Log24

Saturday, September 2, 2017

Knight Moves

Filed under: General — m759 @ 12:42 PM

Ursula K. Le Guin, in Amazing Stories , Sept. 1992, published
"The Rock That Changed Things" (pp. 9-13) and her story from
thirty years earlier, "April in Paris" (Fantastic Stories , Sept. 1962.)
The latter (pp. 14-19) was followed by some brief remarks (p. 19)
comparing the two stories.

For "The Rock," see Le Guin + Rock in this journal.

"April in Paris" is about time travel by means of an alchemist's
pentagram. The following figure from 1962 is in lieu of a pentagram —

'Loop De Loop,' Johnny Thunder, Diamond Records, 1962

See as well a search for 1962 in this journal.

Sunday, August 10, 2014

Knight Moves

Filed under: General,Geometry — m759 @ 7:00 PM

Some illustrations:

Springer logo - A chess knight

Chess Knight
(in German, Springer)

See also…

Katherine Neville's 'The Eight,' edition with knight on cover, on her April 4 birthday

More technically (click image for details):

Friday, November 25, 2011

Knight’s Move* for Wicker

Filed under: General — m759 @ 10:30 PM

Hamlet Fire

The "knight's move" of the title is the supplying of the above link.
For details, click on the link (a search on the link's two words).

* For the meaning of "knight's move," see To Make a Short Story Long.

For the meaning of the phrase  (as opposed to the search ),
    see the birthplace of Tom Wicker, who died today.

Tuesday, June 14, 2016

Model Kit

Filed under: General,Geometry — m759 @ 12:14 PM

The title refers to the previous post, which quotes a 
remark by a poetry critic in the current New Yorker .

Scholia —

From the post Structure and Sense of June 6, 2016 —

Structure

Sense

A set of 7 partitions of the 2x2x2 cube that is invariant under PSL(2, 7) acting on the 'knight' coordinatization

From the post Design Cube of July 23, 2015 —

Broken Symmetries  in  Diamond Space 

Monday, June 6, 2016

Structure and Sense

Filed under: General,Geometry — Tags: — m759 @ 2:01 PM

"… the war of 70-some years ago
has already become something like the Trojan War
had been for the Homeric bards:
a major event in the mythic past
that gives structure and sense to our present reality."

— Justin E. H. Smith, a professor of philosophy at
     the University of Paris 7–Denis Diderot,
     in the New York Times  column "The Stone"
     (print edition published Sunday, June 5, 2016)

In memory of a British playwright who reportedly
died at 90 this morning —

Structure

Sense

A set of 7 partitions of the 2x2x2 cube that is invariant under PSL(2, 7) acting on the 'knight' coordinatization

Sunday, June 5, 2016

Sunday School: Seven Seals

Filed under: General,Geometry — m759 @ 7:00 AM

A set of 7 partitions of the 2x2x2 cube that is invariant under PSL(2, 7) acting on the 'knight' coordinatization

Click image for some background.

See also Standard Disclaimer.

Sunday, December 2, 2012

Realism

Filed under: General — Tags: — m759 @ 4:30 PM

From the Los Angeles Times  yesterday

"Chess player Elena Akhmilovskaya Donaldson sits
in deep concentration at the U.S. chess championship
in Seattle in 2002. (Greg Gilbert / Seattle Times / 
January 5, 2002)"

Linda Shaw, Seattle Times :

"Elena Akhmilovskaya Donaldson, who was once the world's
second-ranked women's chess player and eloped in 1988
with the captain of the U.S. chess team when they were both
playing at a tournament in Greece, has died. She was 55.

Donaldson, who earned the title of international women's
grandmaster, died Nov. 18 in her adopted hometown of Seattle…."

more »

From the Log24 post "Sermon" on the date of Donaldson's death,
Sunday, Nov. 18, 2012—

"You must allow us to play every conceivable combination of chess."
— Marie-Louise von Franz in Number and Time

An October 2011 post titled  Realism in Plato's Cave displays
the following image:

Cover illustration: Durer's 'Knight, Death, and the Devil'

Cover illustration: Knight, Death, and the Devil,
by Albrecht Dürer

George Steiner and myself  in Closing the Circle, a Log24 post
of Sept. 4, 2009: 

“Allegoric associations of death with chess are perennial….”

"Yes, they are."

For related remarks on knight moves and the devil, see
today's previous two posts, Knight's Labyrinth and The Rite.

Monday, April 23, 2012

Mate in Six

Filed under: General — m759 @ 6:29 PM

In memory of Mike Wallace

  1. An April 8 post noting the death of Wallace—
    a NY Times  obituary notice with ad at top— "The North Face"
     
  2. An April 21 post noting the death of Charles Colson,
    reportedly at at 3:12 PM on Saturday, April 21, 2012,
    with ad at top— "Discover New Horizons" 
     
  3. An April 21 post linking to a 3/12 post (this date being
    suggested by the reported time of Colson's death)
    that has a review of the film "The Ninth Configuration"
     
  4. An April 22 post with six lines that some might
    interpret as meeting on a horizon, two lines that might
    be interpreted as meeting at a "depth horizon," and
    a ninth line that emerges from the other eight
     
  5. An April 23 post that combines a passage from
    "The Ninth Configuration" with a detail from
    the North Face ad that appeared above Wallace's
    NY TImes  obituary on April 8 and again above
    Colson's obituary on April 23
     
  6. "The game's…"

See also Knight Moves.

Thursday, December 1, 2011

Dog Tale

Filed under: General — m759 @ 8:20 PM

"I love gazing into things. Can you imagine with me how glorious it is, for example, to see into a dog, in passing— into him… to ease oneself into the dog exactly at his center, the place out of which he exists as a dog, that place in him where God would, so to speak, have sat down for a moment when the dog was complete, in order to watch him at his first predicaments and notions and let him know with a nod that he was good, that he lacked nothing, that no better dog could be made. For a while one can endure being in the middle of the dog, but one has to be sure to jump out in time, before the world closes in around him completely, otherwise one would remain the dog within the dog and be lost to everything else."

— Rainer Maria Rilke, quoted in The New York Times  in 1988

http://www.log24.com/log/pix11C/111201-DogTale.jpg

Omitting unneeded narrative details,
a madman's knight move

A novel search in memory of the late
uncrowned crown prince of Albania—

Nabokov, Pale Fire

Related narratives—

Prose Tale and The Meadow.

Oh, and happy birthday to Woody Allen (76 today)—

"Outside of a dog, a book is man's best friend.
Inside of a dog it's too dark to read." —Groucho Marx

Wednesday, October 26, 2011

Erlanger and Galois

Filed under: General,Geometry — m759 @ 8:00 PM

Peter J. Cameron yesterday on Galois—

"He was killed in a duel at the age of 20…. His work languished for another 14 years until Liouville published it in his Journal; soon it was recognised as the foundation stone of modern algebra, a position it has never lost."

Here Cameron is discussing Galois theory, a part of algebra. Galois is known also as the founder* of group theory, a more general subject.

Group theory is an essential part of modern geometry as well as of modern algebra—

"In der Galois'schen Theorie, wie hier, concentrirt sich das Interesse auf Gruppen von Änderungen. Die Objecte, auf welche sich die Änderungen beziehen, sind allerdings verschieden; man hat es dort mit einer endlichen Zahl discreter Elemente, hier mit der unendlichen Zahl von Elementen einer stetigen Mannigfaltigkeit zu thun."

— Felix Christian Klein, Erlanger Programm , 1872

("In the Galois theory, as in ours, the interest centres on groups of transformations. The objects to which the transformations are applied are indeed different; there we have to do with a finite number of discrete elements, here with the infinite number of elements in a continuous manifoldness." (Translated by M.W. Haskell, published in Bull. New York Math. Soc. 2, (1892-1893), 215-249))

Related material from Hermann Weyl, Symmetry , Princeton University Press, 1952 (paperback reprint of 1982, pp. 143-144)—

"A field is perhaps the simplest algebraic structure we can invent. Its elements are numbers…. Space is another example of an entity endowed with a structure. Here the elements are points…. What we learn from our whole discussion and what has indeed become a guiding principle in modern mathematics is this lesson: Whenever you have to do with a structure-endowed entity  Σ try to determine is group of automorphisms , the group of those element-wise transformations which leave all structural relations undisturbed. You can expect to gain a deep insight into the constitution of Σ in this way."

For a simple example of a group acting on a field (of 8 elements) that is also a space (of 8 points), see Generating the Octad Generator and Knight Moves.

