Log24

Sunday, July 12, 2020

Knight Moves

Filed under: General — Tags: — m759 @ 4:40 PM

As noted in the previous post, the phrase “the ability to jump
in and out of spaces” was quoted in an update this morning to
a July 2 post, “The Maxwell Enticement.” Related jumping —

See also other Log24 posts now tagged Knight Move.

Saturday, September 2, 2017

Knight Moves

Filed under: General — Tags: — 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 — Tags: — 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 — Tags: — 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.

Thursday, July 2, 2020

The Maxwell* Enticement

Filed under: General — Tags: , — m759 @ 5:20 PM

[Update of Sunday morning, July 12, 2020 —

This July 2 post was suggested in part by the July 1 post Magic Child
and in part by the Sept. 15, 1984, date in the image below. For more
details about that date, possibly the death date of author Richard
Brautigan, see “The Life and Death of Richard Brautigan,” by
Lawrence Wright, in Rolling Stone  on April 11, 1985.

From that article:

Marcia called him the next night [Sept. 15, 1984]
in Bolinas. He asked if she liked his mind. “I said,
‘Yes, Richard, I like your mind. You have the ability
to jump in and out of spaces. It’s not linear thinking;
it’s exciting, catalytic, random thinking.’ “

Such thinking, though interesting, is not recommended for the
general public.  Sept. 15, 1984, was perhaps Brautigan’s last day alive.]

* See Maxwell in posts tagged Gods and Giants.

Saturday, December 28, 2019

Caballo Blanco

Filed under: General — Tags: , , , — m759 @ 9:02 AM

The key  is the cocktail that begins the proceedings.”

– Brian Harley, Mate in Two Moves

http://www.log24.com/log/pix18/180809-The_EIght-and-coordinates-for-PSL(2,7)-actions-500w.jpg

“Just as these lines that merge to form a key
Are as chess squares . . . .” — Katherine Neville, The Eight

“The complete projective group of collineations and dualities of the
[projective] 3-space is shown to be of order [in modern notation] 8! ….
To every transformation of the 3-space there corresponds
a transformation of the [projective] 5-space. In the 5-space, there are
determined 8 sets of 7 points each, ‘heptads’ ….”

— George M. Conwell, “The 3-space PG (3, 2) and Its Group,”
The Annals of Mathematics , Second Series, Vol. 11, No. 2 (Jan., 1910),
pp. 60-76.

“It must be remarked that these 8 heptads are the key  to an elegant proof….”

— Philippe Cara, “RWPRI Geometries for the Alternating Group A8,” in
Finite Geometries: Proceedings of the Fourth Isle of Thorns Conference
(July 16-21, 2000), Kluwer Academic Publishers, 2001, ed. Aart Blokhuis,
James W. P. Hirschfeld, Dieter Jungnickel, and Joseph A. Thas, pp. 61-97.

Friday, December 6, 2019

November Seventh Death

Filed under: General — Tags: — m759 @ 5:47 PM

See as well some notes from the date of the above death:

Related material: Endgame (November 7, 1986).

Friday, November 8, 2019

Glitch

Filed under: General — Tags: — m759 @ 7:12 AM

The terms glitch  and cross-carrier  in the previous post
suggest a review

 

Gerard Manley Hopkins in 1888

Cross-Carrier

For some backstory, see GlitchGerard Manley HopkinsInscape
particularly the post A Balliol Star.

Thursday, November 7, 2019

Forgotten Ghosts

Filed under: General — Tags: — m759 @ 9:46 PM

See that tag.

Cross-carrier glitch sent people ancient texts

For Connoisseurs of Insane Fantasy

Filed under: General — Tags: — m759 @ 11:23 AM

From a 1962 young-adult novel —

"There's something phoney in the whole setup, Meg thought.
There is definitely something rotten in the state of Camazotz."

Song adapted from a 1960 musical —

"In short, there's simply not
A more congenial spot
For happy-ever-aftering
Than here in Camazotz!"

Google News 'For you' comic book news item

Jagged Crest

Filed under: General — Tags: — m759 @ 9:01 AM

"The man touched the white bishop, queen and king,
and ran his finger over the jagged crest of the rook.
Then, sitting down before the chess set owner could nod
his head, he made his first move with the white pawn."

