Log24

Thursday, August 5, 2004

Thursday August 5, 2004

Filed under: General — Tags: — m759 @ 4:06 pm

In the beginning
was…
the recursion?

"Words are events."
— The Walter J. Ong Project,
    quoted in Log24 on Aug. 25, 2003 

"Words are events."
— The Walter J. Ong Project,
    quoted in the Heckler & Coch weblog
    on July 17, 2004 as part of a section
    titled "Recursive, Wide, and Loopy"

Walter J. Ong was a Jesuit.  The Feast of St. Ignatius Loyola, founder of the Jesuit order, is celebrated on July 31 each year.

"Recursive, Wide, and Loopy 2", a Heckler & Coch entry dated July 31, 2004, leads to the following:

MSNBC, Jan. 15, 2004:

How humans got
the gift of gab
:

Why do other primates
lag behind in language?
 

"New research may help scientists dissect just what it is about the human brain that endows us with language.

Researchers have found that tamarin monkeys have some distinctly languagelike abilities but that they can’t quite master the more complex rules of human grammar. The findings appear in Friday’s issue of the journal Science, published by AAAS, the non-profit science society.

 The grammatical toolkit

'A relatively open question concerning language evolution is, "What aspects of the language faculty are shared with other animals, and what aspects are unique to humans?" ' said study author Marc Hauser of Harvard University.

To investigate, Hauser and W. Tecumseh Fitch of the University of St. Andrews, in Scotland, devised tests for cotton-top tamarin monkeys and human volunteers. Tamarins have been evolving separately from humans for approximately 40 million years –suggesting that any shared machinery in human and tamarin brains is old enough to be relatively common among primates.

Instead of trying to teach the monkeys real words, Hauser and Fitch generated strings of one-syllable words that followed various grammatical rules.

According to linguistics expert Noam Chomsky, the simplest type of grammar is a 'finite state grammar' or 'FSG,' which dictates which types of words go near each other in a sentence. In English, for example, an adjective like 'fast' must go directly in front of 'car,' the noun it's describing.

Building on previous experiments, Hauser and Fitch recorded word-strings that obeyed a specific FSG, in which any syllable spoken by a female voice was automatically followed by one from a male voice.

Audio: Listen to an FSG word-string.
(Requires Windows Media Player.)

After listening to a series of word-strings, the monkeys were able to distinguish between those that followed this rule and others that didn't. Human test subjects could tell the difference as well, implying that tamarins and humans may share at least some components of what Hauser called 'the universal toolkit underlying all languages.'

Mastering this type of grammar represents the ability to compute some simple statistics, something human infants accomplish early on as they learn to speak. This ability may not be specific to language, however.

'Either the same mechanism or some approximation of it is used in mathematics, vision, music and other activities,' Hauser said.

Upping the Complexity

The grammatical rules of real languages govern more than just the placement of neighboring words, as anyone who had to diagram sentences in English class may remember all too well.

One of the more complex types of grammar is known as a 'phrase structure grammar,' or PSG. These grammars involve relationships between words that aren't next to each other in a sentence and thus allow for a more complex range of expression. The 'if … then' construction is an example of a PSG.

The researchers generated a second set of word-strings that followed a PSG in which a pairing of syllables spoken by a female and a male could be embedded within another pairing. This grammar produces structures like [female [female, male] male].

Audio: Listen to a PSG word-string.
(Requires Windows Media Player)

After playing these recordings repeatedly to the monkeys, the researchers found that the animals didn't seem to notice the difference between word strings that obeyed the PSG and other strings that did not. In contrast, the human volunteers did notice the difference."

— Kathleen Wren

"The grammar or syntax of human language is certainly unique. Like an onion or Russian doll, it is recursive: One instance of an item is embedded in another instance of the same item. Recursion makes it possible for the words in a sentence to be widely separated and yet dependent on one another. 'If-then' is a classic example…. Are animals capable of such recursion? Fitch and Hauser have reported that tamarin monkeys are not capable of recursion. Although the monkeys learned a nonrecursive grammar, they failed to learn a grammar that is recursive. Humans readily learn both."