* Joseph J. Rotman, An Introduction to the Theory of Groups , 4th ed., Springer, 1994, page 2

Thursday, October 8, 2009

Thursday October 8, 2009

Filed under: General — m759 @ 10:30 AM

Knight Moves

Deborah Solomon, New York Times Magazine, Sunday, June 27, 1999:

“While modern art began as an assault on the academy, post-modern art might be described as a return to the academy. Instead of the old academy of rules, now we have the Academy of Cool, schools that treat avant-garde rebellion as a learned occupation.”

Christopher Knight, LA Times art critic, on Solomon:

“Back in the day, Solomon interviewed Knight for a Times Magazine story on Los Angeles art schools. ‘Having been a journalist (at that time) for almost two decades, I also did my homework,’ Knight writes [in a letter to the New York Press]. ‘I prepared a couple of quotable quotes on the subject, which might encapsulate larger ideas.’ One of Knight’s pearls of wisdom, ‘Modern art began as an assault on the academy, but post-modern art might be described as a return to the academy,’ excited Solomon so much that, according to Knight, she printed it as her own observation in her final piece, which bore no mention of the Knight interview. In the final story, a seriously bitter Knight writes, ‘It was not a quote; my words had become her words.'” —Gawker, Oct. 11, 2007

A reference to Solomon’s piece appeared in this journal in 2003.

See also yesterday’s entry, today’s 9 AM entry, and (for the Academy) an example of knight’s move thinking.

Friday, April 3, 2009

Friday April 3, 2009

Filed under: General,Geometry — m759 @ 5:24 AM
Knight Moves

“Lord, I remember”
Bob Seger 


“Philosophers ponder the idea of identity: what it is to give something a name on Monday and have it respond to that name on Friday….”

Bernard Holland in The New York Times of Monday, May 20, 1996

Yesterday’s afternoon entry cited philosopher John Holbo on chess. This, together with Holland’s remark above and Monday’s entries on Zizek, suggests…

Holbo on Zizek
(pdf, 11 pages)

In this excellent analysis,
Holbo quotes Kierkegaard:

“… the knight of faith
‘has the pain of being unable to
make himself intelligible to others'”

(Kierkegaard, Fear and Trembling)

For some material that may serve to illustrate Kierkegaard’s remark, see Log24 on Twelfth Night and Epiphany this year.

“… There was a problem laid out on the board, a six-mover. I couldn’t solve it, like a lot of my problems. I reached down and moved a knight…. I looked down at the chessboard. The move with the knight was wrong. I put it back where I had moved it from. Knights had no meaning in this game. It wasn’t a game for knights.”


— Raymond Chandler, The Big Sleep

Perhaps a game for bishops?

Henry Edward Cardinal Manning

Cardinal Manning

Click on the cardinal
for a link to some remarks
related to the upcoming film
 “Angels & Demons” and to
a Paris “Sein Feld.”


Context: the five entries
ending at 9:26 AM
on March 10, 2009…
and, for Kierkegaard,
Diamonds Are Forever.

Monday, July 21, 2008

Monday July 21, 2008

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

Knight Moves:

The Relativity Theory
of Kindergarten Blocks


(Continued from
January 16, 2008)

“Hmm, next paper… maybe
‘An Unusually Complicated
Theory of Something.'”

Garrett Lisi at
Physics Forums, July 16

Something:

From Friedrich Froebel,
who invented kindergarten:

Froebel's Third Gift: A cube made up of eight subcubes

Click on image for details.

An Unusually
Complicated Theory:


From Christmas 2005:

The Eightfold Cube: The Beauty of Klein's Simple Group

Click on image for details.

For the eightfold cube
as it relates to Klein’s
simple group, see
A Reflection Group
of Order 168
.”

For an even more
complicated theory of
Klein’s simple group, see

Cover of 'The Eightfold Way: The Beauty of Klein's Quartic Curve'

Click on image for details.

Sunday, June 1, 2008

Sunday June 1, 2008

Filed under: General,Geometry — Tags: — m759 @ 2:14 PM
Yet Another
Cartoon Graveyard

The conclusion of yesterday’s commentary on the May 30-31 Pennsylvania Lottery numbers:

Thomas Pynchon, Gravity’s Rainbow:

“The fear balloons again inside his brain. It will not be kept down with a simple Fuck You…. A smell, a forbidden room, at the bottom edge of his memory. He can’t see it, can’t make it out. Doesn’t want to. It is allied with the Worst Thing.

He knows what the smell has to be: though according to these papers it would have been too early for it, though he has never come across any of the stuff among the daytime coordinates of his life, still, down here, back here in the warm dark, among early shapes where the clocks and calendars don’t mean too much, he knows that’s what haunting him now will prove to be the smell of Imipolex G.

Then there’s this recent dream he is afraid of having again. He was in his old room, back home. A summer afternoon of lilacs and bees and

286”

What are we to make of this enigmatic 286? (No fair peeking at page 287.)

One possible meaning, given The Archivists claim that “existence is infinitely cross-referenced”–

Page 286 of Ernest G. Schachtel, Metamorphosis: On the Conflict of Human Development and the Psychology of Creativity (first published in 1959), Hillsdale NJ and London, The Analytic Press, 2001 (chapter– “On Memory and Childhood Amnesia”):

“Both Freud and Proust speak of the autobiographical [my italics] memory, and it is only with regard to this memory that the striking phenomenon of childhood amnesia and the less obvious difficulty of recovering any past experience may be observed.”

The concluding “summer afternoon of lilacs and bees” suggests that 286 may also be a chance allusion to the golden afternoon of Disney’s Alice in Wonderland. (Cf. St. Sarah’s Day, 2008)

Some may find the Disney afternoon charming; others may see it as yet another of Paul Simon’s dreaded cartoon graveyards.

More tastefully, there is poem 286 in the 1919 Oxford Book of English Verse– “Love.”

For a midrash on this poem, see Simone Weil, who became acquainted with the poem by chance:

“I always prefer saying chance rather than Providence.”

— Simone Weil, letter of about May 15, 1942

Weil’s brother André might prefer Providence (source of the Bulletin of the American Mathematical Society.)

Andre Weil and his sister Simone, summer of 1922(Photo from Providence)

 

Related material:


Log24, December 20, 2003–
White, Geometric, and Eternal

A description in Gravity’s Rainbow of prewar Berlin as “white and geometric”  suggested, in combination with a reference elsewhere to “the eternal,” a citation of the following illustration of the concept “white, geometric, and eternal”–

For more on the mathematical significance of this figure, see (for instance) Happy Birthday, Hassler Whitney, and Combinatorics of Coxeter Groups, by Anders Björner and Francesco Brenti, Graduate Texts in Mathematics, vol. 231, Springer, New York, 2005.

This book is reviewed in the current issue (July 2008) of the above-mentioned Providence Bulletin.

The review in the Bulletin discusses reflection groups in continuous spaces.

For a more elementary approach, see Reflection Groups in Finite Geometry and Knight Moves: The Relativity Theory of Kindergarten Blocks.

See also a commentary on
the phrase “as a little child.”

Monday, February 25, 2008

Monday February 25, 2008

Filed under: General,Geometry — Tags: , — m759 @ 4:00 PM
A System of Symbols

A book from
Yale University Press
discussed in Log24
four years ago today:

Inside Modernism: Relativity Theory, Cubism, Narrative

Click on image for details.

The book is titled
Inside Modernism:
Relativity Theory,
 Cubism, Narrative
.

For a narrative about relativity
and cubes, see Knight Moves.

Related material:

Geek chic in
this week's New Yorker

"… it takes a system of symbols  
to make numbers precise–
      to 'crystallize' them…."

— and a mnemonic for three
 days in October 2006
following a memorial to
 the Amish schoolchildren
slain that month:

Seven is Heaven,
Eight is a Gate,
Nine is a Vine.

Saturday, February 23, 2008

Saturday February 23, 2008

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

"An acute study of the links
between word and fact"
Nina daVinci Nichols

 
Thanks to a Virginia reader for a reminder:
 
Virginia /391062427/item.html? 2/22/2008 7:37 PM
 
The link is to a Log24 entry
that begins as follows…

An Exercise

of Power

Johnny Cash:
"And behold,
a white horse."

Springer logo - A chess knight
Chess Knight
(in German, Springer)

This, along with the "jumper" theme in the previous two entries, suggests a search on springer jumper.That search yields a German sports phrase, "Springer kommt!"  A search on that phrase yields the following:
"Liebe Frau vBayern,
mich würde interessieren wie man
mit diesem Hintergrund
(vonbayern.de/german/anna.html)
zu Springer kommt?"

