Log24

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

Birthday Tablet

“Of the world’s countless customs and traditions, perhaps none is as elegant, nor as beautiful, as the tradition of sangaku, Japanese temple geometry.”

— Tony Rothman

Sangaku means “mathematical tablet.”

Here is a sangaku for
Dr. Mary McClintock Dusenbury
on her birthday.

For an explanation,
click here.

Realism in Literature:
Under the Volcano

Here are 3 webcam views of the volcano.   Nothing to see at the moment.

Literary background:

Malcolm Lowry’s Under the Volcano,

Plato, Pegasus, and the Evening Star,

A Mass for Lucero,

Shining Forth,

and, as background for today’s earlier entry on Platonism and Derrida,

The Shining of May 29.

In the above two-step vignette, the part of Gollum is played by the author of “Further Into the Depths of Satan,” who called  C. S. Lewis a fool† “that was and is extremely useful to his father the devil.”

† See Matthew 5:22: “…whosoever shall say, Thou fool, shall be in danger of hell fire.” 

Hideous Strength

On a Report from London:

Assuming rather prematurely that the body found in Oxfordshire today is that of David Kelly, Ministry of Defence germ-warfare expert and alleged leaker of information to the press, the Financial Times has the following:

“Mr Kelly’s death has stunned all the players involved in this drama, resembling as it does a fictitious political thriller.”

— Financial Times, July 18,
   2003, 19:06 London time

I feel it resembles rather a fictitious religious thriller… Namely, That Hideous Strength, by C. S. Lewis.  The use of the word “idea” in my entries’ headlines yesterday was not accidental.  It is related to an occurrence of the word in Understanding: On Death and Truth, a set of journal entries from May 9-12.  The relevant passage on “ideas” is quoted there, within commentary by an Oberlin professor:

“That the truth we understand must be a truth we stand under is brought out nicely in C. S. Lewis’ That Hideous Strength when Mark Studdock gradually learns what an ‘Idea’ is. While Frost attempts to give Mark a ‘training in objectivity’ that will destroy in him any natural moral sense, and while Mark tries desperately to find a way out of the moral void into which he is being drawn, he discovers what it means to under-stand.

‘He had never before known what an Idea meant: he had always thought till now that they were things inside one’s own head. But now, when his head was continually attacked and often completely filled with the clinging corruption of the training, this Idea towered up above him-something which obviously existed quite independently of himself and had hard rock surfaces which would not give, surfaces he could cling to.’

This too, I fear, is seldom communicated in the classroom, where opinion reigns supreme. But it has important implications for the way we understand argument.”

— “On Bringing One’s Life to a Point,” by Gilbert Meilaender, First Things,  November 1994

The old philosophical conflict between realism and nominalism can, it seems, have life-and-death consequences.  I prefer Plato’s realism, with its “ideas,” such as the idea of seven-ness.  A reductio ad absurdum of nominalism may be found in the Stanford Encyclopedia of Philosophy under Realism:

“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….”

The claim that 7 is not prime is, regardless of its motives, dangerously stupid… A quality shared, it seems, by many in power these days.

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.

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.

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.

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.

Stoicheia:

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

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“

Born on this date:
William Butler Yeats.

“Surely some revelation
  is at hand” — W. B. Yeats

Behold a Pale Horse:
A link in memory of Gregory Peck.

In Slouching Towards Bethlehem, Joan Didion wrote that

“The oral history of Los Angeles
is written in piano bars.”

Today’s site music, a piano rendition of “Speak Low,” from “One Touch of Venus,” was suggested by

• the “black triangle” theme of Wednesday’s entry and by
• the name “Amy Hollywood.” 

Ms. Hollywood has an essay in the April 2003 Princeton journal Theology Today.

My own theological interests (besides those expressed in the “black triangle” link above) are much closer to those in a 2001 First Things essay, The End of Endings. 

 Washington Square Press paperback, 1981, page 222

Mental Health Month, Day 26:

Many Dimensions,

Part III — Why 26?