The late Stephen Dixon, "The Chess House," in
The Paris Review Winter-Spring 1963 (early in 1963).

I Ching chessboard (original 1989 arrangement)

Thursday, May 9, 2019

The Birdseye Requiem

Filed under: General — Tags: — m759 @ 9:10 AM

From The Boston Globe  yesterday evening —

" Ms. Adams 'had this quiet intelligence that made you feel like
she understood you and she loved you. She was a true friend —
a true generous, generous friend. This is the kind of person
you keep in your life,' Birdseye added.

'And she had such a great sense of humor,' Birdseye said.
“She would always have the last laugh. She wasn’t always
the loudest, but she was always the funniest, and in the
smartest way.' "

"Ms. Adams, who lived in Waltham, was 55 when she died April 9 . . . ."

See as well April 9 in the post Math Death and a post from April 8,
also now tagged "Berlekamp's Game" — Horses of a Dream.

"When logic and proportion have fallen sloppy dead
And the white knight is talking backwards . . . ."

— Grace Slick in a song from yesterday's post "When the Men"

Tuesday, June 14, 2016

Model Kit

Filed under: General,Geometry — Tags: — 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 — Tags: — 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, March 1, 2015

Sunday School

Filed under: General — Tags: — 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 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 — Tags: — 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 — Tags: — 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 — Tags: — 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 — Tags: — 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.

Thursday, April 23, 2009

Thursday April 23, 2009

Filed under: General,Geometry — Tags: , — m759 @ 10:00 AM

 

The Geometry
of Language

(continued from April 16)

Background:

Professor Arielle Saiber with chess set

Click on the image for an
interview with the author of
Giordano Bruno and
the Geometry of Language
.

Related material:

Joyce on language —

The sigla of 'Finnegans Wake'

Bruno, Joyce, and coincidentia oppositorum

Cullinane on geometry —

Geometry of the I Ching (for comparison to Joyce's 'sigla')

Click on images for details.
 

Friday, April 3, 2009

Friday April 3, 2009

Filed under: General,Geometry — Tags: , — 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.

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.)

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 — Tags: — 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).

Friday, January 18, 2008

Friday January 18, 2008

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

Front page top center, online NY Times: Bobby Fischer Dead at 64

Friday January 18, 2008

Filed under: General — Tags: — m759 @ 4:00 AM

Nativity

… Todo lo sé por el lucero puro
que brilla en la diadema de la Muerte.

Rubén Darío,
born January 18, 1867

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.

Wednesday, June 20, 2007

Wednesday June 20, 2007

Filed under: General,Geometry — Tags: — m759 @ 1:06 AM

Kernel

Mathematical Reviews citation:

MR2163497 (2006g:81002) 81-03 (81P05)
Gieser, Suzanne The innermost kernel. Depth psychology and quantum physics. Wolfgang Pauli's dialogue with C. G. Jung. Springer-Verlag, Berlin, 2005. xiv+378 pp. ISBN: 3-540-20856-9

A quote from MR at Amazon.com:

"This revised translation of a Swedish Ph. D. thesis in philosophy offers far more than a discussion of Wolfgang Pauli's encounters with the psychoanalyst Carl Gustav Jung…. Here the book explains very well how Pauli attempted to extend his understanding beyond superficial esotericism and spiritism…. To understand Pauli one needs books like this one, which… seems to open a path to a fuller understanding of Pauli, who was seeking to solve a quest even deeper than quantum physics." (Arne Schirrmacher, Mathematical Reviews, Issue 2006g)
 

An excerpt:

 

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

I do not yet know what Gieser means by "the innermost kernel." The following is my version of a "kernel" of sorts– a diagram well-known to students of anthropologist Claude Levi-Strauss and art theorist Rosalind Krauss:

The four-group is also known as the Vierergruppe or Klein group.  It appears, notably, as the translation subgroup of A, the group of 24 automorphisms of the affine plane over the 2-element field, and therefore as the kernel of the homomorphism taking A to the group of 6 automorphisms of the projective line over the 2-element field. (See Finite Geometry of the Square and Cube.)