Background of "Frau vBayern" from thePeerage.com:

Anna-Natascha Prinzessin zu Sayn-Wittgenstein-Berleburg 

F, #64640, b. 15 March 1978Last Edited=20 Oct 2005

     Anna-Natascha Prinzessin zu Sayn-Wittgenstein-Berleburg was born on 15 March 1978. She is the daughter of Ludwig Ferdinand Prinz zu Sayn-Wittgenstein-Berleburg and Countess Yvonne Wachtmeister af Johannishus. She married Manuel Maria Alexander Leopold Jerg Prinz von Bayern, son of Leopold Prinz von Bayern and Ursula Mohlenkamp, on 6 August 2005 at Nykøping, Södermanland, Sweden.

 

The date of the above "Liebe Frau vBayern" inquiry, Feb. 1, 2007, suggests the following:

From Log24 on
St. Bridget's Day, 2007:

The quotation
"Science is a Faustian bargain"
and the following figure–

Change

The 63 yang-containing hexagrams of the I Ching as a Singer 63-cycle

From a short story by
the above Princess:

"'I don't even think she would have wanted to change you. But she for sure did not want to change herself. And her values were simply a part of her.' It was true, too. I would even go so far as to say that they were her basis, if you think about her as a geometrical body. That's what they couldn't understand, because in this age of the full understanding for stretches of values in favor of self-realization of any kind, it was a completely foreign concept."

To make this excellent metaphor mathematically correct,
change "geometrical body" to "space"… as in

"For Princeton's Class of 2007"

Review of a 2004 production of a 1972 Tom Stoppard play, "Jumpers"–

John Lahr on Tom Stoppard's play Jumpers

Related material:

Knight Moves (Log24, Jan. 16),
Kindergarten Theology (St. Bridget's Day, 2008),
and

The image “My space -(the affine space of six dimensions over the two-element field
(Click on image for details.)

Friday, February 1, 2008

Friday February 1, 2008

Filed under: General,Geometry — m759 @ 5:01 AM
Kindergarten Theology

On the late James Edwin Loder,
a Presbyterian minister and
a professor of Christian education
at Princeton Theological Seminary,
co-author of The Knight’s Move (1992):

“At his memorial service his daughter Tami told the story of ‘little Jimmy,’ whose kindergarten teacher recognized a special quality of mind that set him apart. ‘Every day we read a story, and after the story is over, Jimmy gets up and wants to tell us what the story means.'” — Dana R. Wright

For a related story about
knight moves and kindergarten,
see Knight Moves: The Relativity
Theory of Kindergarten Blocks
,
and Log24, Jan. 16, 17, and 18.

See also Loder’s book
(poorly written, but of some
interest in light of the above):

The Knight's Move, by Loder and Neidhardt

Opening of The Knight’s Move

“In a game of chess, the knight’s move is unique because it alone goes around corners. In this way, it combines the continuity of a set sequence with the discontinuity of an unpredictable turn in the middle. This meaningful combination of continuity and discontinuity in an otherwise linear set of possibilities has led some to refer to the creative act of discovery in any field of research as a ‘knight’s move’ in intelligence.

The significance of the title of this volume might stop there but for Kierkegaard’s use of the ‘knight’ image. The force of Kierkegaards’s usage might be described in relation to the chess metaphor by saying that not merely does Kierkegaard’s ‘knight of faith’ undertake a unique move within the rules of the human game, but faith transposes the whole idea of a ‘knight’s move’ into the mind of the Chess Master Himself. That is to say, chess is a game of multiple possibilities and interlocking strategies, so a chess master must combine the  continuity represented by the whole complex of the game with the unpredictable decision he must make every time it is his turn. A master chess player, then, does not merely follow the rules; in him the game becomes a construct of consciousness. The better the player the more fully the game comes into its own as a creation of human intelligence. Similarly, for Kierkegaard, the knight of faith is a unique figure in human experience. The knight shows how, by existing in faith as a creative act of Christ’s Spirit, human existence comes into its own as an expression of the mind of Christ. Thus, the ultimate form of a ‘knight’s move’ is a creative act raised to the nth power by Spiritus Creator, but it still partakes fully in the concrete pieces and patterns that comprise the nature of the human game and the game of nature.”

— James E. Loder and W. Jim Neidhardt (Helmers & Howard Publishing, 1992)

For a discussion, see Triplett’s
Thinking Critically as a Christian.”

Many would deny that such
a thing is possible; let them
read the works of T. S. Eliot.

Related material:

The Knight’s Move
discusses (badly) Hofstadter’s
“strange loop” concept; see
Not Mathematics but Theology
(Log24, July 12, 2007).

Thursday, January 17, 2008

Thursday January 17, 2008

Filed under: General — Tags: — m759 @ 5:24 PM
Well, she was
   just seventeen…
 
(continued)

"Mazur introduced the topic of prime numbers with a story from Don Quixote in which Quixote asked a poet to write a poem with 17 lines. Because 17 is prime, the poet couldn't find a length for the poem's stanzas and was thus stymied."

— Undated American Mathematical Society news item about a Nov. 1, 2007, event

Related material:

Desconvencida,
Jueves, Enero 17, 2008

Horses of a Dream
(Log24, Sept. 12, 2003)

Knight Moves
(Log24 yesterday–
anniversary of the
Jan. 16 publication
of Don Quixote)

Windmill and Diamond
(St. Cecilia's Day 2006)

Wednesday, January 16, 2008

Wednesday January 16, 2008

Filed under: General,Geometry — Tags: — m759 @ 12:25 PM
Knight Moves:
Geometry of the
Eightfold Cube

Actions of PSL(2, 7) on the eightfold cube

Click on the image for a larger version
and an expansion of some remarks
quoted here on Christmas 2005.

Tuesday, October 12, 2004

Tuesday October 12, 2004

Filed under: General — Tags: — m759 @ 4:30 PM

The Last Enemy
(See April 30)

  

"I was also impressed… by the intensity of Continental modes of literary-critical thought….

On the Continent, studies of Hölderlin and Rousseau, of Poe, Baudelaire, Mallarmé and Rilke, of Rabelais, Nietzsche, Kafka, and Joyce, challenged not only received ideas on the unity of the work of art but many aspects of western thought itself. Derrida, at the same time, who for nearly a decade found a home in Yale's Comparative Literature Department, expanded the concept of textuality to the point where nothing could be demarcated as 'hors d'œuvre' and escape the literary-critical eye. It was uncanny to feel hierarchic boundaries waver until the commentary entered the text—not literally, of course, but in the sense that the over-objectified work became a reflection on its own status, its stability as an object of cognition. The well-wrought urn contained mortal ashes."

— Geoffrey Hartman, A Life of Learning

In memory of
Jacques Derrida and James Chace,
both of whom died in Paris on
Friday, Oct. 8, 2004… continued…
(See previous three entries.)

The image “http://www.log24.com/log/pix04A/041010-Welles.gif” cannot be displayed, because it contains errors.

Orson Welles


Mate in 2
V. Nabokov, 1919

"The last enemy
that shall be destroyed is death."
— Saul of Tarsus, 1 Cor. 15:26

The image “http://www.log24.com/log/pix04A/041012-Welles.gif” cannot be displayed, because it contains errors.

Knight move,
courtesy of V. Nabokov:

Nfe5 mate

The image “http://www.log24.com/log/pix04A/041012-Kt.gif” cannot be displayed, because it contains errors.

Knight:

Sir John Falstaff
(See Chimes at Midnight.)

Sunday, August 19, 2018

Possible Permutations

Filed under: General,Geometry — Tags: — m759 @ 9:18 PM

John Calder, an independent British publisher who built a prestigious list
of authors like Samuel Beckett and Heinrich Böll and spiritedly defended
writers like Henry Miller against censorship, died on Aug. 13 in Edinburgh.
He was 91.

— Richard Sandomir in the online New York Times  this evening

On Beckett —

http://www.log24.com/log/pix18/180819-Joyce-Possible_Permutations-Cambridge_Companion-2004-p168.gif

Also on August 13th

http://www.log24.com/log/pix18/180813-Knight_Moves-080116-page-top.gif

Wednesday, August 15, 2018

An Exceptional Isomorphism

Filed under: General,Geometry — Tags: , , — m759 @ 5:40 AM

Why PSL(2,7) is isomorphic to GL(3.2)

From previous posts on this topic —

http://www.log24.com/log/pix18/180813-Structure_and_Sense-post-160606.gif

http://www.log24.com/log/pix18/180813-Knight_Moves-080116-page-top.gif

Tuesday, August 14, 2018

D8ing

Filed under: General,Geometry — Tags: , , , — m759 @ 12:06 AM

(Continued)

http://www.log24.com/log/pix18/180813-Ezra_Brown-PSL(2,7)-GL(2,3)-Oct-2009.jpg

http://www.log24.com/log/pix18/180814-Knight_Moves-archived-May_9_2008-500w.jpg

Monday, August 13, 2018

Trojan Horsitude

Filed under: General,Geometry — Tags: — m759 @ 3:33 AM
 

"It is said that the students of medieval Paris came to blows in the streets over the question of universals. The stakes are high, for at issue is our whole conception of our ability to describe the world truly or falsely, and the objectivity of any opinions we frame to ourselves. It is arguable that this is always the deepest, most profound problem of philosophy. It structures Plato's (realist) reaction to the sophists (nominalists). What is often called 'postmodernism' is really just nominalism, colourfully presented as the doctrine that there is nothing except texts. It is the variety of nominalism represented in many modern humanities, paralysing appeals to reason and truth."