— David Premack (Science 2004 303:318, quoted in ScienceWeek)

These citations by Heckler & Coch show that inability to understand complex language is not limited to monkeys.

The examples given by Wren in the audio samples are of alternating female (Hi) and male (Lo) voices, thus —

FSG:  Hi Lo Hi Lo Hi Lo

PSG:  Hi Hi Hi Lo Lo Lo

As these examples show, neither monkeys nor humans heard the sound of parentheses (or square brackets) as Wren describes them:

"structures like [female [female, male] male]."

There of course is, in ordinary language (which does not include the monologues of Victor Borge), no such thing as the sound of parentheses.

Thus the research of Hauser and Fitch is not only invalid, but ridiculous.

This point is driven strongly home by the following two articles:

Greg Kochanski, Research Fellow,
 Oxford University Phonetics Lab
:

Is a Phrase Structure Grammar
the Important Difference
between Humans and Monkeys?
,

and

Mark Liberman, Professor,
University of Pennsylvania

Departments of Linguistics
and of Computer Science,
and co-director of the
Institute for Research
in Cognitive Science,
in his

Language Log,
January 17, 2004:

Hi Lo Hi Lo,
it's off to
formal language theory
we go
.

Thursday, May 27, 2004

Thursday May 27, 2004

Filed under: General — Tags: , — m759 @ 10:10 am

Ineluctable

On the poetry of Geoffrey Hill:

"… why read him? Because of the things he writes about—war and peace and sacrifice, and the search for meaning and the truths of the heart, and for that haunting sense that, in spite of war and terror and the indifferences that make up our daily hells, there really is some grander reality, some ineluctable presence we keep touching. There remains in Hill the daunting possibility that it may actually all cohere in the end, or at least enough of it to keep us searching for more.

There is a hard edge to Hill, a strong Calvinist streak in him, and an intelligence that reminds one of Milton….."

— Paul Mariani, review in America of Geoffrey Hill's The Orchards of Syon

"Hello! Kinch here. Put me on to Edenville. Aleph, alpha: nought, nought, one." 

"A very short space of time through very short times of space…. Am I walking into eternity along Sandymount strand?"

James Joyce, Ulysses, Proteus chapter

"Time has been unfolded into space."

James O. Coplien, Bell Labs

"Pattern and symmetry are closely related."

James O. Coplien on Symmetry Breaking

"… as the critic S. L. Goldberg puts it, 'the chapter explores the Protean transformations of matter in time . . . apprehensible only in the condition of flux . . . as object . . . and Stephen himself, as subject. In the one aspect Stephen is seeking the principles of change and the underlying substance of sensory experience; in the other, he is seeking his self among its temporal manifestations'….

— Goldberg, S.L. 'Homer and the Nightmare of History.' Modern Critical Views: James Joyce. Ed. Harold Bloom. New York: Chelsea House, 1986. 21-38."

from the Choate site of David M. Loeb

In summary:

 

James Joyce
Joyce

Aleph,
alpha:
nought,
nought,
one
:

See also Time Fold.

(By the way, Jorn Barger seems
to have emerged from seclusion.)

 

Friday, February 20, 2004

Friday February 20, 2004

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

The Da Vinci Code
and
Symbology at Harvard

The protagonist of the recent bestseller The Da Vinci Code is Robert Langdon, "a professor of Religious Symbology at Harvard University."  A prominent part in the novel is played by the well-known Catholic organization Opus Dei.  Less well known (indeed, like Langdon, nonexistent) is the academic discipline of "symbology."  (For related disciplines that do exist, click here.) What might a course in this subject at Harvard be like?

Harvard Crimson, April 10, 2003:

While Opus Dei members said that they do not refer to their practices of recruitment as "fishing," the Work’s founder does describe the process of what he calls "winning new apostles" with an aquatic metaphor.