Related material:

The "chessboard" of
   Nov. 7, 2006
(as revised Nov. 7, 2012)–

I Ching chessboard. Previous version replaced on Nov. 7, 2012, by original 1989 chessboard arrangement

I Ching chessboard

None of this material really has much to do with the history of physics, except for its relation to the life and thought of physicist Wolfgang Pauli— the "Mephistopheles" of the new book Faust in Copenhagen. (See previous entry.)

"Only gradually did I discover
what the mandala really is:
'Formation, Transformation,
Eternal Mind's eternal recreation'"

(Faust, Part Two, as
quoted by Jung in
Memories, Dreams, Reflections)
 

Tuesday, November 7, 2006

Tuesday November 7, 2006

Filed under: General,Geometry — Tags: — m759 @ 12:00 AM
A Game of Chess

The image “http://www.log24.com/log/pix06B/061107-McQueen.jpg” cannot be displayed, because it contains errors.

"And these chessmen are men and women as they appear to themselves and to one another in this world. And the silver table is Time. And those who stand and watch are the immortal souls of these same men and women."

— C. S. Lewis,
The Great Divorce

I Ching chessboard

I Ching chessboard

Related material:

"At the still point,
there the dance is
"

and

Number and Time, by Marie-Louise von Franz
 

Sunday, September 10, 2006

Sunday September 10, 2006

Filed under: General,Geometry — Tags: , — m759 @ 4:00 AM

And the
"
Meet Max Black"
Award goes to…

The image “http://www.log24.com/log/pix06A/060910-Obits.jpg” cannot be displayed, because it contains errors.

"For the Aeron and other designs,
Mr. Stumpf won this year’s
National Design Award
in Product Design
,
which is to be presented
posthumously on Oct. 18
by the Cooper-Hewitt
National Design Museum
in Manhattan."

— Today's New York Times

Stumpf died on August 30,
the date of the Log24 entry
"The Seventh Symbol."

Related material:

From
Geometry of the I Ching,
chessboard:

I Ching chessboard (original 1989 arrangement)

From the
 National Design Museum:

The image “http://www.log24.com/log/pix06A/060910-DesignAwards.gif” cannot be displayed, because it contains errors.

 From Log24 on the
date of Stumpf's death,

The Seventh Symbol:

The image “http://www.log24.com/log/pix06A/060910-Box121.jpg” cannot be displayed, because it contains errors.

Pictorial version of
Hexagram 20,
Contemplation (View)

See also
Fearful Symmetry
and
Symmetry Framed.

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.

Sunday, November 20, 2005

Sunday November 20, 2005

Filed under: General,Geometry — Tags: — m759 @ 4:04 PM

An Exercise
of Power

Johnny Cash:
“And behold,
a white horse.”

The image “http://www.log24.com/log/pix05B/051120-SpringerLogo9.gif” cannot be displayed, because it contains errors.
Adapted from
illustration below:

The image “http://www.log24.com/log/pix05B/051120-NonEuclideanRev.jpg” cannot be displayed, because it contains errors.

“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

“A new epistemology is emerging to replace the Diamond Theory of truth. I will call it the ‘Story Theory’ of truth: There are no diamonds. People make up stories about what they experience. Stories that catch on are called ‘true.’ The Story Theory of truth is itself a story that is catching on. It is being told and retold, with increasing frequency, by thinkers of many stripes*….”

Richard J. Trudeau in
The Non-Euclidean Revolution

“‘Deniers’ of truth… insist that each of us is trapped in his own point of view; we make up stories about the world and, in an exercise of power, try to impose them on others.”

— Jim Holt in The New Yorker.

(Click on the box below.)

The image “http://www.log24.com/log/pix05B/050819-Critic4.jpg” cannot be displayed, because it contains errors.

Exercise of Power:

Show that a white horse–

A Singer 7-Cycle

a figure not unlike the
symbol of the mathematics
publisher Springer–
is traced, within a naturally
arranged rectangular array of
polynomials, by the powers of x
modulo a polynomial
irreducible over a Galois field.

This horse, or chess knight
“Springer,” in German–
plays a role in “Diamond Theory”
(a phrase used in finite geometry
in 1976, some years before its use
by Trudeau in the above book).

Related material

On this date:

 In 1490, The White Knight
 (Tirant lo Blanc The image “http://www.log24.com/images/asterisk8.gif” cannot be displayed, because it contains errors. )–
a major influence on Cervantes–
was published, and in 1910

The image “http://www.log24.com/log/pix05B/051120-Caballo1.jpg” cannot be displayed, because it contains errors.

the Mexican Revolution began.

