Sunday, October 18, 2009 11:00 AM
ART WARS for St. Luke's Day:
Friday, October 16, 2009 11:30 AM
Noncontinuous Groups:A page with this title has been added to my finite-geometry site.
Wednesday, October 14, 2009 9:29 AM
Tuesday, October 13, 2009 7:00 AM
Man and His Symbols:
Monday, October 12, 2009 7:00 AM
Annals of Aesthetics:
Sunday, October 11, 2009 7:00 PM
Saturday, October 10, 2009 7:00 AM
ART WARS review:
Friday, October 9, 2009 9:00 AM
Thursday, October 8, 2009 10:30 AM
Aesthetics continued...Knight Moves
Thursday, October 8, 2009 9:00 AM
Wednesday, October 7, 2009 7:59 AM
A Passage:Finucane's Wake
Tuesday, October 6, 2009 11:07 AM
Annals of Aesthetics, continued--
Monday, October 5, 2009 4:00 AM
Annals of Aesthetics:
Saturday, October 3, 2009 3:31 AM
Annals of Aesthetics:
Friday, October 2, 2009 6:00 AM
Some Finite Geometry:
|Edge on Heptads|
Part I: Dye on Edge
....we obtain various orbits of partitions of quadrics over GF(2a) by their maximal totally singular subspaces; the corresponding stabilizers in the relevant orthogonal groups are investigated. It is explained how some of these partitions naturally generalize Conwell's heptagons for the Klein quadric in PG(5,2)."
In 1910 Conwell... produced his heptagons in PG(5,2) associated with the Klein quadric K whose points represent the lines of PG(3,2).... Edge... constructed the 8 heptads of complexes in PG(3,2) directly. Both he and Conwell used their 8 objects to establish geometrically the isomorphisms SL(4,2)=A8 and O6(2)=S8 where O6(2) is the group of K...."
-- "Partitions and Their Stabilizers for Line Complexes and Quadrics," by R.H. Dye, Annali di Matematica Pura ed Applicata, Volume 114, Number 1, December 1977, pp. 173-194
Part II: Edge on Heptads
"The Geometry of the Linear Fractional Group LF(4,2)," by W.L. Edge, Proc. London Math Soc., Volume s3-4, No. 1, 1954, pp. 317-342. See the historical remarks on the first page.
Note added by Edge in proof:
"Since this paper was finished I have found one by G. M. Conwell: Annals of Mathematics (2) 11 (1910), 60-76...."
Wednesday, September 30, 2009 9:48 AM
Art and Religion:
I know more than Apollo,
For oft when he lies sleeping
I see the stars at mortal wars
In the wounded welkin weeping.
Wednesday, September 30, 2009 2:02 AM
Synchronistic Recognition:Not So Second-Rate
Another discovery reportedly also made last Thursday, Sept. 24:
Picture from Fox News:
Tuesday, September 29, 2009 3:00 AM
Annals of Religion:From Harper's Magazine
Monday, September 28, 2009 3:00 AM
Annals of Aesthetics, continued:
Sunday, September 27, 2009 3:00 AM
"In sixteenth-century Transylvania, Unitarian congregations were established for the first time in history."Gravity's Rainbow--
"For every kind of vampire, there is a kind of cross."Unitarian minister Richard Trudeau--
"... I called the belief thatH. S. M. Coxeter, 1987, introduction to Trudeau's book--
(1) Diamonds-- informative, certain truths about the world-- exist
the 'Diamond Theory' of truth. I said that for 2200 years the strongest evidence for the Diamond Theory was the widespread perception that
(2) The theorems of Euclidean geometry are diamonds....
As the news about non-Euclidean geometry spread-- first among mathematicians, then among scientists and philosophers-- the Diamond Theory began a long decline that continues today.
Factors outside mathematics have contributed to this decline. Euclidean geometry had never been the Diamond Theory's only ally. In the eighteenth century other fields had seemed to possess diamonds, too; when many of these turned out to be man-made, the Diamond Theory was undercut. And unlike earlier periods in history, when intellectual shocks came only occasionally, received truths have, since the eighteenth century, been found wanting at a dizzying rate, creating an impression that perhaps no knowledge is stable.
Other factors notwithstanding, non-Euclidean geometry remains, I think, for those who have heard of it, the single most powerful argument against the Diamond Theory*-- first, because it overthrows what had always been the strongest argument in favor of the Diamond Theory, the objective truth of Euclidean geometry; and second, because it does so not by showing Euclidean geometry to be false, but by showing it to be merely uncertain." --The Non-Euclidean Revolution, p. 255
As noted here on Oct. 8, 2008 (A Yom Kippur Meditation), Coxeter was aware in 1987 of a more technical use of the phrase "diamond theory" that is closely related to..."There is a pleasantly discursive treatment of Pontius Pilate's unanswered question 'What is truth?'."