Point #978 of The Way invokes a passage in the New Testament in which Jesus tells Peter that he will make him a "fisher of men." The point reads:

" ‘Follow me, and I will make you into fishers of men.’ Not without reason does our Lord use these words: men—like fish—have to be caught by the head. What evangelical depth there is in the ‘intellectual apostolate!’ ”

IMAGE- Escher, 'Fishes and Scales'

IMAGE- Cullinane, 'Invariance'

Exercise for Symbology 101:

Describe the symmetry
in each of the pictures above.
Show that the second picture
retains its underlying structural
symmetry under a group of
322,560 transformations.

Having reviewed yesterday's notes
on Gombrich, Gadamer, and Panofsky,
discuss the astrological meaning of
the above symbols in light of
today's date, February 20.

Extra credit:

Relate the above astrological
symbolism to the four-diamond
symbol in Jung's Aion.

Happy metaphors!

Robert Langdon

Thursday, November 6, 2003

Thursday November 6, 2003

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

Legacy Codes:

The Most Violent Poem

Lore of the Manhattan Project:

From The Trinity Site

“I imagined Oppenheimer saying aloud,
‘Batter my heart, three person’d God,”
unexpectedly recalling John Donne’s ‘Holy Sonnet [14],’
and then he knew, ‘ “Trinity” will do.’
Memory has its reasons.

‘Batter my heart’ — I remember these words.
I first heard them on a fall day at Duke University in 1963.
Inside a classroom twelve of us were
seated around a long seminar table
listening to Reynolds Price recite this holy sonnet….

I remember Reynolds saying, slowly, carefully,
‘This is the most violent poem in the English language.’ ”

Related Entertainment

Today’s birthday:
director Mike Nichols

From a dead Righteous Brother:

“If you believe in forever
Then life is just a one-night stand.”

Bobby Hatfield, found dead
in his hotel room at
7 PM EST Wednesday, Nov. 5, 2003,
before a concert scheduled at
Western Michigan University, Kalamazoo
.

From a review of The Matrix Revolutions:

“You’d have to be totally blind at the end
to miss the Christian symbolism….
Trinity gets a glimpse of heaven…. And in the end…
God Put A Rainbow In The Clouds.”

Moral of the
Entertainment:

According to Chu Hsi [Zhu Xi],

“Li” is
“the principle or coherence
or order or pattern
underlying the cosmos.”

— Smith, Bol, Adler, and Wyatt,
Sung Dynasty Uses of the I Ching,
Princeton University Press, 1990

Related Non-Entertainment

Symmetry and a Trinity
(for the dotting-the-eye symbol above)

Introduction to Harmonic Analysis
(for musical and historical background)

Mathematical Proofs
(for the spirit of Western Michigan
University, Kalamazoo)

Moral of the
Non-Entertainment:

“Many kinds of entity
become easier to handle
by decomposing them into
components belonging to spaces
invariant under specified symmetries.”

The importance of
mathematical conceptualisation

by David Corfield,
Department of History and
Philosophy of Science,
University of Cambridge

See, too,
Symmetry of Walsh Functions and
Geometry of the I Ching.

Sunday, August 17, 2003

Sunday August 17, 2003

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

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

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

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

Further Reading:

Thursday, March 6, 2003

Thursday March 6, 2003

Filed under: General,Geometry — Tags: , — m759 @ 2:35 am

ART WARS:

Geometry for Jews

Today is Michelangelo's birthday.

Those who prefer the Sistine Chapel to the Rothko Chapel may invite their Jewish friends to answer the following essay question:

Discuss the geometry underlying the above picture.  How is this geometry related to the work of Jewish artist Sol LeWitt? How is it related to the work of Aryan artist Ernst Witt?  How is it related to the Griess "Monster" sporadic simple group whose elements number 

808 017 424 794 512 875 886 459 904 961 710 757 005 754 368 000 000 000?

Some background:

Friday, January 31, 2003

Friday January 31, 2003

Filed under: General — Tags: , , , — m759 @ 6:20 pm

Irish Fourplay

"…something I once heard Charles M. Schulz say, 'Don't worry about the world coming to an end today. It's already tomorrow in Australia.'"

 — William F. House

"Forewarned is four-armed."

— Folk saying

The painting at left is by Mary B. Kelly, a 1958 graduate of Saint Mary-of-the-Woods College.

