Log24

For example, Plato's diamond as an object to be transformed —

Versions of the transformed object —

See also The 4×4 Relativity Problem in this journal.

… Or:  A Funny Thing Happened
     on the Way to the Embedding

This journal on the morning of Saturday, Dec. 8, 2012:

Marilyn Monroe and her music coach in 1954,
from last night's online New York Times :

" 'We were very close to making love; I don’t remember
the stage we were at, but I would say half-dressed,'
Mr. Schaefer recalled. He added: 'And all of a sudden
for some reason, Marilyn got these vibrations, and
we went over to the window….' "  more »

"Mr. Schaefer died on Saturday at 87 at his home in
Fort Lauderdale, Fla. ….

He [had] coached Monroe through 'Diamonds Are
a Girl’s Best Friend,' her signature number in the
1953 movie 'Gentlemen Prefer Blondes' (he arranged
the music as well)…."

Perhaps on Saturday she returned the favor.

"Imbedding the God character in a holy book's very detailed narrative
and building an entire culture around this narrative
seems by itself to confer a kind of existence on Him."

— John Allen Paulos in the philosophy column "The Stone,"
     New York Times  online, Oct. 24, 2010

A related post from Log24 later that year—

"Next come the crown of thorns and Jesus' agonized crawl across the stage,
bearing the weight of his own crucifix. And at last, after making
yet another entrance, Mr. Nolan strikes the pose immortalized
in centuries of art, clad in a demure loincloth, arms held out to his sides,
one leg artfully bent in front of the other, head hanging down
in tortured exhaustion. Gently spotlighted, he rises from the stage
as if by magic, while a giant cross, pulsing with hot gold lights,
descends from above to meet him. Mr. Lloyd Webber's churning guitar rock
hits a climactic note, and the audience erupts in excited applause."

— Charles Isherwood, review of "Jesus Christ Superstar" in today's  New York Times

Other remarks on embedding —

Part I

Review of a new book on linguistics, embedding, and a South American tribe—

"Imagine a linguist from Mars lands on Earth to survey the planet's languages…."
— Chronicle of Higher Education , March 20, 2012

Part II

The Embedding , by Ian Watson (Review of a 1973 novel from Shakespeare's birthday, 2006)

The New York Times Magazine  this morning on a seminar on film theory at Columbia University—

"When the seminar reconvened after the break, Schamus said, 'Let’s dive into the Meno,' a dialogue in which Plato and Socrates consider virtue. 'The heart of it is the mathematical proof.' He rose from his seat and went to the whiteboard, where he drew figures and scribbled numbers as he worked through the geometry. 'You can only get the proof visually,' he concluded, stepping back and gazing at it. Plato may be skeptical about the category of the visual, he said, but 'you are confronted with a visual proof that gets you back to the idea embedded in visuality.'"

The Meno Embedding

See also Plato's Code and
 Plato Thanks the Academy.

Hoax and Hype 
Four Years Ago Today—

There is Plato's diamond—

and there is diamond theory—

… but there is no "Plato's Diamond Theory."

See, however, today's noon entry, "Plato's Code."

"You gotta be true to your code…" —Sinatra

“The present study is closely connected with a lecture* given by Prof. Ernst Cassirer at the Warburg Library whose subject was ‘The Idea of the Beautiful in Plato’s Dialogues’…. My investigation traces the historical destiny of the same concept….”

* See Cassirer’s Eidos und Eidolon : Das Problem des Schönen und der Kunst in Platons Dialogen, in Vorträge der Bibliothek Warburg II, 1922/23 (pp. 1–27). Berlin and Leipzig, B.G. Teubner, 1924.

— Erwin Panofsky, Idea: A Concept in Art Theory, foreword to the first German edition, Hamburg, March 1924

On a figure from Plato’s Meno—

The above figures illustrate Husserl’s phrase  “eidetic variation”—
a phrase based on Plato’s use of eidos, a word
closely related to the word “idea” in Panofsky’s title.