Saturday, September 26, 2009 8:28 AM
ART WARS continued--
Friday, September 25, 2009 3:09 AM
Thursday, September 24, 2009 2:01 AM
Aesthetics for King, continued:Who Knows
"All the thoughts you never seeThe song title is from a book, The Shadow of the Object (Columbia U. Press, 1987), by psychoanalyst Christopher Bollas.
You are always thinking
Brain is wide, the brain is deep
Oh, are you sinking?"
Wednesday, September 23, 2009 9:26 AM
Aesthetics for King, continued:Backspacing
See PearlJamEvolution.com, Aug. 3, 2009, and Aug. 6, 2009.
Monday, September 21, 2009 2:56 AM
Aesthetics for King:
Sunday, September 20, 2009 9:00 AM
Sunday, September 20, 2009 12:00 AM
Saturday, September 19, 2009 4:23 PM
Block that Metaphor, continued:
Old Year, Raus!
Also in today's New York Times obituaries index:
John T. Elson, Editor Who Asked
"Is God Dead?" at Time, Dies at 78
Wikipedia article on George Polya:
Saturday, September 19, 2009 2:22 AM
A New Year's Prayer...
Jerusalem Post Interview
by Hilary Leilea Krieger, JPost Correspondent, Washington
Can you talk a little bit about your own Jewish upbringing and sense of Jewishness, and how that influences you? I assume it's a factor in this particular project.
I grew up in a Modern Orthodox home [in Montreal]. I went to Jewish day school right through high school, so half of my day was spent speaking Hebrew from age six to 16. I studied thousands of hours of Talmud. My father thought I didn't get enough Talmud at school, so I took the extra Talmud class at school and he had a rabbi come to the house three nights a week. One of those nights was Saturday night, so in synagogue Saturday morning my brother and I would pray very hard for snow so he wouldn't be able to come on Saturday night and we could watch hockey night in Canada. That's where I learned about prayer.
That didn't seem to you to be a prayer that was likely to go unanswered?
Yeah, I was giving it a shot to see what side God was on.
And what did you determine?
It rarely snowed.************************************
More on Krauthammer's Canadian childhood:
"His parents were Orthodox and sent him to
Also in the Jerusalem Post interview:
.... What, then, did you mean by a Jewish sensibility?
".... In literature it's an interesting question, what's a Jewish novel?"
Private Gomorrah lessons
"Heaven Can Wait"
Happy Rosh Hashanah
Update, 5:01 AM Sept. 19
Before becoming a writer,
Krauthammer was, his
Washington Post biography says,
a resident and then chief resident
in psychiatry at
Massachusetts General Hospital.
This morning's New York Times:
MicheleBachmann.com this morning:
James Hillman's "acorn theory"
of personality development
Friday, September 18, 2009 2:22 PM
Annals of Aesthetics
Thursday, September 17, 2009 8:00 PM
Instant Review Department:
"Watch it, forget it, move on."A perhaps more enduring tribute:
Thursday, September 17, 2009 11:07 AM
Words and Music:
Thursday, September 17, 2009 2:45 AM
On Wings of Song:
Wednesday, September 16, 2009 11:07 AM
For Dan Brown, continued:
Tuesday, September 15, 2009 2:02 PM
"...one has to be willing
to tolerate ambiguity,
even to be crazy."
"The party line...."
Tuesday, September 15, 2009 12:00 AM
For Dan Brown:
Monday, September 14, 2009 3:09 PM
Back-to-School Special, continued:
Friday, September 11, 2009 1:00 PM
Friday, September 11, 2009 2:56 AM
Annals of Aesthetics:
Cf. August 5, 2009.
Tuesday, September 8, 2009 12:25 PM
"Your mind has been conditioned to Euclid," Holloway said. "So this-- thing-- bores us, and seems pointless. But a child knows nothing of Euclid. A different sort of geometry from ours wouldn't impress him as being illogical. He believes what he sees."
"Are you trying to tell me that this gadget's got a fourth dimensional extension?" Paradine demanded.
"Not visually, anyway," Holloway denied. "All I say is that our minds, conditioned to Euclid, can see nothing in this but an illogical tangle of wires. But a child-- especially a baby-- might see more. Not at first. It'd be a puzzle, of course. Only a child wouldn't be handicapped by too many preconceived ideas."
"Hardening of the thought-arteries," Jane interjected.
