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Friday, May 9, 2003

Friday May 9, 2003

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

ART WARS:
The Religion of Cubism

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

 


logo

 

Die

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

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

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

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

See, too, Nabokov's Transparent Things :

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

Monday, February 17, 2003

Monday February 17, 2003

Filed under: General — m759 @ 3:36 pm

Saint Faggot’s Day

“During the European Inquisitions, faggot referred to the sticks used to set fires for burning heretics, or people who opposed the teachings of the Catholic Church. Heretics were required to gather bundles of sticks (‘faggots’) and carry them to the fire that was being built for them. Heretics who changed their beliefs to avoid being killed were forced to wear a faggot design embroidered on their sleeve, to show everyone that they had opposed the Church.”

— Handout


Cover illustration
by Stephen Savage

N.Y. Times Feb. 2, 2003



‘A Box of Matches’:
A Miniaturist’s
Novel of Details

In Nicholson Baker’s novel,
things not worth noticing
eventually become
all there is to notice.

Head White House speechwriter Michael Gerson:

“In the last two weeks, I’ve been returning to Hopkins.  Even in the ‘world’s wildfire,’ he asserts that ‘this Jack, joke, poor potsherd, patch, matchwood, immortal diamond,/Is immortal diamond.’ A comfort.”
— Vanity Fair, May 2002, page 162

“At midnight on the Emperor’s pavement flit
Flames that no faggot feeds….”

— William Butler Yeats, “Byzantium”

On this date in 1600, Saint Giordano Bruno was burned at the stake for heresy by the Roman Catholic Church.

He was resurrected by Saint Frances Yates, who went to her reward on the feast day of Saint Michael and All Angels, 1981.

Wednesday, January 29, 2003

Wednesday January 29, 2003

Filed under: General — m759 @ 6:09 pm

Inaugural Address
for Cullinane College

(undelivered):

The Prisoner

Cullinane College was scheduled to open its doors officially on January 29, 2003.  The following might have been an appropriate inaugural address.

From The Prisoner: Comments
 on the Final Episode, “Fall Out”
:

“When the President asks for a vote, he says: ‘All in favor.’ But he never asks for those opposed. (Though it appears that none will be opposed — and though he says its a democratic assembly, it is hardly that. The President even says that the society is in a ‘democratic crisis,’ though without democracy present, it’s just a sham.)

#48/Young Man sings ‘Dry Bones,’, which is his rebellion (notice its chaotic effect on ‘society’). But then the song gets taken over, ‘polished,’ and sung by a voice-over (presumably set up by #1). Does this mean that society is stealing the thunder (i.e. the creative energy) of youth, and cheapening it, or does it mean that youth is just rebelling in the same way that their fathers did (with equal ineffectiveness)? Perhaps it is simply a comment on the ease with which society can deal with the real rebellion of the 1960’s, which purported to be led by musicians; one that even the Beatles said was impossible in ‘Revolution.'”

President: Guilty! Read the Charge!

#48 is guilty, of something, and then the society pins something on him.”

The Other Side of the Coin

The Weinman Dime

From the CoinCentric website:

In 1916, sculptor Adolph A. Weinman produced a new design for the dime called the Liberty Head type. The motif features Miss Liberty facing left, wearing a Phrygian cap with wings, symbolizing “liberty of thought”. The word “LIBERTY” encircles her head, with “IN GOD WE TRUST” and the date below her head.

The reverse depicts Roman fasces, a bundle of rods with the center rod being an ax, against a branch in the background. It is a symbol of state authority, which offers a choice: “by the rod or by the ax”. The condemned was either beaten to death with the rods or allowed the mercy of the ax. The words “UNITED STATES OF AMERICA” and “ONE DIME” surround the border. “E PLURIBUS UNUM” appears at the lower right.

Excerpt from the poem that Robert Frost (who died on this date in 1963) meant to read at the 1961 inauguration of John F. Kennedy:

It makes the prophet in us all presage
The glory of a next Augustan age
Of a power leading from its strength and pride,
Of young ambition eager to be tried,
Firm in our free beliefs without dismay,
In any game the nations want to play.
A golden age of poetry and power
Of which this noonday’s the beginning hour.

I greatly prefer Robinson Jeffers’s “Shine, Perishing Republic“:

While this America settles in the mould of its vulgarity,
    heavily thickening to empire,
And protest, only a bubble in the molten mass, pops and sighs out, 
    and the mass hardens,
I sadly smiling remember….