At first blush, it seems unlikely that the number 26=2×13, as a product of only two small primes (and those distinct) has any purely mathematical properties of interest. (On the other hand, consider the number 6.)  Parts I and II of “Many Dimensions,” notes written earlier today, deal with the struggles of string theorists to justify their contention that a space of 26 dimensions may have some significance in physics.  Let them struggle.  My question is whether there are any interesting purely mathematical properties of 26, and it turns out, surprisingly, that there are some such properties. All this is a longwinded way of introducing a link to the web page titled “Info on M13,” which gives details of a 1997 paper by J. H. Conway*.

There is no obvious connection to physics, although the physics writer John Baez quoted in my previous two entries shares Conway’s interest in the Mathieu groups. 

 * J. H. Conway, “M₁₃,” in Surveys in Combinatorics, 1997, edited by R. A. Bailey, London Mathematical Society Lecture Note Series, 241, Cambridge University Press, Cambridge, 1997. 338 pp. ISBN 0 521 59840 0.

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.
 

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

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.”

Related material (update of May 12, 2003):

1. Pilate, Truth, and Friday the Thirteenth
2. The Diamond Theory of Truth
3. Understanding

Question:

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

ART WARS:

Graphical Password

From a summary of “The Design and Analysis of Graphical Passwords“:

“Results from cognitive science show that people can remember pictures much better than words….

The 5×5 grid creates a good balance between security and memorability.”

— Ian Jermyn, New York University; Alain Mayer, Fabian Monrose, Michael K. Reiter, Bell Labs, Lucent Technologies; Aviel Rubin, AT&T Labs — Research

Illustration — Warren Beatty as
a graphical password:

“Town & Country,”
released April 27, 2001

Those who prefer the simplicity of a 3×3 grid are referred to my entry of Jan. 9, 2003, Balanchine’s Birthday.  For material related to the “Town & Country” theme and to Balanchine, see Leadbelly Under the Volcano (Jan. 27, 2003). (“Sometimes I live in the country, sometimes I live in town…” – Huddie Ledbetter).  Those with more sophisticated tastes may prefer the work of Stephen Ledbetter on Gershwin’s piano preludes or, in view of Warren Beatty’s architectural work in “Town & Country,” the work of Stephen R. Ledbetter on window architecture.

As noted in Balanchine’s Birthday, Apollo (of the Balanchine ballet) has been associated by an architect with the 3×3, or “ninefold” grid.  The reader who wishes a deeper meditation on the number nine, related to the “Town & Country” theme and more suited to the fact that April is Poetry Month, is referred to my note of April 27 two years ago, Nine Gates to the Temple of Poetry.

Intermediate between the simplicity of the 3×3 square and the (apparent) complexity of the 5×5 square, the 4×4 square offers an introduction to geometrical concepts that appears deceptively simple, but is in reality fiendishly complex.  See Geometry for Jews.  The moral of this megilla?

3² + 4² = 5².

But that is another story.

Aptheker

A Look at the Rat

In memory of Herbert Aptheker, theoretician of the American Communist Party, who died on St. Patrick’s Day, 2003 —

From The New Yorker, issue dated March 24, 2003, Louis Menand on Edmund Wilson’s To the Finland Station:

“Wilson did know what was going on in the Soviet Union in the nineteen-thirties, as his pages on Stalin in To the Finland Station make clear. The problem wasn’t with Stalin; the problem was with Lenin, the book’s ideal type of the intellectual as man of action. Wilson admitted that he had relied on publications controlled by the Party for his portrait of Lenin. (Critical accounts were available; for example, the English translation of the émigré Mark Landau-Aldanov’s Lenin was published, by Dutton, in 1922.) Lenin could create an impression of selfless humanitarianism; he was also a savage and ruthless politician—a ‘pail of milk of human kindness with a dead rat at the bottom,’ as Vladimir Nabokov put it to Wilson in 1940, after reading To the Finland Station.  In the introduction to the 1972 edition, Wilson provided a look at the rat. He did not go on to explain in that introduction that the most notorious features of Stalin’s regime—the use of terror, the show trials, and the concentration camps—had all been inaugurated by Lenin. To the Finland Station begins with Napoleon’s betrayal of the principles of the French Revolution; it should have ended with Lenin’s betrayal of European socialism.” 

