Log24

"[The applet] Syntheme illustrates ways of partitioning the 12 vertices of an icosahedron into 3 sets of 4, so that each set forms the corners of a rectangle in the Golden Ratio. Each such rectangle is known as a duad. The short sides of a duad are opposite edges of the icosahedron, and there are 30 edges, so there are 15 duads.

Each partition of the vertices into duads is known as a syntheme. There are 15 synthemes; 5 consist of duads that are mutually perpendicular, while the other 10 consist of duads that share a common line of intersection."

— Greg Egan, Syntheme

The above note shows
duads and synthemes related
to the diamond theorem.

See also John Baez's essay
"Some Thoughts on the Number 6."
That essay was written 15 years
ago today– which happens
to be the birthday of
Sir Laurence Olivier, who,
were he alive today, would
be 100 years old.

"Is it safe?"

Venus at
St. Anne's,
continued

In honor of
the film "Bobby,"
now playing.

("Venus at St. Anne's"
is the title of the final
chapter of
the C. S. Lewis classic
That Hideous Strength.)

Symbol of Venus
and
Symbol of Plato

Related symbols:

Click on pictures
for details related tp
the Feast of St. Anne
(July 26).

"The best theology today,
in its repudiation of a
rhetorical religious idealism,
finds itself in agreement
with a recurrent note
in contemporary poetry….

We keep coming back
and coming back/
To the real: to the hotel
instead of the hymns/
That fall upon it
out of the wind.  We seek/
… Nothing beyond reality.
Within it/
Everything,
the spirit’s alchemicana….

(From 'An Ordinary Evening
in New Haven,'
in The Collected Poems
of Wallace Stevens….)

… Not grim/
Reality, but reality grimly seen….

(Ibid.)"

— "The Church's
New Concern with the Arts,"
by Amos N. Wilder,
Hollis Professor
of Divinity, Emeritus,
at Harvard Divinity School,
in Christianity and Crisis,
February 18, 1957.

 

 

"All the truth in the world
adds up to one big lie."

— Dylan, "Things Have Changed"
 

Windmill and Diamond

From “Today in History,”
by The Associated Press:

On this date:

In 1965, the musical
“Man of La Mancha”
opened in New York.

In 1975, Juan Carlos
was proclaimed
King of Spain.

Today’s birthdays:

… Movie director
Arthur Hiller is 83….

Hiller directed the 1972 film
of “Man of La Mancha.”

A quotation from that film:

“When life itself seems lunatic,
who knows where madness lies?”

One can approach these symbols in either a mathematical or a literary fashion. For a mathematical discussion of the symbols’ structure, see Theme and Variations. Those who prefer literary discussions may make up their own stories.

Venus at St. Anne’s
(Title of the closing chapter
of That Hideous Strength)

Symbol of Venus
and
Symbol of Plato

“What do they
teach them
   at these schools?”
— C. S. Lewis

Today is the
feast of St. Anne.

From Wikipedia, an unsigned story:

“In 1923 Alfred Teitelbaum and his brother Wacław changed their surnames to Tarski, a name they invented because it sounded very Polish, was simple to spell and pronounce, and was unused. (Years later, he met another Alfred Tarski in northern California.) The Tarski brothers also converted to Roman Catholicism, the national religion of the Poles. Alfred did so, even though he was an avowed atheist, because he was about to finish his Ph.D. and correctly anticipated that it would be difficult for a Jew to obtain a serious position in the new Polish university system.”

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

The Quality of Diamond

On February 3, 2004, archivist and abstract painter Ward Jackson died at 75.  From today’s New York Times:

“Inspired by painters like Piet Mondrian and Josef Albers, Mr. Jackson made austere, hard-edged geometric compositions, typically on diamond-shaped canvases.”

On February 3, the day that Jackson died, there were five different log24.net entries:

1. The Quality with No Name 
2. Speaking Globally
3. Lila
4. Theory of Design
5. Retiring Faculty.

Parallels with the Helguera exhibit:

Florence Foster Jenkins: Janet Jackson in (2) above.

Giulio Camillo: Myself as compiler of the synchronistic excerpts in (5).

Friedrich Froebel: David Wade in (4).

The last Shakers: Christopher Alexander and his acolytes in (1).

Ward Jackson: On Feb. 3, Jackson became a permanent part of Quality — i.e., Reality — itself, as described in (3).

Some thoughts of Hans-Georg Gadamer
relevant to Jackson’s death:

As noted in entry (3) above
on the day that Jackson died,

“All the world’s a stage.”

— William Shakespeare

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.

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.

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.

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.

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.