— Simon Blackburn, Think,
    Oxford University Press, 1999, page 268


". . . a perfect triptych of horsitude"

James Parker on the 2007 film "Michael Clayton"

Related material —

Horsitude in the 4×2 grid, and

http://www.log24.com/log/pix18/180813-Structure_and_Sense-post-160606.gif

http://www.log24.com/log/pix18/180813-Knight_Moves-080116-page-top.gif

Sunday, October 15, 2017

The Strength at the Centre

Filed under: General — Tags: , , — m759 @ 1:00 PM

The title, a phrase from a poem by Wallace Stevens,
was suggested by the previous post, "Center."

See posts tagged May 19 Gestalt in particular, 
May 19, 2007 — "Point of View."

Saturday, October 14, 2017

Center

Rosalind Krauss in 1978

"To get inside the systems of this work,
whether LeWitt's or Judd's or Morris's,
is precisely to enter
a world without a center,
a world of substitutions and transpositions
nowhere legitimated by the revelations
of a transcendental subject. This is the strength
of this work, its seriousness, and its claim to modernity." 

Wikipedia

"The center of
the quaternion group,
Q8 = {1, −1, i, −i, j, −j, k, −k} ,
is {1, −1}."

Illustration from a post of Feb. 3,  2011

http://www.log24.com/log/pix11/110203-Scholia.jpg.

In Principio:

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

Red October  continues …

See also Molloy in this  journal.

Related art  theory —

Geometry of the 4×4 Square 

Friday, October 13, 2017

Sicut Erat

Filed under: General — Tags: — m759 @ 9:26 PM

Smith College in 2011 on some music by Dan Brown's brother —

"Using the conventions of a traditional five-movement
Roman Catholic Mass to revere Darwin’s body of work,
Gregory Brown, Smith’s assistant director of choral
activities and a composer of choral music, is
collaborating with Craig Phillips, an early music specialist
and member of the classical a cappella male quartet
New York Polyphony, to create the piece Missa Charles Darwin . 
Brown is building the work in three large-scale sections and
scoring it for a male vocal quartet, which will be performed by
New York Polyphony."

https://www.smith.edu/insight/stories/darwin.php

Dan Brown has said his brother's Missa  helped suggest his new novel Origin .

Material from Smith College related to a performance of
Missa Charles Darwin  at the college on Feb. 4, 2011 —

Dan Brown, in the following passage, claims that an eight-ray star with arrowheads
at the rays' ends is "the mathematical symbol for entropy."  Brown may have first
encountered this symbol at a questionable "Sacred Science" website.  Wikipedia
discusses some even less  respectable uses of the symbol.

My own version of the above symbol (from the pure mathematics of group actions
on a 3×3 square) appeared here the day before  the Friday, Feb. 4, 2011,
Smith College Darwin Mass . . .

See posts now tagged The Next Thing.

Speak, Memra

Filed under: General,Geometry — Tags: — m759 @ 11:59 AM

The above was suggested by a Log24 review of October 13, 2002,
which in turn suggested a Log24 search for Carousel that yielded
(from Bloomsday Lottery) —

See as well Asimov's "prime radiant," and an illustration
of the number 13 as a radiant prime

"The Prime Radiant can be adjusted to your mind,
and all corrections and additions can be made
through mental rapport. There will be nothing to
indicate that the correction or addition is yours.
In all the history of the Plan there has been no
personalization. It is rather a creation of all of us 
together. Do you understand?"  

"Yes, Speaker!"

— Isaac Asimov, 
    Second Foundation , Ch. 8: Seldon's Plan

"Before time began, there was the Cube."
— Optimus Prime

See also Transformers in this journal.

Tuesday, June 7, 2016

Art and Space…

Filed under: General,Geometry — Tags: — m759 @ 6:00 AM

Continues, in memory of chess grandmaster Viktor Korchnoi,
who reportedly died at 85 yesterday in Switzerland —

IMAGE- Spielfeld (1982-83), by Wolf Barth

The coloring of the 4×4 "base" in the above image
suggests St. Bridget's cross.

From this journal on St. Bridget's Day this year —

"Possible title: 

A new graphic approach 
to an old geometric approach
to a new combinatorial approach
to an old algebraic approach
to M24
"

The narrative leap from image to date may be regarded as
an example of "knight's move" thinking.

Tuesday, June 9, 2015

Colorful Song

Filed under: General,Geometry — Tags: — m759 @ 8:40 PM

For geeks* —

Domain, Domain on the Range , "

where Domain = the Galois tesseract  and
Range = the four-element Galois field.

This post was suggested by the previous post,
by a Log24 search for Knight + Move, and by
the phrase "discouraging words" found in that search.

* A term from the 1947 film "Nightmare Alley."

Sunday, March 1, 2015

Sunday School

Filed under: General — m759 @ 9:00 AM

The previous post suggests a review of
the phrase "strange loop" in this journal.

The Knight's Move, by Loder and Neidhardt

Sunday, December 22, 2013

The Bronfman Catechism

Filed under: General — m759 @ 10:21 AM

Meanwhile

Log24 on Sunday, October 5, 2008

Theologian James Edwin Loder:

“In a game of chess, the knight’s move is unique because it alone goes around corners. In this way, it combines the continuity of a set sequence with the discontinuity of an unpredictable turn in the middle. This meaningful combination of continuity and discontinuity in an otherwise linear set of possibilities has led some to refer to the creative act of discovery in any field of research as a ‘knight’s move’ in intelligence.”

Related material:

Terence McKenna:

“Schizophrenia is not a psychological disorder peculiar to human beings. Schizophrenia is not a disease at all but rather a localized traveling discontinuity of the space time matrix itself. It is like a travelling whirl-wind of radical understanding that haunts time. It haunts time in the same way that Alfred North Whitehead said that the color dove grey ‘haunts time like a ghost.’”

Anonymous author:

“‘Knight’s move thinking’ is a psychiatric term describing a thought disorder where in speech the usual logical sequence of ideas is lost, the sufferer jumping from one idea to another with no apparent connection. It is most commonly found in schizophrenia.”

Related journalism

IMAGE- Scene from a blackboard jungle

"What's the 'S'  stand for?" — Amy Adams

Monday, March 25, 2013

Brightness at Noon

Filed under: General — Tags: — m759 @ 12:00 PM

The two symbols on the monolith from yesterday

Images of time and eternity in a 1x4x9 black monolith

— may, if one likes, be interpreted as standing for
Damnation Morning and for the Windmill of Time
(alternately, as motifs for a ukara cloth).

The above explanation may help those confused by
knight's-move discourse like that described by
Jemima in The Eiger Sanction .

Wednesday, June 20, 2012

Discouraging Words

Filed under: General — m759 @ 4:01 PM

A quote from The Oxford Murders , 
a novel by Guillermo Martinez

"Anyone can follow the path once it’s been marked out.
But there is of course an earlier moment of illumination, 
what you called the knight’s move. Only a few people, 
sometimes only one person in many centuries, 
manage to see the correct first step in the darkness.”

“A good try,” said Seldom.

Saturday, October 8, 2011

For Whom the Bell

Filed under: General — Tags: — m759 @ 8:00 PM

In memory of "Mr. Piano" Roger Williams, who died today

Flashback:

http://www.log24.com/log/pix11/110320-OmegaHaiku.jpg

http://www.log24.com/log/pix11/110320-TempleBellHaiku.jpg

http://www.log24.com/log/pix11/110313-BellOnGauss.jpg

Related material— A quote from The Oxford Murders ,
a novel by Guillermo Martinez

"Anyone can follow the path once it’s been marked out.
But there is of course an earlier moment of illumination,
what you called the knight’s move. Only a few people,
sometimes only one person in many centuries,
manage to see the correct first step in the darkness.”

“A good try,” said Seldom.

Friday, July 15, 2011

Crossing the Bridge

Filed under: General — m759 @ 11:59 PM

King Arthur and the Deathly Hallows

BLACK KNIGHT: None shall pass.

ARTHUR: What?

BLACK KNIGHT: None shall pass.

ARTHUR: I have no quarrel with you, good Sir Knight,
                but I must cross this bridge.