From Herbert Aptheker, “More Comments on Howard Fast“:

“We observe that in the list of teachers whom Howard Fast names as most influential in his own life there occur the names of fourteen individuals from Jefferson to Bernard Shaw, Upton Sinclair to Marx, Douglass to Engels, but there is no room for Lenin.
   He is, I think, an important teacher, too; indeed, in my view, Lenin is the greatest figure in the whole galaxy of world revolutionary leaders. He is, certainly, the greatest analyzer of and fighter against imperialism.”

For more on Howard Fast, see my entry
“Death Knell” of March 13, 2003. 

For a look at the pail of milk, see
the New Yorker cover in Geometry for Jews.

For a more cheerful look at geometry
on this St. Joseph’s Day, see
Harry J. Smith’s

Tesseract Site.

“There is such a thing as a tesseract.”
— A Wrinkle in Time

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.” © 1997-2003 Annenberg/CPB. All rights reserved. Legal Policy

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

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.

Symbols —

Broadway:
The Sound of Silence

See also the footnote on the Halmos “tombstone” symbol in the previous entry, the entry “Dustin in Wonderland” of Feb. 24, the film “Marathon Man,” and the entry “Geometry for Jews” of March 6.

Ash Wednesday

Brace Yourself, Maureen

From Maureen Dowd’s New York Times column today:

“During the innocent summer before 9/11, the defense secretary’s office sponsored a study of ancient empires — Macedonia, Rome, the Mongols — to figure out how they maintained dominance.

What tips could Rummy glean from Alexander the Great, Julius Caesar and Genghis Khan?”

Saddle up!

Background briefing, added at 6:29 AM:

See also the use of the hyperbolic paraboloid in Mexican church architecture by Félix Candela and an essay on saddle surfaces by Joseph F. MacDonnell, Society of Jesus, who spent eight years in Iraq teaching physics and mathematics at two Jesuit schools in Baghdad: Baghdad College and Al Hikma University.  He writes that “since the 1968 Baathi takeover of the two Jesuit schools and expulsion of all Jesuits from Iraq in 1969” he has been teaching mathematics at Fairfield University. 

MacDonnell notes that there are only three doubly ruled surfaces (in real 3-space): the hyperboloid (used for nuclear cooling towers), the hyperbolic paraboloid (used, as noted, for Mexican churches), and the plane (used widely).  The geometry here is perhaps less relevant than the existence of the Society of Jesus as a sort of intelligence agency within the Church — an agency the current Pope has never understood how to use.  Opus Dei is a greatly inferior substitute.

Diamond Life
(Von Neumann’s Song, Part II)

A reader of yesterday’s entry “St. John von Neumann’s Song” suggested the relevance of little Dougie Hofstadter‘s book Gödel, Escher, Bach: An Eternal Golden Braid.  While the title of this work does continue the “golden” theme of my last three entries, Dougie is not playing in von Neumann’s league.  The nature of this league is suggested by yesterday’s citation of

Abstract Harmonic Analysis. 

For work that is more in von Neumann’s league than in Hofstadter’s, see the following

harmonic analysis abstract:

VECTOR-VALUED EXTENSIONS
OF SOME CLASSICAL THEOREMS
IN HARMONIC ANALYSIS

Maria Girardi and Lutz Weis

Abstract:
…. The approach used combines methods from Fourier analysis and the geometry of Banach spaces, such as R-boundedness.

A related paper by the same authors:

CRITERIA FOR R-BOUNDEDNESS
OF OPERATOR FAMILIES

Abstract:
…smooth operator-valued functions have a R-bounded range, where the degree of smoothness depends on the geometry of the Banach space.

Those who would like to make a connection to music in the charmingly childlike manner of Hofstadter are invited to sing a few choruses of “How do you solve a problem like Maria?“

Personally, I prefer the following lyrics:

Diamond life, lover boy;
We move in space with minimum waste and maximum joy.
City lights and business nights
When you require streetcar desire for higher heights.

No need to ask.
He’s a smooth operator….

Words and Music: Sade Adu and Ray St. John

Some may wish to alter the last five syllables of these lyrics in accordance with yesterday’s entry on another St. John.