Kelly is an expert on portrayals of Goddess figures in art

Today in Australia is February First, the feast of St. Bridget.  As several websites note, St. Bridget is a combination of Christian saint and Goddess figure… rather like St. Sara (patron saint of Gypsies, also known as Kali) or like Sara Pezzini in the classic TV series "Witchblade."

"Aww… Irish foreplay."

— Sara Pezzini in Witchblade, Episode 6

"Mighty in the gift of purity
She was pleasing unto the Bridegroom on high."

Song of St. Bridget

"Brace yourself, Bridget."

— Definition of Irish foreplay

Saint Bridget's Cross:

Four people can form this cross by joining hands as shown.  Of course, a Goddess like Kali (shown above) or Sara Pezzini could do it all by herself.

 

For futher details, see The Swastika Goddess,  the history of Jews and the Roman Catholic Church, and the history of Irish neutrality in World War II.

Postscript  of 11 PM

The Goddess Bridget in Literature

The Goddess Bridget (or Brigid) is incarnated in two classic works of American literature —

  • The American patriot and Communist Party supporter Dashiell Hammett gave an unflattering portrayal of Brigid (O'Shaughnessy) in The Maltese Falcon.  For a Jungian analysis of the relationship between Sam Spade and Brigid, see the perceptive remarks of Ryan Benedetti:

"In Jungian terms, Brigid becomes a projection of Spade's anima, a contrasexual replica of his own face as expressed in someone of the opposite sex.

Spade wears a variety of masks in his work. Masking allows him to get underneath the scam most clients lay on him. He is closer to the darker side of his unconscious than any of the other characters in the book, and he is so, because of his role as shamus. His function in his society is to expose all of the underlying darkness of the human psyche."

One way of looking at animus and anima is through the following archetypes:

A diamond and its dual "whirl" figure —
or a "jewel-box and its mate"

  • Mark Twain, in Life on the Mississippi, describes the way Goddess Bridget (again, O'Shaughnessy) arranged the conveyance of her late husband to the next world:

 "D'ye mane to soy that Bridget O'Shaughnessy bought the mate to that joo-ul box to ship that dhrunken divil to Purgatory in?"

"Yes, madam."

"Then Pat shall go to heaven in the twin to it, if it takes the last rap the O'Flaherties can raise!"

Thursday, January 9, 2003

Thursday January 9, 2003

Filed under: General — Tags: , , , , — m759 @ 4:48 pm

Balanchine's Birthday

Today seems an appropriate day to celebrate Apollo and the nine Muses.

From a website on Balanchine's and Stravinsky's ballet, "Apollon Musagete":

In his Poetics of Music (1942) Stravinsky says: "Summing up: What is important for the lucid ordering of the work– for its crystallization– is that all the Dionysian elements which set the imagination of the artist in motion and make the life-sap rise must be properly subjugated before they intoxicate us, and must finally be made to submit to the law: Apollo demands it."  Stravinsky conceived Apollo as a ballet blanc– a "white ballet" with classical choreography and monochromatic attire. Envisioning the work in his mind's eye, he found that "the absence of many-colored hues and of all superfluities produced a wonderful freshness." Upon first hearing Apollo, Diaghilev found it "music somehow not of this world, but from somewhere else above." The ballet closes with an Apotheosis in which Apollo leads the Muses towards Parnassus. Here, the gravely beautiful music with which the work began is truly recapitulated "on high"– ceaselessly recycled, frozen in time.

— Joseph Horowitz

 

Another website invoking Apollo:

The icon that I use… is the nine-fold square…. The nine-fold square has centre, periphery, axes and diagonals.  But all are present only in their bare essentials.  It is also a sequence of eight triads.  Four pass through the centre and four do not.  This is the garden of Apollo, the field of Reason…. 

In accordance with these remarks, here is the underlying structure for a ballet blanc:

A version of 'grid3x3.gif.'

This structure may seem too simple to support movements of interest, but consider the following (click to enlarge):

As Sir Arthur Quiller-Couch, paraphrasing Horace, remarks in his Whitsun, 1939, preface to the new edition of the Oxford Book of English Verse, "tamen usque recurret Apollo."