BLACK KNIGHT: Then you shall die.

ARTHUR: I command you as King of the Britons
                to stand aside!

BLACK KNIGHT: I move for no man.

ARTHUR: So be it!

— Monty Python

http://www.log24.com/log/pix11B/110715-Bushell-BlackKnight1954-Sm.jpg

Above: Anthony Bushell as King Arthur in 1954.
Bushell died on April 2, 1997 (4/2/97).
"Well, she was just seventeen…."

http://www.log24.com/log/pix11B/110715-NYlotteryEvening.jpg

Thursday, September 3, 2009

Thursday September 3, 2009

Filed under: General,Geometry — m759 @ 11:07 AM
Autistic Enchantment

“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 8×8 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

Quaternion rotations in a finite geometry
Click above images for some context.

See also:

Log24 entries of May 30, 2006, as well as “For John Cramer’s daughter Kathryn”– August 27, 2009— and related material at Wikipedia (where Kathryn is known as “Pleasantville”).

Sunday, April 12, 2009

Sunday April 12, 2009

Filed under: General — m759 @ 3:09 AM
Where Entertainment
Is God
, continued

Dialogue from the classic film Forbidden Planet

“… Which makes it a gilt-edged priority that one of us gets into that Krell lab and takes that brain boost.”

— Taken from a video (5:18-5:24 of 6:09) at David Lavery’s weblog in the entry of Tuesday, April 7.

(Cf. this journal on that date.)

Thanks to Professor Lavery for his detailed notes on his viewing experiences.

My own viewing recently included, on the night of Good Friday, April 10, the spiritually significant film Indiana Jones and the Kingdom of the Crystal Skull.

The mystic circle of 13 aliens at the end of that film, together with Leslie Nielsen’s Forbidden Planet remark quoted above, suggests the following:

“The aim of Conway’s game M13 is to get the hole at the top point and all counters in order 1,2,…,12 when moving clockwise along the circle.” —Lieven Le Bruyn

http://www.log24.com/log/pix09/090411-M13.gif

The illustration is from the weblog entry by Lieven Le Bruyn quoted below. The colored circles represent 12 of the 13 projective points described below, the 13 radial strokes represent the 13 projective lines, and the straight lines in the picture, including those that form the circle, describe which projective points are incident with which projective lines. The dot at top represents the “hole.”

From “The Mathieu Group M12 and Conway’s M13-Game” (pdf), senior honors thesis in mathematics by Jeremy L. Martin under the supervision of Professor Noam D. Elkies, Harvard University, April 1, 1996–

“Let P3 denote the projective plane of order 3. The standard construction of P3 is to remove the zero point from a three-dimensional vector space over the field F3 and then identify each point x with -x, obtaining a space with (33 – 1)/2 = 13 points. However, we will be concerned only with the geometric properties of the projective plane. The 13 points of P3 are organized into 13 lines, each line containing four points. Every point lies on four lines, any two points lie together on a unique line, and any two lines intersect at a unique point….

Conway [3] proposed the following game…. Place twelve numbered counters on the points… of P3 and leave the thirteenth point… blank. (The empty point will be referred to throughout as the “hole.”) Let the location of the hole be p; then a primitive move of the game consists of selecting one of the lines containing the hole, say {p, q, r, s}. Move the counter on q to p (thus moving the hole to q), then interchange the counters on r and s….

There is an obvious characterization of a move as a permutation in S13, operating on the points of P3. By limiting our consideration to only those moves which return the hole to its starting point…. we obtain the Conway game group. This group, which we shall denote by GC, is a subgroup of the symmetric group S12 of permutations of the twelve points…, and the group operation of GC is concatenation of paths. Conway [3] stated, but did not prove explicitly, that GC is isomorphic to the Mathieu group M12. We shall subsequently verify this isomorphism.

The set of all moves (including those not fixing the hole) is given the name M13 by Conway. It is important that M13 is not a group….”

[3] John H. Conway, “Graphs and Groups and M13,” Notes from New York Graph Theory Day XIV (1987), pp. 18–29.


Another exposition (adapted to Martin’s notation) by Lieven le Bruyn (see illustration above):

“Conway’s puzzle M13 involves the 13 points and 13 lines of P3. On all but one point numbered counters are placed holding the numbers 1,…,12 and a move involves interchanging one counter and the ‘hole’ (the unique point having no counter) and interchanging the counters on the two other points of the line determined by the first two points. In the picture [above] the lines are represented by dashes around the circle in between two counters and the points lying on this line are those that connect to the dash either via a direct line or directly via the circle. In the first part we saw that the group of all reachable positions in Conway’s M13 puzzle having the hole at the top position contains the sporadic simple Mathieu group M12 as a subgroup.”

For the religious significance of the circle of 13 (and the “hole”), consider Arthur and the 12 knights of the round table, et cetera.

But seriously…
 
Delmore Schwartz, 'Starlight Like Intuition Pierced the Twelve'

Thursday, April 2, 2009

Thursday April 2, 2009

Filed under: General,Geometry — Tags: — m759 @ 12:25 PM

Transformative
Hermeneutics

In memory of
physics historian
Martin J. Klein,
(June 25, 1924-
March 28, 2009)

"… in physics itself, there was what appeared, briefly, to be an ending, which then very quickly gave way to a new beginning: The quest for the ultimate building-blocks of the universe had been taken down to the molecular level in nineteenth-century kinetic theory… and finally to the nuclear level in the second and third decades of the twentieth century. For a moment in the 1920s the quest appeared to have ended…. However… this paradise turned out to be, if not exactly a fool's paradise, then perhaps an Eden lost."

No Truth Except in the Details: Essays in Honor of Martin J. Klein, introduction by A.J. Kox and Daniel Siegel, June 25, 1994

New York Times obituary dated April 1, 2009:

"Martin J. Klein, a historian of modern physics…. died Saturday, [March 28, 2009] in Chapel Hill, N.C. He was 84 and lived in Chapel Hill."

Klein edited, among other things, Paul Ehrenfest: Collected Scientific Papers (publ. by North-Holland, Amsterdam, 1959).

"It seems, as one becomes older,
 That the past has another pattern,
 and ceases to be a mere sequence…."

— T. S. Eliot, Four Quartets

A Walsh function and a corresponding finite-geometry hyperplane

"Note that at first, you can see
 the 'arrow of time.'
 After a long period, however,
 the direction of time
 is no longer evident."

— "The Ehrenfest Chains,"
     by Kyle Siegrist, ex. 16

Related material:

"Almost every famous chess game
is a well-wrought urn
in Cleanth Brooks’ sense."

— John Holbo,
Now We See
Wherein Lies the Pleasure

"The entire sequence of moves in these… chapters reminds one– or should remind one– of a certain type of chess problem where the point is not merely the finding of a mate in so many moves, but what is termed 'retrograde analysis'…."

— Vladimir Nabokov, foreword to The Defense

Tuesday, March 10, 2009

Tuesday March 10, 2009

Filed under: General,Geometry — m759 @ 9:26 AM
Language Game

“Music, mathematics, and chess are in vital respects dynamic acts of location. Symbolic counters are arranged in significant rows. Solutions, be they of a discord, of an algebraic equation, or of a positional impasse, are achieved by a regrouping, by a sequential reordering of individual units and unit-clusters (notes, integers, rooks or pawns). The child-master, like his adult counterpart, is able to visualize in an instantaneous yet preternaturally confident way how the thing should look several moves hence. He sees the logical, the necessary harmonic and melodic argument as it arises out of an initial key relation or the preliminary fragments of a theme. He knows the order, the appropriate dimension, of the sum or geometric figure before he has performed the intervening steps. He announces mate in six because the victorious end position, the maximally efficient configuration of his pieces on the board, lies somehow ‘out there’ in graphic, inexplicably clear sight of his mind….”

“… in some autistic enchantment, pure as one of Bach’s inverted canons or Euler’s formula for polyhedra.”

— George Steiner, “A Death of Kings,” in The New Yorker, issue dated Sept. 7, 1968

Related material:

“Classrooms are filled with discussions not of the Bible and Jesus but of 10 ‘core values’– perseverance and curiosity, for instance– that are woven into the curriculum.”

— “Secular Education, Catholic Values,” by Javier C. Hernandez, The New York Times, Sunday, March 8, 2009

“… There was a problem laid out on the board, a six-mover. I couldn’t solve it, like a lot of my problems. I reached down and moved a knight…. I looked down at the chessboard. The move with the knight was wrong. I put it back where I had moved it from. Knights had no meaning in this game. It wasn’t a game for knights.”


— Raymond Chandler, The Big Sleep

The Chandler quotation appears in “Language Game,” an entry in this journal on April 7, 2008.