Illustration:
Zapata by Diego Rivera,
Museum of Modern Art,
New York

The image “http://www.log24.com/images/asterisk8.gif” cannot be displayed, because it contains errors. Description from Amazon.com

“First published in the Catalan language in Valencia in 1490…. Reviewing the first modern Spanish translation in 1969 (Franco had ruthlessly suppressed the Catalan language and literature), Mario Vargas Llosa hailed the epic’s author as ‘the first of that lineage of God-supplanters– Fielding, Balzac, Dickens, Flaubert, Tolstoy, Joyce, Faulkner– who try to create in their novels an all-encompassing reality.'”

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, September 22, 2002

Sunday September 22, 2002

Filed under: General,Geometry — Tags: , , , — m759 @ 8:02 PM

Force Field of Dreams

Metaphysics and chess in today’s New York Times Magazine:

  • From “Must-See Metaphysics,” by Emily Nussbaum:

    Joss Whedon, creator of a new TV series —

    “I’m a very hard-line, angry atheist” and
    “I want to invade people’s dreams.”

  • From “Check This,” by Wm. Ferguson:

    Garry Kasparov on chess

    “When the computer sees forced lines,
    it plays like God.”

Putting these quotations together, one is tempted to imagine God having a little game of chess with Whedon, along the lines suggested by C. S. Lewis:

As Lewis tells it the time had come for his “Adversary [as he was wont to speak of the God he had so earnestly sought to avoid] to make His final moves.” (C. S. Lewis, Surprised by Joy, Harcourt, Brace, and World, Inc., 1955, p. 216) Lewis called them “moves” because his life seemed like a chess match in which his pieces were spread all over the board in the most disadvantageous positions. The board was set for a checkmate….

For those who would like to imagine such a game (God vs. Whedon), the following may be helpful.

George Steiner has observed that

The common bond between chess, music, and mathematics may, finally, be the absence of language.

This quotation is apparently from

Fields of Force:
Fischer and Spassky at Reykjavik
. by George Steiner, Viking hardcover, June 1974.

George Steiner as quoted in a review of his book Grammars of Creation:

“I put forward the intuition, provisional and qualified, that the ‘language-animal’ we have been since ancient Greece so designated us, is undergoing mutation.”

The phrase “language-animal” is telling.  A Google search reveals that it is by no means a common phrase, and that Steiner may have taken it from Heidegger.  From another review, by Roger Kimball:

In ”Grammars of Creation,” for example, he tells us that ”the classical and Judaic ideal of man as ‘language animal,’ as uniquely defined by the dignity of speech . . . came to an end in the antilanguage of the death camps.”

This use of the Holocaust not only gives the appearance of establishing one’s credentials as a person of great moral gravity; it also stymies criticism. Who wants to risk the charge of insensitivity by objecting that the Holocaust had nothing to do with the ”ideal of man as ‘language animal’ ”?

Steiner has about as clear an idea of the difference between “classical” and “Judaic” ideals of man as did Michael Dukakis. (See my notes of September 9, 2002.)

Clearly what music, mathematics, and chess have in common is that they are activities based on pure form, not on language. Steiner is correct to that extent. The Greeks had, of course, an extremely strong sense of form, and, indeed, the foremost philosopher of the West, Plato, based his teachings on the notion of Forms. Jews, on the other hand, have based their culture mainly on stories… that is, on language rather than on form. The phrase “language-animal” sounds much more Jewish than Greek. Steiner is himself rather adept at the manipulation of language (and of people by means of language), but, while admiring form-based disciplines, is not particularly adept at them.

I would argue that developing a strong sense of form — of the sort required to, as Lewis would have it, play chess with God — does not require any “mutation,” but merely learning two very powerful non-Jewish approaches to thought and life: the Forms of Plato and the “archetypes” of Jung as exemplified by the 64 hexagrams of the 3,000-year-old Chinese classic, the I Ching.

For a picture of how these 64 Forms, or Hexagrams, might function as a chessboard,

click here.

Other relevant links:

“As you read, watch for patterns. Pay special attention to imagery that is geometric…”

and


from Shakhmatnaia goriachka

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