Paradine was not convinced. "Then a baby could work calculus better than Einstein? No, I don't mean that. I can see your point, more or less clearly. Only--"
"Well, look. Let's suppose there are two kinds of geometry-- we'll limit it, for the sake of the example. Our kind, Euclidean, and another, which we'll call x. X hasn't much relationship to Euclid. It's based on different theorems. Two and two needn't equal four in it; they could equal y, or they might not even equal. A baby's mind is not yet conditioned, except by certain questionable factors of heredity and environment. Start the infant on Euclid--"
"Poor kid," Jane said.
Holloway shot her a quick glance. "The basis of Euclid. Alphabet blocks. Math, geometry, algebra-- they come much later. We're familiar with that development. On the other hand, start the baby with the basic principles of our x logic--"
"Blocks? What kind?"
Holloway looked at the abacus. "It wouldn't make much sense to us. But we've been conditioned to Euclid."
-- "Mimsy Were the Borogoves," Lewis Padgett, 1943
Monday, September 7, 2009 12:00 AM
Midnight in the Garden, continued:
"... he lifted a square, transparent crystal block, small enough to cup in his palm-- much too small to contain the maze of apparatus within it. In a moment Scott had solved that problem. The crystal was a sort of magnifying glass, vastly enlarging the things inside the block. Strange things they were, too. Miniature people, for example-- They moved. Like clockwork automatons, though much more smoothly. It was rather like watching a play."Part III: A Crystal Block --
Sunday, September 6, 2009 11:07 AM
Sunday, September 6, 2009 11:00 AM
Saturday, September 5, 2009 10:31 PM
Annals of Aesthetics:
Friday, September 4, 2009 2:02 PM
April is Math Awareness Month.
This year's theme is "mathematics and art."
Cf. both of yesterday's entries.
Thursday, September 3, 2009 2:02 PM
Painting the Mystical:
Thursday, September 3, 2009 11:07 AM
Annals of Aesthetics:
"Music and mathematics are among the pre-eminent wonders of the race. Levi-Strauss sees in the invention of melody 'a key to the supreme mystery' of man-- a clue, could we but follow it, to the singular structure and genius of the species. The power of mathematics to devise actions for reasons as subtle, witty, manifold as any offered by sensory experience and to move forward in an endless unfolding of self-creating life is one of the strange, deep marks man leaves on the world. Chess, on the other hand, is a game in which thirty-two bits of ivory, horn, wood, metal, or (in stalags) sawdust stuck together with shoe polish, are pushed around on sixty-four alternately coloured squares. To the addict, such a description is blasphemy. The origins of chess are shrouded in mists of controversy, but unquestionably this very ancient, trivial pastime has seemed to many exceptionally intelligent human beings of many races and centuries to constitute a reality, a focus for the emotions, as substantial as, often more substantial than, reality itself. Cards can come to mean the same absolute. But their magnetism is impure. A mania for whist or poker hooks into the obvious, universal magic of money. The financial element in chess, where it exists at all, has always been small or accidental.
To a true chess player, the pushing about of thirty-two counters on 8x8 squares is an end in itself, a whole world next to which that of a mere biological or political or social life seems messy, stale, and contingent. Even the patzer, the wretched amateur who charges out with his knight pawn when the opponent’s bishop decamps to R4, feels this daemonic spell. There are siren moments when quite normal creatures otherwise engaged, men such as Lenin and myself, feel like giving up everything-- marriage, mortgages, careers, the Russian Revolution-- in order to spend their days and nights moving little carved objects up and down a quadrate board. At the sight of a set, even the tawdriest of plastic pocket sets, one’s fingers arch and a coldness as in a light sleep steals over one’s spine. Not for gain, not for knowledge or reknown, but in some autistic enchantment, pure as one of Bach’s inverted canons or Euler’s formula for polyhedra."
-- George Steiner in "A Death of Kings," The New Yorker, issue dated September 7, 1968, page 133"Examples are the stained-glass windows of knowledge." --Nabokov
Wednesday, September 2, 2009 9:48 PM
Wednesday, September 2, 2009 6:00 PM
Chronicles of Boston:
Wednesday, September 2, 2009 11:09 AM
Soul of the Party, continued:
Friday, August 28, 2009,
in this journal
Rites of Passage
"Things fall apart;
the centre cannot hold...."
"Inside the church,
the grief was real...."
|Sunday, August 30, 2009,|
in The New York Times
"Things fall apart; the center cannot hold. Facebook, the online social grid, could not command loyalty forever."
-- Virginia Heffernan, "Facebook Exodus," NY Times Magazine, Sunday, August 30, 2009
Tuesday, September 1, 2009 11:30 PM
Soul of the Party, continued:Back to the Garden