Some say the “Language Game” date, April 7, is the true date (fixed, permanent) of the Crucifixion– by analogy, Eliot’s “still point” and Jung’s “centre.” (See yesterday, noon.)

Thursday, January 15, 2009

Thursday January 15, 2009

Filed under: General — m759 @ 6:00 PM
Harvard, Magic,
and The New York Times

The New York Times Magazine for next Sunday:

The Edge of the Mystery, by Matt Bai–

“Weeks before the election of 1960, Norman Mailer, already an accomplished novelist, sat down to write his first major work of political journalism, an essay for Esquire in which he argued that only John F. Kennedy could save America… the only kind of leader who could rescue it, who could sweep in an era of what Mailer called ‘existential’ politics, was a ‘hipster’ hero– someone who welcomed risk and adventure, someone who sought out new experience, both for himself and for the country….

… Mailer essentially created a new genre for a generation of would-be literary philosophers covering politics….  By 1963, Mailer and other idealists were crushed to discover that Kennedy was in fact a fairly conventional and pragmatic politician, more Harvard Yard than Fortress of Solitude.”

The New York Times today:

Magic and Realism, by Roger Cohen–

“… what I want from the Obama administration is something more than Harvard-to-the-Beltway smarts. I want magical realism.”

Mailer and Cohen, taken together, suggest I should review two authors– Picard and Hesse– I encountered as a Harvard freshman in 1960.

Max Picard:

“In the ‘Prologue in Heaven’ in Goethe’s Faust a powerful silence is produced by the powerful word after each verse. There is an active, audible silence after every verse. The things that were moved into position by the word stand motionless in the silence, as if they were waiting to be called back into the silence and to disappear therein. The word not only brings the things out of silence; it also produces the silence in which they can disappear again.”

Goethe:

DER HERR:
Kennst du den Faust?

MEPHISTOPHELES:
Den Doktor?

DER HERR:
Meinen Knecht!

Online Etymology Dictionary:

knight
O.E. cniht “boy, youth, servant,” common W.Gmc. (cf. O.Fris. kniucht, Du. knecht, kneht “boy, youth, lad,” Ger. Knecht “servant, bondsman, vassal”), of unknown origin. Meaning “military follower of a king or other superior” is from c.1100. Began to be used in a specific military sense in Hundred Years War, and gradually rose in importance through M.E. period until it became a rank in the nobility 16c. The verb meaning “to make a knight of (someone)” is from c.1300. Knighthood is O.E. cnihthad M.H.G. “the period between childhood and manhood;” sense of “rank or dignity of a knight” is from c.1300. The chess piece so called from c.1440.

Further background on the word “Knecht”–

'Magister Ludi,' or 'The Glass Bead Game,' by Hermann Hesse

Epigraph to Magister Ludi
(Joseph Knecht’s translation):


“… For although in a certain sense and for light-minded persons non-existent things can be more easily and irresponsibly represented in words than existing things, for the serious and conscientious historian it is just the reverse. Nothing is harder, yet nothing is more necessary, than to speak of certain things whose existence is neither demonstrable nor probable. The very fact that serious and conscientious men treat them as existing things brings them a step closer to existence and to the possibility of being born.”

Sunday, October 5, 2008

Sunday October 5, 2008

Filed under: General — Tags: — m759 @ 4:23 PM
Nash Equilibrium or:
To Make a Short Story Long

Last night's entry presented a
short story summarized by
four lottery numbers.

Today's mid-day lotteries
and associated material:

Pennsylvania, 201– i.e., 2/01:
Kindergarten Theology

Theologian James Edwin Loder:

"In a game of chess, the knight's move is unique because it alone goes around corners. In this way, it combines the continuity of a set sequence with the discontinuity of an unpredictable turn in the middle. This meaningful combination of continuity and discontinuity in an otherwise linear set of possibilities has led some to refer to the creative act of discovery in any field of research as a 'knight's move' in intelligence."

New York, 229– i.e., 2/29:
I Have a Dreamtime

"One must join forces with friends of like mind"

Related material:

Terence McKenna:

"Schizophrenia is not a psychological disorder peculiar to human beings. Schizophrenia is not a disease at all but rather a localized traveling discontinuity of the space time matrix itself. It is like a travelling whirl-wind of radical understanding that haunts time. It haunts time in the same way that Alfred North Whitehead said that the color dove grey 'haunts time like a ghost.'"

Anonymous author:

"'Knight's move thinking' is a psychiatric term describing a thought disorder where in speech the usual logical sequence of ideas is lost, the sufferer jumping from one idea to another with no apparent connection. It is most commonly found in schizophrenia."

Star Wars:
 
John Nash, as portrayed by Russell Crowe

I know more than Apollo,
For oft when he lies sleeping
I see the stars at mortal wars
In the wounded welkin weeping.

Tom O'Bedlam's Song

For more on the sleep of Apollo,
see the front page of today's
New York Times Book Review.

Garrison Keillor's piece there,
"Dying of the Light," is
about the fear of death felt
by an agnostic British twit.

For relevant remarks by
a British non-twit, see
William Dunbar–

Timor Mortis conturbat me.

Monday, April 7, 2008

Monday April 7, 2008

Filed under: General,Geometry — m759 @ 11:07 PM
A year ago…

  (Holy Saturday, 2007) —

From Friedrich Froebel,
who invented kindergarten:

Froebel's Third Gift

For further details, see
Gift of the Third Kind
and
  Kindergarten Relativity.

Related material:

“… There was a problem laid out on the board, a six-mover. I couldn’t solve it, like a lot of my problems. I reached down and moved a knight…. I looked down at the chessboard. The move with the knight was wrong. I put it back where I had moved it from. Knights had no meaning in this game. It wasn’t a game for knights.”


— Raymond Chandler, The Big Sleep

Perhaps, instead,
a game for jumpers?

The image “http://www.log24.com/images/IChing/hexagram35.gif” cannot be displayed, because it contains errors.

See
Tom Stoppard’s Progress.

Sunday, December 10, 2006

Sunday December 10, 2006

Filed under: General,Geometry — m759 @ 9:00 AM
The Librarian

"Like all men of the Library,
I have traveled in my youth."
— Jorge Luis Borges,
The Library of Babel

"Papá me mandó un artículo
de J. G. Ballard en el que
se refiere a cómo el lugar
de la muerte es central en
nuestra cultura contemporánea
."

— Sonya Walger,
interview dated September 14
(Feast of the Triumph of the Cross),
Anno Domini 2006

The image “http://www.log24.com/log/pix06B/061210-Quest.gif” cannot be displayed, because it contains errors.

Sonya Walger,
said to have been
born on D-Day,
the sixth of June,
in 1974

 

Walger's father is, like Borges,
from Argentina.
She "studied English Literature
at Christ Church College, Oxford,
where she received
    a First Class degree…. "

Wikipedia

"… un artículo de J. G. Ballard…."–

A Handful of Dust
, by J. G. Ballard

(The Guardian, March 20, 2006):

"… The Atlantic wall was only part of a huge system of German fortifications that included the Siegfried line, submarine pens and huge flak towers that threatened the surrounding land like lines of Teutonic knights. Almost all had survived the war and seemed to be waiting for the next one, left behind by a race of warrior scientists obsessed with geometry and death.

Death was what the Atlantic wall and Siegfried line were all about….

… modernism of the heroic period, from 1920 to 1939, is dead, and it died first in the blockhouses of Utah beach and the Siegfried line…

Modernism's attempt to build a better world with the aid of science and technology now seems almost heroic. Bertolt Brecht, no fan of modernism, remarked that the mud, blood and carnage of the first world war trenches left its survivors longing for a future that resembled a white-tiled bathroom.  Architects were in the vanguard of the new movement, led by Le Corbusier and the Bauhaus design school. The old models were thrown out. Function defined form, expressed in a pure geometry that the eye could easily grasp in its entirety."

The image “http://www.log24.com/theory/images/motto2.jpg” cannot be displayed, because it contains errors.
 
The image “http://www.log24.com/theory/images/grid3x3.gif” cannot be displayed, because it contains errors.

"This is the garden of Apollo,
the field of Reason…."
John Outram, architect 

(Click on picture for details.)

The image “http://www.log24.com/log/pix06B/061210-Holl.gif” cannot be displayed, because it contains errors.
The Left Hand of God, by Adolf Holl

Related material:

The Lottery of Babylon
and
the previous entry.
 

Friday, July 7, 2006

Friday July 7, 2006

Filed under: General — m759 @ 9:00 AM
Born (some say)
on this date:
Yul Brynner

The image “http://www.log24.com/log/pix06A/060707-Brynner1.jpg” cannot be displayed, because it contains errors.