The alert reader will note that in the above diagrams, only eight of the positions move.

Which muse remains at the center?

Consider the remark of T. S. Eliot, "At the still point, there the dance is," and the fact that on the day Eliot turned 60, Olivia Newton-John was born.  How, indeed, in the words of another "sixty-year-old smiling public man," can we know the dancer from the dance?

Saturday, December 7, 2002

Saturday December 7, 2002

Filed under: General — m759 @ 9:30 pm

ART WARS:

Shall we read?

From Contact, by Carl Sagan:

  “You mean you could decode a picture hiding in pi
and it would be a mess of Hebrew letters?”
  “Sure.  Big black letters, carved in stone.”
  He looked at her quizzically.
  “Forgive me, Eleanor, but don’t you think
you’re being a mite too… indirect? 
You don’t belong to a silent order of Buddhist nuns. 
Why don’t you just tell your
story?”

From The Nation – Thailand
Sat Dec 7 19:36:00 EST 2002:

New Jataka books
blend ethics and art

Published on Dec 8, 2002

“The Ten Jataka, or 10 incarnations of the Lord Buddha before his enlightenment, are among the most fascinating religious stories….

His Majesty the King wrote a marvellous book on the second incarnation of the Lord Buddha…. It has become a classic, with the underlying aim of encouraging Thais to pursue the virtue of perseverance.

For her master’s degree at Chulalongkorn University’s Faculty of Arts, Her Royal Highness Princess Maha Chakri Sirindhorn wrote a dissertation related to the Ten Jataka of the Buddha. Now with the 4th Cycle Birthday of Princess Sirindhorn approaching on April 2, 2003, a group of artists, led by prominent painter Theeraphan Lorpaiboon, has produced a 10-volume set, the “Ten Jataka of Virtues”, as a gift to the Princess.

Once launched on December 25, the “Ten Jataka of Virtues” will rival any masterpiece produced in book form….”

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

Friday, December 6, 2002

Friday December 6, 2002

Filed under: General — m759 @ 1:06 pm

Great Simplicity

Frank Tall

Iaido

 

Daisetsu

 

 

Today

is the day that Daisetsu Suzuki attained satori,
according to the Zen Calendar.  “Daisetsu” is
said to mean “Great Simplicity.”

For those who prefer Harry Potter and
Diagon Alley, here is another calendar:


To Have and Have Not

Those who prefer traditional Western religions may like a site on the Trinity that contains this:

“Zen metaphysics is perhaps most succinctly set forth in the words ‘not-two.”  But even when he uses this expression, Suzuki is quick to assert that it implies no monism.  Not-two, it is claimed, is not the same as one.*  But when Suzuki discusses the relationship of Zen with Western mysticism, it is more difficult to escape the obvious monistic implications of his thinking.  Consider the following:

We are possessed of the habit of looking at Reality by dividing it into two… It is all due to the human habit of splitting one solid Reality into two, and the result is that my ‘have’ is no ‘have’ and my ‘have not’ is no ‘have not.’  While we are actually passing, we insist that the gap is impassable.**”

*See: Daisetz T. Suzuki, ‘Basic Thoughts Underlying  Eastern Ethical and Social Practice’ in Philosophy and Culture  East and West: East-West Philosophy in Practical Perspective, ed. Charles A. Moore (Honolulu: University of Hawaii Press, 1968), p. 429

** Daisetsu Teitaro Suzuki, Mysticism Christian and Buddhist (London: George Allen & Unwin, 1957, Unwin paperback, 1979), p. 57.


Personally, I am reminded by Suzuki’s satori on this date that today is the eve of the anniversary of Pearl Harbor.  I am also reminded by the rather intolerant tract on the Trinity quoted above that the first atomic bomb was exploded in the New Mexico desert at a test site named Trinity.  Of course, sometimes intolerance is justified.

Concluding unscientific postscript:

On the same day in 1896 that D. T. Suzuki attained satori,
lyricist Ira Gershwin was born.

Dies irae, dies illa.

Sunday, September 15, 2002

Sunday September 15, 2002

Filed under: General,Geometry — Tags: — 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.

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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