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:

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.

Symmetry and a Trinity

From a web page titled Spectra:

"What we learn from our whole discussion and what has indeed become a guiding principle in modern mathematics is this lesson:

Whenever you have to do with a structure-endowed entity  S try to determine its group of automorphisms, the group of those element-wise transformations which leave all structural relations undisturbed. You can expect to gain a deep insight into the constitution of S in this way. After that you may start to investigate symmetric configurations of elements, i.e., configurations which are invariant under a certain subgroup of the group of all automorphisms . . ."

— Hermann Weyl in Symmetry, Princeton University Press, 1952, page 144

"… any color at all can be made from three different colors, in our case, red, green, and blue lights. By suitably mixing the three together we can make anything at all, as we demonstrated . . .

Further, these laws are very interesting mathematically. For those who are interested in the mathematics of the thing, it turns out as follows. Suppose that we take our three colors, which were red, green, and blue, but label them A, B, and C, and call them our primary colors. Then any color could be made by certain amounts of these three: say an amount a of color A, an amount b of color B, and an amount c of color C makes X:

Now suppose another color Y is made from the same three colors:

Then it turns out that the mixture of the two lights (it is one of the consequences of the laws that we have already mentioned) is obtained by taking the sum of the components of X and Y:

It is just like the mathematics of the addition of vectors, where (a, b, c ) are the components of one vector, and (a', b', c' ) are those of another vector, and the new light Z is then the "sum" of the vectors. This subject has always appealed to physicists and mathematicians."

— According to the author of the Spectra site, this is Richard Feynman in Elementary Particles and the Laws of Physics, The 1986 Dirac Memorial Lectures, by Feynman and Steven Weinberg, Cambridge University Press, 1989.

These two concepts — symmetry as invariance under a group of transformations, and complicated things as linear combinations (the technical name for Feynman's sums) of simpler things — underlie much of modern mathematics, both pure and applied.      

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.”

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.

In memory of Arthur T. Winfree:
Time, Eternity, and Grace

Professor Arthur T. Winfree died on November 5, 2002. 
He was the author of “The Geometry of Biological Time.”

• Charles Small (see the earlier entry “Hope of Heaven,” November 21):

“I’ve always been enthralled by the notion that Time is an illusion, a trick our minds play in an attempt to keep things separate, without any reality of its own. My experience suggests that this is literally true….”

• Kylie Minogue in my November 5 entry:

“Time disappears with Tequila.
It goes elastic, then vanishes.”

• Octavio Paz on his childhood in Mexico

(Nobel Prize lecture):

“All time, past or future, real or imaginary, was pure presence.”

• A colleague on Professor Winfree:

“He just wanted to get to the truth.”

• Octavio Paz, condensed and translated 

  Nobel Prize lecture:

“Gracias.”

Birthdate of Hermann Weyl

Result of a Google search.

Category:  Science > Math > Algebra > Group Theory 

Quotation from Weyl’s Symmetry:

“Symmetry is a vast subject, significant in art and nature. Mathematics lies at its root, and it would be hard to find a better one on which to demonstrate the working of the mathematical intellect.”

In honor of Princeton University, of Sylvia Nasar (see entries of Nov, 6), of the Presbyterian Church (see entry of Nov. 8), and of Professor Weyl (whose work partly inspired the website Diamond Theory), this site’s background music is now Pink Floyd’s

Updates of Friday, November 15, 2002:

In order to clarify the meaning of “Shine” and “Crazy” in the above, consult the following —

• “Shine” … The Shining, by Stephen King
• “Crazy” … The Hollow Man, by Dan Simmons 

To accompany this detailed exegesis of Pink Floyd, click here for a reading by Marlon Brando.

For a related educational experience, see pages 126-127 of The Book of Sequels, by Henry Beard, Christopher Cerf, Sarah Durkee, and Sean Kelly (Random House paperback, 1990).

Speaking of sequels, be on the lookout for Annie Dillard’s sequel to Teaching a Stone to Talk, titled Teaching a Brick to Sing.