For remarks by Cassirer on the theory of groups, a part of
mathematics underlying the above diamond variations, see
his “The Concept of Group and the Theory of Perception.”

Sketch of some further remarks—

The Waterfield question in the sketch above
is from his edition of Plato’s Theaetetus
(Penguin Classics, 1987).

The “design theory” referred to in the sketch
is that of graphic  design, which includes the design
of commercial logos. The Greek  word logos
has more to do with mathematics and theology.

“If there is one thread of warning that runs
through this dialogue, from beginning to end,
it is that verbal formulations as such are
shot through with ambiguity.”

— Rosemary Desjardins, The Rational Enterprise:
Logos in Plato’s Theaetetus, SUNY Press, 1990

Related material—

(Click to enlarge.)

A supplement to yesterday's post on variation of an eidos—

Enlarge.

Frame Tales

From June 30 —

("Will this be on the test?")

Frame Tale One:

Frame Tale Two:

Barry Sharples
on his version of the
  Kaleidoscope Puzzle —

Background:

"A possible origin of this puzzle is found in a dialogue
 between Socrates and Meno written by the Greek philosopher,
 Plato, where a square is drawn inside a square such that
the blue square is twice the area  of the yellow square.

Colouring the triangles produces a starting pattern
which is a one-diamond figure made up of four tiles
and there are 24 different possible arrangements."

The King and the Corpse  —

"The king asked, in compensation for his toils during this strangest
of all the nights he had ever known, that the twenty-four riddle tales
told him by the specter, together with the story of the night itself,
should be made known over the whole earth
and remain eternally famous among men."

Frame Tale Three:

Finnegans Wake —

"The quad gospellers may own the targum
but any of the Zingari shoolerim may pick a peck
of kindlings yet from the sack of auld hensyne."

“… he triumphed again with The Goldberg Variations. Mr Jay, assisted by Goldberg, a concentration camp survivor, is rehearsing a montage of biblical scenes in Jerusalem. It is inspired satire, laced with Jewish and Christian polemics, sparkling wit and dazzlingly simple effects. For Golgotha a stagehand brings on three crosses. ‘Just two,’ says Jay. ‘The boy is bringing his own.’ Tabori often claimed that the joke was the most perfect literary form.”

George Tabori

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

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

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

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

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

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

From Year of Jewish Culture:

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

From the related page Programme (October-December):

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

Variations on
Birth and Death

From Log24 on the date of the Prague production of the Tabori “Goldberg Variations,” an illustration in honor of Sir Thomas Browne, who was born, and died, on that date: The above is from Variable Resolution 4–k Meshes: Concepts and Applications (pdf), by Luiz Velho and Jonas Gomes. See also Symmetry Framed and The Garden of Cyrus.

From Log24 on the date of Tabori’s death: Theme (Plato, Meno) and Variations: Click on “variations” above for some material on the “Goldberg Variations” of Johann Sebastian Bach.

In 1946, Robert Graves published “King Jesus, an historical novel based on the theory and Graves’ own historical conjecture that Jesus was, in fact, the rightful heir to the Israelite throne… written while he was researching and developing his ideas for The White Goddess.”

In 1948, C. S. Lewis finished the first draft of The Lion, The Witch, and The Wardrobe, a novel in which one of the main characters is “the White Witch.”

In 1948, Robert Graves published The White Goddess.

In 1949, Robert Graves published Seven Days in New Crete [also titled Watch the North Wind Rise], “a novel about a social distopia in which Goddess worship is (once again?) the dominant religion.”

Lewis died on November 22, 1963, the day John F. Kennedy was killed.

Related material:
Log24, December 10, 2005

Graves died on December 7 (Pearl Harbor Day), 1985.

Related material:
Log24, December 7, 2005, and
Log24, December 11, 2005

Jesus died, some say, on April 7 in the year 30 A.D.

Related material:

Art Wars, April 7, 2003:
Geometry and Conceptual Art,

Eight is a Gate, and

Mathematics and Narrative.

Plato’s Diamond

— Motto of
Plato’s Academy

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.

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.

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.