Mate in 6
(White moves.)

(White: Ke8, Nd7, Be5,
b5, e4, f2. Black: Ke6.
)

The image “http://www.log24.com/log/pix06A/060707-Problem.jpg” cannot be displayed, because it contains errors.

Sevitov, 1938

(For solution, click here.)

Log24, July 3, 2006:

“… There was a problem laid out on the board, a six-mover. I couldn’t solve it, like a lot of my problems. I reached down and moved a knight….
I looked down at the chessboard. The move with the knight was wrong. I put it back where I had moved it from. Knights had no meaning in this game. It wasn’t a game for knights.”

— Raymond Chandler, The Big Sleep, begun in the summer of 1938

Log24, July 2, 2006:

Is a puzzlement!

Log24, July 5, 2006:

“In this way we are offered a formidable lesson for every Christian community.”

— Pope Benedict XVI on Pentecost, June 4, 2006, St. Peter’s Square.

Monday, July 3, 2006

Monday July 3, 2006

Filed under: General — m759 @ 11:07 PM

Culture War

The New York Times, August 6, 2003,
on its executive editor Bill Keller:

“‘It is past time for our magnificent coverage of culture and lifestyles, so essential to our present allure and to our future growth, to get the kind of attention we routinely bestow on hard news,’ Mr. Keller wrote in an e-mail message to the staff.”

The New York Times, June 25, 2006,
on art in Mexico:

“At the Hilario Galguera gallery, newly opened in a fortresslike, century-old building, was Damien Hirst’s gory new series ‘The Death of God– Towards a Better Understanding of Life Without God Aboard the Ship of Fools.’  He conceived the work at his part-time home in the Mexican surf town Troncones.”

Raymond Chandler in The Big Sleep:

   “I went over to a floor lamp and pulled the switch, went back to put off the ceiling light, and went across the room again to the chessboard on a card table under the lamp. There was a problem laid out on the board, a six-mover.  I couldn’t solve it, like a lot of my problems.  I reached down and moved a knight, then pulled my hat and coat off and threw them somewhere.  All this time the soft giggling went on from the bed, that sound that made me think of rats behind a wainscoting in an old house.

…………

    I looked down at the chessboard.  The move with the knight was wrong.  I put it back where I had moved it from.  Knights had no meaning in this game.  It wasn’t a game for knights.”

Tuesday, January 4, 2005

Tuesday January 4, 2005

Filed under: General — m759 @ 7:00 AM
The Romantic School

Today’s New York Times:

“Mr. Denker was of the romantic school
of chess – always looking to attack.”

The image “http://www.log24.com/log/pix05/050104-Chess.jpg” cannot be displayed, because it contains errors.

Related material:

From Endgame:

The image “http://www.log24.com/log/pix05/050104-Endgame.jpg” cannot be displayed, because it contains errors.

Black the knight upon that ocean,
Bright the sun upon the king.
Dark the queen that stands beside him,
White his castle, threatening.

In the shadows’ see a bishop
Guards his queen of love and hate.
Another move, the game will be up;
Take the queen, her knight will mate.

The knight said “Move, be done.  It’s over.”
“Love and resign,” the bishop cried.
“When it’s done you’ll stand forever
By the darkest beauty’s side.”

From Log24.net, Feb. 18, 2003:

The image “http://www.log24.com/log/pix05/050104-Kali.jpg” cannot be displayed, because it contains errors.

Kali, a goddess sometimes depicted
as a dancing girl; Kali is related to kAla,
time, according to one website,  as
“the force which governs and stops time.”
See also the novel The Fermata,
by Nicholson Baker.

The image “http://www.log24.com/log/pix05/050104-Ferm.jpg” cannot be displayed, because it contains errors.

From an entry of Sunday, Jan. 2,
the day Denker died:

“Time had been canceled.”    
— Stephen King, The Shining

From Truth and Style, a tribute
to the late Amy Spindler, style editor
of the New York Times Magazine:

“I don’t believe in truth. I believe in style.”
— Hugh Grant in Vogue magazine, July 1995

From a related page,
The Crimson Passion:

“He takes us to the central activity
of mathematics—which is imagining….”

Harvard Magazine on
Harvard mathematician
and author Barry Mazur.

For related material on Mazur, see

A Mathematical Lie.

“The teenagers aren’t all bad.
I love ’em if nobody else does.
There ain’t nothing wrong
with young people.
Jus’ quit lyin’ to ’em.”

Jackie “Moms” Mabley

Saturday, May 22, 2004

Saturday May 22, 2004

Filed under: General — m759 @ 6:14 AM

Star Wars

In memory of Melvin J. Lasky, editor, 1958-1990, of the CIA-funded journal Encounter:

“Once called as lively, and as bitchy, as a literary cocktail party, Encounter published articles of unrivalled authority on politics, history and literature.”

— Obituary in the Telegraph 

Lasky died on Wednesday, May 19, 2004.  From a journal entry of my own on that date:

This newly-digitized diagram is from a
paper journal note of October 21, 1999.

Note that the diagram’s overall form is that of an eight-point star.  Here is an excerpt from a Fritz Leiber story dealing with such a star, the symbol of a fictional organization:

Time traveling, which is not quite the good clean boyish fun it’s cracked up to be, started for me when this woman with the sigil on her forehead looked in on me from the open doorway of the hotel bedroom where I’d hidden myself and the bottles and asked me, “Look, Buster, do you want to live?”
….

Her right arm was raised and bent, the elbow touching the door frame, the hand brushing back the very dark bangs from her forehead to show me the sigil, as if that had a bearing on her question.

The sigil was an eight-limbed asterisk made of fine dark lines and about as big as a silver dollar.  An X superimposed on a plus sign.  It looked permanent.
….

… “Here is how it stacks up:  You’ve bought your way with something other than money into an organization of which I am an agent….”
….

“It’s a very big organization,” she went on, as if warning me.  “Call it an empire or a power if you like.  So far as you are concerned, it has always existed and always will exist.  It has agents everywhere, literally.  Space and time are no barriers to it.  Its purpose, so far as you will ever be able to know it, is to change, for its own aggrandizement, not only the present and the future, but also the past.  It is a ruthlessly competitive organization and is merciless to its employees.”

“I. G. Farben?” I asked grabbing nervously and clumsily at humor.

She didn’t rebuke my flippancy, but said, “And it isn’t the Communist Party or the Ku Klux Klan, or the Avenging Angels or the Black Hand, either, though its enemies give it a nastier name.”

“Which is?” I asked.

“The Spiders,” she said.

That word gave me the shudders, coming so suddenly.  I expected the sigil to step off her forehead and scuttle down her face and leap at me—something like that.

She watched me.  “You might call it the Double Cross,” she suggested, “if that seems better.”

— Fritz Leiber,
   “Damnation Morning,” 1959

From last year’s entry,
Indiana Jones and the Hidden Coffer,
of 6/14:

From Borges’s “The Aleph“:

“The Faithful who gather at the mosque of Amr, in Cairo, are acquainted with the fact that the entire universe lies inside one of the stone pillars that ring its central court…. The mosque dates from the seventh century; the pillars come from other temples of pre-Islamic religions…. Does this Aleph exist in the heart of a stone?”

(“Los fieles que concurren a la mezquita de Amr, en el Cairo, saben muy bien que el universo está en el interior de una de las columnas de piedra que rodean el patio central…. la mezquita data del siglo VII; las columnas proceden de otros templos de religiones anteislámicas…. ¿Existe ese Aleph en lo íntimo de una piedra?”)

From The Hunchback of Notre Dame:

Un cofre de gran riqueza
Hallaron dentro un pilar,
Dentro del, nuevas banderas
Con figuras de espantar.

A coffer of great richness
In a pillar’s heart they found,
Within it lay new banners,
With figures to astound.

See also the figures obtained by coloring and permuting parts of the above religious symbol.

Lena Olin and Harrison Ford
in “Hollywood Homicide

Finally, from an excellent site
on the Knights Templar,
a quotation from Umberto Eco:

When all the archetypes burst out shamelessly, we plumb the depths of Homeric profundity. Two cliches make us laugh but a hundred cliches move us because we sense dimly that the cliches are talking among themselves, celebrating a reunion . . . Just as the extreme of pain meets sensual pleasure, and the extreme of perversion borders on mystical energy, so too the extreme of banality allows us to catch a glimpse of the Sublime.

— “Casablanca: Cult Movies and Intertextual Collage” (1984) from Travels in Hyperreality

Monday, April 28, 2003

Monday April 28, 2003

Filed under: General,Geometry — Tags: — m759 @ 12:07 AM

ART WARS:

Toward Eternity

April is Poetry Month, according to the Academy of American Poets.  It is also Mathematics Awareness Month, funded by the National Security Agency; this year's theme is "Mathematics and Art."