Religious Symbolism
at Princeton

In memory of Steve McQueen (“The Great Escape” and “The Thomas Crown Affair”… see preceding entry) and of Rudolf Augstein (publisher of Der Spiegel), both of whom died on November 7 (in 1980 and 2002, respectively), in memory of the following residents of

The Princeton Cemetery
of the Nassau Presbyterian Church
Established 1757

and of the long and powerful association of Princeton University with the Presbyterian Church, as well as the theological perspective of Carl Jung in Man and His Symbols, I offer the following “windmill,” taken from the Presbyterian Creedal Standards website, as a memorial:

The background music Les Moulins de Mon Coeur, selected yesterday morning in memory of Steve McQueen, continues to be appropriate.

“A is for Anna.”
— James Joyce

From The Unknowable (1999), by Gregory J. Chaitin, who has written extensively about his constant, which he calls Omega:

"What is Omega? It's just the diamond-hard distilled and crystallized essence of mathematical truth! It's what you get when you compress tremendously the coal of redundant mathematical truth…" 

Charles H. Bennett has written about Omega as a cabalistic number.

Here is another result with religious associations which, historically, has perhaps more claim to be called the "diamond-hard essence" of mathematical truth: The demonstration in Plato's Meno that a diamond inscribed in a square has half the area of the square (or that, vice-versa, the square has twice the area of the diamond).

From Ivars Peterson's discussion of Plato's diamond and the Pythagorean theorem:

"In his textbook The History of Mathematics, Roger Cooke of the University of Vermont describes how the Babylonians might have discovered the Pythagorean theorem more than 1,000 years before Pythagoras.

Basing his account on a passage in Plato's dialogue Meno, Cooke suggests that the discovery arose when someone, either for a practical purpose or perhaps just for fun, found it necessary to construct a square twice as large as a given square…."

From "Halving a Square," a presentation of Plato's diamond by Alexander Bogomolny, the moral of the story:

SOCRATES: And if the truth about reality is always in our soul, the soul must be immortal….

From "Renaissance Metaphysics and the History of Science," at The John Dee Society website:

Galileo on Plato's diamond:

"Cassirer, drawing attention to Galileo's frequent use of the Meno, particularly the incident of the slave's solving without instruction a problem in geometry by 'natural' reason stimulated by questioning, remarks, 'Galileo seems to accept all the consequences drawn by Plato from this fact…..'"

Roger Bacon on Plato's diamond:

"Fastening on the incident of the slave in the Meno, which he had found reproduced in Cicero, Bacon argued from it 'wherefore since this knowledge (of mathematics) is almost innate and as it were precedes discovery and learning or at least is less in need of them than other sciences, it will be first among sciences and will precede others disposing us towards them.'"

It is perhaps appropriate to close this entry, made on All Hallows' Eve, with a link to a page on Dr. John Dee himself.

Introduction to
Harmonic Analysis

From Dr. Mac’s Cultural Calendar for Oct. 22:

• The French actress Catherine Deneuve was born on this day in Paris in 1943….
• The Beach Boys released the single “Good Vibrations” on this day in 1966.

“I hear the sound of a    gentle word On the wind that lifts    her perfume    through the air.” — The Beach Boys

Fermat’s Sombrero

Mexican singer Vincente Fernandez holds up the Latin Grammy award (L) for Best Ranchero Album he won for “Mas Con El Numero Uno” and the Latin Grammy Legend award at the third annual Latin Grammy Awards September 18, 2002 in Hollywood. REUTERS/Adrees Latif

From a (paper) journal note of January 5, 2002:

Princeton Alumni Weekly 
January 24, 2001 

The Sound of Math:
Turning a mathematical theorem
 and proof into a musical

Wallace Stevens:
Poet of the American Imagination

Consider these lines from
“Six Significant Landscapes” part VI:

Rationalists, wearing square hats,
Think, in square rooms,
Looking at the floor,
Looking at the ceiling.
They confine themselves
To right-angled triangles.
If they tried rhomboids,
Cones, waving lines, ellipses-
As, for example, the ellipse of the half-moon-
Rationalists would wear sombreros.

Addendum of 9/19/02: See also footnote 25 in

Theological Method and Imagination

by Julian N. Hartt

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.