Sept. 11, 2001. But there’s one possible explanation for his distraction when he testified Wednesday morning to the Sept. 11 commission: He was thinking about his luncheon plans.

Tenet was spotted around 12:30 at the Hay-Adams, sitting at a window table for two with none other than Jack Valenti, outgoing head of the Motion Picture Association of America….”

Hey, that’s why
they make erasers.

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

“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….

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

Friday, March 28, 2003

Friday March 28, 2003

Filed under: General — m759 @ 10:16 AM

Bright Star

From a Spanish-English dictionary:

lucero m. morning or evening star: any bright star….

 Today is Reba McEntire’s birthday.

” ‘I know what it is you last saw,’ she said; ‘for that is also in my mind. Do not be afraid! But do not think that only by singing amid the trees, nor even by the slender arrows of elven-bows, is this land of Lothlórien maintained and defended against the Enemy. I say to you, Frodo, that even as I speak to you, I perceive the Dark Lord and know his mind, or all his mind that concerns the Elves. And he gropes ever to see me and my thought. But still the door is closed!’

      She lifted up her white arms, and spread out her hands towards the East in a gesture of rejection and denial. Eärendil, the Evening Star, most beloved of the Elves, shone clear above. So bright was it that the figure of the Elven-lady cast a dim shadow on the ground. Its ray glanced upon a ring about her finger; it glittered like polished gold overlaid with silver light, and a white stone in it twinkled as if the Even-star had come to rest upon her hand. Frodo gazed at the ring with awe; for suddenly it seemed to him that he understood. 

      ‘Yes’, she said, divining his thought, ‘it is not permitted to speak of it, and Elrond could not do so. But it cannot be hidden from the Ring-Bearer, and one who has seen the Eye. Verily it is in the land of Lórien upon the finger of Galadriel that one of the Three remains. This is Nenya, the Ring of Adamant, and I am its keeper.’ “

— J. R. R. Tolkien, The Lord of the Rings

Related material on telepathy:

Shining Forth and Naturalized Epistemology

Related material on rings, and another musical Reba:

Leonard Gillman interview, Part I and Part II

Gillman, a pianist, is co-author of Rings of Continuous Functions.

Sunday, September 15, 2002

Sunday September 15, 2002

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

Evariste Galois and 
The Rock That Changed Things

An article in the current New York Review of Books (dated Sept. 26) on Ursula K. Le Guin prompted me to search the Web this evening for information on a short story of hers I remembered liking.  I found the following in the journal of mathematician Peter Berman:

  • A Fisherman of the Inland Sea, Ursula K. Le Guin, 1994:
    A book of short stories. Good, entertaining. I especially liked “The Rock That Changed Things.” This story is set in a highly stratified society, one split between elite and enslaved populations. In this community, the most important art form is a type of mosaic made from rocks, whose patterns are read and interpreted by scholars from the elite group. The main character is a slave woman who discovers new patterns in the mosaics. The story is slightly over-the-top but elegant all the same.

I agree that the story is elegant (from a mathematician, a high compliment), so searched Berman’s pages further, finding this:

A table of parallels

between The French Mathematician (a novel about Galois) and Harry Potter and the Sorcerer’s Stone

My own version of the Philosopher’s Stone (the phrase used instead of “Sorcerer’s Stone” in the British editions of Harry Potter) appears in my profile picture at top left; see also the picture of Plato’s diamond figure in my main math website.  The mathematics of finite (or “Galois”) fields plays a role in the underlying theory of this figure’s hidden symmetries.  Since the perception of color plays a large role in the Le Guin story and since my version of Plato’s diamond is obtained by coloring Plato’s version, this particular “rock that changes things” might, I hope, inspire Berman to extend his table to include Le Guin’s tale as well.

Even the mosaic theme is appropriate, this being the holiest of the Mosaic holy days.

Dr. Berman, G’mar Chatimah Tova.

Tuesday, July 30, 2002

Tuesday July 30, 2002

Filed under: General — Tags: — m759 @ 12:12 AM

Aesthetics of Madness

Admirers of the film "A Beautiful Mind" may be interested in the thoughts of psychotherapist Eric Olson on what he calls the "collage method" of therapy.  The fictional protagonist of "A Beautiful Mind," very loosely based on the real-life mathematician John Nash, displays his madness in a visually striking manner (as required by cinematic art).  He makes enormous collages of published matter in which he believes he has found hidden patterns. 

This fictional character is in some ways more like the real-life therapist Olson than like the real-life schizophrenic Nash.  For an excellent introduction to Olson's world, see the New York Times Magazine article of April 1, 2001, on Olson and on the mysterious death of Olson's father Frank, who worked for the CIA.  Here the plot thickens… the title of the article is "What Did the C.I.A. Do to Eric Olson's Father?

For Olson's own website, see The Frank Olson Legacy Project, which has links to Olson's work on collage therapy.   Viewed in the context of this website, the resemblance of Olson's collages to the collages of "A Beautiful Mind" is, to borrow Freud's expression, uncanny.  Olson's own introduction to his collage method is found on the web page "Theory and therapy."

All of the above resulted from a Google search to see if Arlene Croce's 1993 New Yorker article on Balanchine and Stravinsky, "The Spelling of Agon," could be found online.   I did not find Arlene, but I did find the following, from a collage of quotations assembled by Eric Olson —

"There might be a game in which paper figures were put together to form a story, or at any rate were somehow assembled. The materials might be collected and stored in a scrap-book, full of pictures and anecdotes. The child might then take various bits from the scrap-book to put into the construction; and he might take a considerable picture because it had something in it which he wanted and he might just include the rest because it was there.”

— Ludwig Wittgenstein,
Lectures and Conversations on Aesthetics, Psychology and Religious Belief, 1943/1978

“Not games. Puzzles. Big difference. That’s a whole other matter. All art — symphonies, architecture, novels — it’s all puzzles. The fitting together of notes, the fitting together of words have by their very nature a puzzle aspect. It’s the creation of form out of chaos. And I believe in form.”

Stephen Sondheim
in Stephen Schiff, “Deconstructing Sondheim,”
The New Yorker, March 8, 1993, p. 76.

“God creates, I assemble.”

— George Balenchine [sic]
in Arlene Croce, “The Spelling of Agon,”
The New Yorker, July 12, 1993, p. 91

The aesthetics of collage is, of course, not without its relevance to the creation (or assembly) of weblogs.

Saturday, July 20, 2002

Saturday July 20, 2002

Filed under: General,Geometry — Tags: , — 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.


For an animated version, click here.


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

Walsh Functions


The Diamond Theory of Truth

Geometry of the I Ching

Solomon's Cube and The Eightfold Way

Crystal and Dragon in Diamond Theory

The Form, the Pattern

The Grid of Time

Block Designs

Finite Relativity

Theme and Variations

Models of Finite Geometries

Quilt Geometry

Pattern Groups

The Fano Plane Revisualized,
or the Eightfold Cube

The Miracle Octad Generator


Visualizing GL(2,p)

Jung's Imago

Author's home page

AMS Mathematics Subject Classification:

20B25 (Group theory and generalizations :: Permutation groups :: Finite automorphism groups of algebraic, geometric, or combinatorial structures)

05B25 (Combinatorics :: Designs and configurations :: Finite geometries)

51E20 (Geometry :: Finite geometry and special incidence structures :: Combinatorial structures in finite projective spaces)

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Page created Jan. 6, 2006, by Steven H. Cullinane      diamondtheorem.com


Initial Xanga entry.  Updated Nov. 18, 2006.

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