See also the thoughts on Republic vs. Empire in the work of Alec Guinness (as Marcus Aurelius and as Obi-Wan Kenobi).

Sunday, September 22, 2002

Sunday September 22, 2002

Filed under: General,Geometry — Tags: , , , , — m759 @ 8:02 pm

Force Field of Dreams

Metaphysics and chess in today’s New York Times Magazine:

  • From “Must-See Metaphysics,” by Emily Nussbaum:

    Joss Whedon, creator of a new TV series —

    “I’m a very hard-line, angry atheist” and
    “I want to invade people’s dreams.”

  • From “Check This,” by Wm. Ferguson:

    Garry Kasparov on chess —

    “When the computer sees forced lines,
    it plays like God.”

Putting these quotations together, one is tempted to imagine God having a little game of chess with Whedon, along the lines suggested by C. S. Lewis:

As Lewis tells it the time had come for his “Adversary [as he was wont to speak of the God he had so earnestly sought to avoid] to make His final moves.” (C. S. Lewis, Surprised by Joy, Harcourt, Brace, and World, Inc., 1955, p. 216) Lewis called them “moves” because his life seemed like a chess match in which his pieces were spread all over the board in the most disadvantageous positions. The board was set for a checkmate….

For those who would like to imagine such a game (God vs. Whedon), the following may be helpful.

George Steiner has observed that

The common bond between chess, music, and mathematics may, finally, be the absence of language.

This quotation is apparently from

Fields of Force:
Fischer and Spassky at Reykjavik
. by George Steiner, Viking hardcover, June 1974.

George Steiner as quoted in a review of his book Grammars of Creation:

“I put forward the intuition, provisional and qualified, that the ‘language-animal’ we have been since ancient Greece so designated us, is undergoing mutation.”

The phrase “language-animal” is telling.  A Google search reveals that it is by no means a common phrase, and that Steiner may have taken it from Heidegger.  From another review, by Roger Kimball:

In ”Grammars of Creation,” for example, he tells us that ”the classical and Judaic ideal of man as ‘language animal,’ as uniquely defined by the dignity of speech . . . came to an end in the antilanguage of the death camps.”

This use of the Holocaust not only gives the appearance of establishing one’s credentials as a person of great moral gravity; it also stymies criticism. Who wants to risk the charge of insensitivity by objecting that the Holocaust had nothing to do with the ”ideal of man as ‘language animal’ ”?

Steiner has about as clear an idea of the difference between “classical” and “Judaic” ideals of man as did Michael Dukakis. (See my notes of September 9, 2002.)

Clearly what music, mathematics, and chess have in common is that they are activities based on pure form, not on language. Steiner is correct to that extent. The Greeks had, of course, an extremely strong sense of form, and, indeed, the foremost philosopher of the West, Plato, based his teachings on the notion of Forms. Jews, on the other hand, have based their culture mainly on stories… that is, on language rather than on form. The phrase “language-animal” sounds much more Jewish than Greek. Steiner is himself rather adept at the manipulation of language (and of people by means of language), but, while admiring form-based disciplines, is not particularly adept at them.

I would argue that developing a strong sense of form — of the sort required to, as Lewis would have it, play chess with God — does not require any “mutation,” but merely learning two very powerful non-Jewish approaches to thought and life: the Forms of Plato and the “archetypes” of Jung as exemplified by the 64 hexagrams of the 3,000-year-old Chinese classic, the I Ching.

For a picture of how these 64 Forms, or Hexagrams, might function as a chessboard,

click here.

Other relevant links:

“As you read, watch for patterns. Pay special attention to imagery that is geometric…”

and


from Shakhmatnaia goriachka

Tuesday, September 3, 2002

Tuesday September 3, 2002

Filed under: G-Notes,General,Geometry — Tags: , — m759 @ 6:00 pm

Today's birthday: James Joseph Sylvester

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

Sylvester, a nineteenth-century mathematician, coined the phrase "synthematic totals" to describe some structures based on 6-element sets that R. T. Curtis has called "rather unwieldy objects." See Curtis's abstract, Symmetric Generation of Finite Groups, John Baez's essay, Some Thoughts on the Number 6, and my website, Diamond Theory. See also the abstract of a December 7, 2000, talk, Mathematics and the Art of M. C. Escher, in which Curtis notes that graphic designs can "often convey a mathematical idea more eloquently than pages of symbolism."