Some previous journal entries for this month seem to be summarized by Emily Dickinson's remarks:

"Because I could not stop for Death–
He kindly stopped for me–
The Carriage held but just Ourselves–
And Immortality.

………………………
Since then–'tis Centuries–and yet
Feels shorter than the Day
I first surmised the Horses' Heads
Were toward Eternity– "

 

Consider the following journal entries from April 7, 2003:
 

Math Awareness Month

April is Math Awareness Month.
This year's theme is "mathematics and art."


 

An Offer He Couldn't Refuse

Today's birthday:  Francis Ford Coppola is 64.

"There is a pleasantly discursive treatment
of Pontius Pilate's unanswered question
'What is truth?'."


H. S. M. Coxeter, 1987, introduction to Richard J. Trudeau's remarks on the "Story Theory" of truth as opposed to the "Diamond Theory" of truth in The Non-Euclidean Revolution

 

From a website titled simply Sinatra:

"Then came From Here to Eternity. Sinatra lobbied hard for the role, practically getting on his knees to secure the role of the street smart punk G.I. Maggio. He sensed this was a role that could revive his career, and his instincts were right. There are lots of stories about how Columbia Studio head Harry Cohn was convinced to give the role to Sinatra, the most famous of which is expanded upon in the horse's head sequence in The Godfather. Maybe no one will know the truth about that. The one truth we do know is that the feisty New Jersey actor won the Academy Award as Best Supporting Actor for his work in From Here to Eternity. It was no looking back from then on."

From a note on geometry of April 28, 1985:

 
The "horse's head" figure above is from a note I wrote on this date 18 years ago.  The following journal entry from April 4, 2003, gives some details:
 

The Eight

Today, the fourth day of the fourth month, plays an important part in Katherine Neville's The Eight.  Let us honor this work, perhaps the greatest bad novel of the twentieth century, by reflecting on some properties of the number eight.  Consider eight rectangular cells arranged in an array of four rows and two columns.  Let us label these cells with coordinates, then apply a permutation.

 


 Decimal 
labeling

 
Binary
labeling


Algebraic
labeling


Permutation
labeling

 

The resulting set of arrows that indicate the movement of cells in a permutation (known as a Singer 7-cycle) outlines rather neatly, in view of the chess theme of The Eight, a knight.  This makes as much sense as anything in Neville's fiction, and has the merit of being based on fact.  It also, albeit rather crudely, illustrates the "Mathematics and Art" theme of this year's Mathematics Awareness Month.

The visual appearance of the "knight" permutation is less important than the fact that it leads to a construction (due to R. T. Curtis) of the Mathieu group M24 (via the Curtis Miracle Octad Generator), which in turn leads logically to the Monster group and to related "moonshine" investigations in the theory of modular functions.   See also "Pieces of Eight," by Robert L. Griess.

Friday, April 4, 2003

Friday April 4, 2003

Filed under: General,Geometry — Tags: — m759 @ 3:33 PM

The Eight

Today, the fourth day of the fourth month, plays an important part in Katherine Neville's The Eight.  Let us honor this work, perhaps the greatest bad novel of the twentieth century, by reflecting on some properties of the number eight.  Consider eight rectangular cells arranged in an array of four rows and two columns.  Let us label these cells with coordinates, then apply a permutation.


Decimal 
labeling


Binary
labeling


Algebraic
labeling

IMAGE- Knight figure for April 4
Permutation
labeling

 

The resulting set of arrows that indicate the movement of cells in a permutation (known as a Singer 7-cycle) outlines rather neatly, in view of the chess theme of The Eight, a knight.  This makes as much sense as anything in Neville's fiction, and has the merit of being based on fact.  It also, albeit rather crudely, illustrates the "Mathematics and Art" theme of this year's Mathematics Awareness Month.  (See the 4:36 PM entry.)

 

 

The visual appearance of the "knight" permutation is less important than the fact that it leads to a construction (due to R. T. Curtis) of the Mathieu group M24 (via the Curtis Miracle Octad Generator), which in turn leads logically to the Monster group and to related "moonshine" investigations in the theory of modular functions.   See also "Pieces of Eight," by Robert L. Griess.
 

Thursday, November 7, 2002

Thursday November 7, 2002

Filed under: General — m759 @ 5:24 AM

16 Years Ago Today:

Endgame

Metaphor for Morphean morphosis,
Dreams that wake, transform, and die,
Calm and lucid this psychosis,
Joyce’s nightmare in Escher’s eye.

At the end there is a city
With cathedral bright and sane
Facing inward from the pity
On the endgame’s wavy plane.

Black the knight upon that ocean,
Bright the sun upon the king.
Dark the queen that stands beside him,
White his castle, threatening.

In the shadows’ see a bishop
Guards his queen of love and hate.
Another move, the game will be up;
Take the queen, her knight will mate.

The knight said “Move, be done.  It’s over.”
“Love and resign,” the bishop cried.
“When it’s done you’ll stand forever
By the darkest beauty’s side.”

Dabo claves regni caelorum.  By silent shore
Ripples spread from castle rock.  The metaphor
For metamorphosis no keys unlock.

— Steven H. Cullinane, November 7, 1986

Accompaniment from
“The Thomas Crown Affair”:
Michel Legrand, “Les Moulins de Mon Coeur”

Lyrics by Eddy Marnay:

Comme une pierre que l’on jette
Dans l’eau vive d’un ruisseau
Et qui laisse derrière elle
Des milliers de ronds dans l’eau….

Saturday, September 7, 2002

Saturday September 7, 2002

Filed under: General — m759 @ 4:44 PM

The Boys from Uruguay

If one were to write a “secret history” of the twentieth century, one possible organizing theme might be the religious struggle between worshippers of the Semitic deity (variously known as Yahweh, God, and Allah) and worshippers of the Aryan deities… notably, the Aryan god of music, light, and reason, Apollo.

(See my jounal notes of Monday, Sept. 2, 2002, below.)

In perhaps the best academic website I have ever seen, Karey L. Perkins quotes Walker Percy:

“The truth is that man’s capacity for symbol-mongering in general and language in particular is…intimately part and parcel of his being human, of his perceiving and knowing, of his very consciousness…”

The greatest symbol-monger of the twentieth century was, of course, Adolf Hitler. His use of the Aryan sun-wheel symbol rose to the level of genius. Of course, it ultimately failed to win the approval of the sun god himself, Apollo, who is also the god of reason.

Since symbol-mongering cannot be avoided, let us hope that it can be done in a somewhat more reasonable way than that of the National Socialist movement. Two examples suggest themselves.

  1. From Peggy Noonan’s column of Friday, Sept. 6:

    The cross, the heart, and the flag.

  2. From Karey Perkins’s website:

    A brain, a heart, and courage

On this Rosh Hashanah, the cross as a symbol of intelligence may be offensive to some worshippers of Yahweh. Let them read The Archivist, a novel by Martha Cooley, and then my journal note The Matthias Defense.

They might also contemplate the biblical quotation in the musical “Contact” broadcast from Lincoln Center on September 1, 2002: “Let there be light!”

Three Jews named Paul have been associated with light…

  1. Saul of Tarsus, who later assumed an alias.

  2. Paul Newman, whose performance in “The Verdict” continues, indirectly, to trouble Cardinal Law of Boston.

  3. Paul R. Halmos, a personal hero of mine ever since I saw his Finite Dimensional Vector Spaces and Measure Theory as an ignorant young undergraduate browsing the bookstores of Harvard Square.

In accordance with the “secret history” theme mentioned above, the struggle between Aryan and Semitic religions may also be viewed in the light of the struggle between Christianity and Communism. Hitler exploited this viewpoint very successfully, pretending to be the champion of the Christians against the godless Reds. Peggy Noonan also successfully uses this strategy. Both Hitler and Noonan manage to ignore the fact that Christianity is itself one of the Semitic religions, and that at least two of its three deities are Jewish.

As for me, I rather identify with the young Hitler clone at the end of the film “The Boys from Brazil.” Forced to decide between Gregory Peck and Sir Laurence Olivier, he sides with Olivier. His reason? Peck lied.

In a similar situation, forced to decide between Peggy Noonan and the Jew Halmos, I would probably side with Halmos. Halmos, who should, if not a saint, be at least dubbed a knight, does not, unlike the great majority of the damned human race, lie.

See Halmos’s memoir, I Want to Be a Mathematician. In particular, see the single index entry “communist by allegation” and the 29 entries under “Uruguay.”

Happy birthday to Elia Kazan and Peggy Noonan, and a happy and prosperous New Year to should-be-Sir Paul R. Halmos. 

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