Saturday, July 20, 2002

Saturday July 20, 2002

 

ABSTRACT: Finite projective geometry explains the surprising symmetry properties of some simple graphic designs– found, for instance, in quilts. Links are provided for applications to sporadic simple groups (via the "Miracle Octad Generator" of R. T. Curtis), to the connection between orthogonal Latin squares and projective spreads, and to symmetry of Walsh functions.

We regard the four-diamond figure D above as a 4×4 array of two-color diagonally-divided square tiles.

Let G be the group of 322,560 permutations of these 16 tiles generated by arbitrarily mixing random permutations of rows and of columns with random permutations of the four 2×2 quadrants.

THEOREM: Every G-image of D (as at right, below) has some ordinary or color-interchange symmetry.

Example:


For an animated version, click here.

Remarks:

Some of the patterns resulting from the action of G on D have been known for thousands of years. (See Jablan, Symmetry and Ornament, Ch. 2.6.) It is perhaps surprising that the patterns' interrelationships and symmetries can be explained fully only by using mathematics discovered just recently (relative to the patterns' age)– in particular, the theory of automorphism groups of finite geometries.

Using this theory, we can summarize the patterns' properties by saying that G is isomorphic to the affine group A on the linear 4-space over GF(2) and that the 35 structures of the 840 = 35 x 24 G-images of D are isomorphic to the 35 lines in the 3-dimensional projective space over GF(2).

This can be seen by viewing the 35 structures as three-sets of line diagrams, based on the three partitions of the four-set of square two-color tiles into two two-sets, and indicating the locations of these two-sets of tiles within the 4×4 patterns. The lines of the line diagrams may be added in a binary fashion (i.e., 1+1=0). Each three-set of line diagrams sums to zero– i.e., each diagram in a three-set is the binary sum of the other two diagrams in the set. Thus, the 35 three-sets of line diagrams correspond to the 35 three-point lines of the finite projective 3-space PG(3,2).

For example, here are the line diagrams for the figures above:

 
Shown below are the 15 possible line diagrams resulting from row/column/quadrant permutations. These 15 diagrams may, as noted above, be regarded as the 15 points of the projective 3-space PG(3,2).


The symmetry of the line diagrams accounts for the symmetry of the two-color patterns. (A proof shows that a 2nx2n two-color triangular half-squares pattern with such line diagrams must have a 2×2 center with a symmetry, and that this symmetry must be shared by the entire pattern.)

Among the 35 structures of the 840 4×4 arrays of tiles, orthogonality (in the sense of Latin-square orthogonality) corresponds to skewness of lines in the finite projective space PG(3,2). This was stated by the author in a 1978 note. (The note apparently had little effect. A quarter-century later, P. Govaerts, D. Jungnickel, L. Storme, and J. A. Thas wrote that skew (i.e., nonintersecting) lines in a projective space seem "at first sight not at all related" to orthogonal Latin squares.)

We can define sums and products so that the G-images of D generate an ideal (1024 patterns characterized by all horizontal or vertical "cuts" being uninterrupted) of a ring of 4096 symmetric patterns. There is an infinite family of such "diamond" rings, isomorphic to rings of matrices over GF(4).

The proof uses a decomposition technique for functions into a finite field that might be of more general use.

The underlying geometry of the 4×4 patterns is closely related to the Miracle Octad Generator of R. T. Curtis– used in the construction of the Steiner system S(5,8,24)– and hence is also related to the Leech lattice, which, as Walter Feit has remarked, "is a blown up version of S(5,8,24)."

For a movable JavaScript version of these 4×4 patterns, see The Diamond 16 Puzzle.

The above is an expanded version of Abstract 79T-A37, "Symmetry invariance in a diamond ring," by Steven H. Cullinane, Notices of the American Mathematical Society, February 1979, pages A-193, 194.

For a discussion of other cases of the theorem, click here.

Related pages:

The Diamond 16 Puzzle

Diamond Theory in 1937:
A Brief Historical Note

Notes on Finite Geometry

Geometry of the 4×4 Square

Binary Coordinate Systems

The 35 Lines of PG(3,2)

Map Systems:
Function Decomposition over a Finite Field

The Diamond Theorem–
The 2×2, the 2x2x2, the 4×4, and the 4x4x4 Cases

Diamond Theory

Latin-Square Geometry

Walsh Functions

Inscapes

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

Kaleidoscope

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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