Log24

Monday, June 17, 2019

The Callahan Turtle

Filed under: General — Tags: , — m759 @ 10:54 PM

By Stephen King

IMAGE- Herbert John Ryser, 'Combinatorial Mathematics' (1963), page 1

Salvation by Grace

Filed under: General — Tags: , — m759 @ 3:18 PM

High Society

Filed under: General — Tags: , — m759 @ 1:16 PM

From the date in the above Google search result —

 See as well Jeremy Gray in this  journal.

Sunday, June 16, 2019

Master Plan from Outer Space

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

IMAGE- The large Desargues configuration and Desargues's theorem in light of Galois geometry

Eliot’s Perpetual Motion Structure*

Filed under: General — Tags: , — m759 @ 11:00 AM

From a date described by Peter Woit in his post
"Not So Spooky Action at a Distance" (June 11) —

See also The Lost Well.

 * "As a Chinese jar…." — Four Quartets

Saturday, May 4, 2019

The Chinese Jars of Shing-Tung Yau

Filed under: General — Tags: , , — m759 @ 11:00 AM

The title refers to Calabi-Yau spaces.

T. S. Eliot —

Four Quartets

. . . Only by the form, the pattern,
Can words or music reach
The stillness, as a Chinese jar still
Moves perpetually in its stillness.

A less "cosmic" but still noteworthy code — The Golay code.

This resides in a 12-dimensional space over GF(2).

Related material from Plato and R. T. Curtis

Counting symmetries with the orbit-stabilizer theorem

A related Calabi-Yau "Chinese jar" first described in detail in 1905

Illustration of K3 surface related to Mathieu moonshine

A figure that may or may not be related to the 4x4x4 cube that
holds the classical  Chinese "cosmic code" — the I Ching

ftp://ftp.cs.indiana.edu/pub/hanson/forSha/AK3/old/K3-pix.pdf

Wednesday, May 1, 2019

For the First of May

Filed under: General — Tags: — m759 @ 9:00 AM

"The purpose of mathematics cannot be derived from an activity 
inferior to it but from a higher sphere of human activity, namely,
religion."

 Igor Shafarevitch in 1973

"The hint half guessed, the gift half understood, is Incarnation."

— T. S. Eliot in Four Quartets

See also Ultron Cube.

Wednesday, January 16, 2019

Permutahedron Dream

Filed under: General — Tags: , , , — m759 @ 3:21 PM

The geometric object of the title appears in a post mentioning Bourgain 
in this journal.  Bourgain appears also in today's online New York Times —

https://www.nytimes.com/2019/01/16/
obituaries/jean-bourgain-dead.html
 .

Bourgain reportedly died on December 22.

An image from this journal on that date

Related poetic meditations —

IMAGE- Herbert John Ryser, 'Combinatorial Mathematics' (1963), page 1

Monday, October 9, 2017

Still Point for a Dance

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

"At the still point of the turning world. Neither flesh nor fleshless;
Neither from nor towards; at the still point, there the dance is,
But neither arrest nor movement. And do not call it fixity,
Where past and future are gathered. Neither movement from nor towards,
Neither ascent nor decline. Except for the point, the still point,
There would be no dance, and there is only the dance."

— T. S. Eliot, Four Quartets

See also a recurrent image
from this journal —

IMAGE- The ninefold square .

Wednesday, April 12, 2017

Contracting the Spielraum

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

The contraction of the title is from group actions on
the ninefold square  (with the center subsquare fixed)
to group actions on the eightfold cube.

From a post of June 4, 2014

At math.stackexchange.com on March 1-12, 2013:

Is there a geometric realization of the Quaternion group?” —

The above illustration, though neatly drawn, appeared under the
cloak of anonymity.  No source was given for the illustrated group actions.
Possibly they stem from my Log24 posts or notes such as the Jan. 4, 2012,
note on quaternion actions at finitegeometry.org/sc (hence ultimately
from my note “GL(2,3) actions on a cube” of April 5, 1985).

Saturday, May 1, 2010

An Education

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

天鈞

 

Made famous by Ursula K. Le Guin
as the book title "Lathe of Heaven,"
this Chinese phrase, tianjun, apparently
means something more like "Scales of Heaven"–
an appropriate image for Law Day 2010.

Image--Scales (the legal symbol)

An anonymous forum user says that

"…if you switch the two characters around,
you get: 鈞天, which is one of
the nine heavens, more specifically,
the middle heaven."

This is supported by a
non-anonymous source:

"I follow A.C. Graham’s translation of
Juntian as 'Level Heaven (the innermost
of the nine divisions of heaven)';
he renders Juntian guangyue as
'the mighty music of the innermost heaven.'"

— "Music in the World of Su Shi (1037-1101):
Terminology
," by Stuart H. Sargent,
Colorado State University,
Journal of Sung-Yuan Studies 32 (2002), 39-81

The Nine Divisions of Heaven–

Image-- Routledge Encyclopedia of Taoism, Vol. I, on the Nine Heavens, 'jiutian,' ed. by Fabrizio Pregadio

Some context–

The 3x3 ('ninefold') square

"This pattern is a square divided into nine equal parts.
It has been called the 'Holy Field' division and
was used throughout Chinese history for many
different purposes, most of which were connected
with things religious, political, or philosophical."

The Magic Square: Cities in Ancient China,
by Alfred Schinz, Edition Axel Menges, 1996, p. 71

Monday, October 31, 2005

Monday October 31, 2005

Filed under: General — Tags: , , — m759 @ 2:00 AM
Balance

The image “http://log24.com/log/pix03/030109-gridsmall.gif” cannot be displayed, because it contains errors.

"An asymmetrical balance is sought since it possesses more movement. This is achieved by the imaginary plotting of the character upon a nine-fold square, invented by some ingenious writer of the Tang dynasty. If the square were divided in half or in four, the result would be symmetrical, but the nine-fold square permits balanced asymmetry."

— Chiang Yee, Chinese Calligraphy,
   
quoted in Aspen no. 10, item 8

"'Burnt Norton' opens as a meditation on time. Many comparable and contrasting views are introduced. The lines are drenched with reminiscences of Heraclitus' fragments on flux and movement….  the chief contrast around which Eliot constructs this poem is that between the view of time as a mere continuum, and the difficult paradoxical Christian view of how man lives both 'in and out of time,' how he is immersed in the flux and yet can penetrate to the eternal by apprehending timeless existence within time and above it. But even for the Christian the moments of release from the pressures of the flux are rare, though they alone redeem the sad wastage of otherwise unillumined existence. Eliot recalls one such moment of peculiar poignance, a childhood moment in the rose-garden– a symbol he has previously used, in many variants, for the birth of desire. Its implications are intricate and even ambiguous, since they raise the whole problem of how to discriminate between supernatural vision and mere illusion. Other variations here on the theme of how time is conquered are more directly apprehensible. In dwelling on the extension of time into movement, Eliot takes up an image he had used in 'Triumphal March': 'at the still point of the turning world.' This notion of 'a mathematically pure point' (as Philip Wheelwright has called it) seems to be Eliot's poetic equivalent in our cosmology for Dante's 'unmoved Mover,' another way of symbolising a timeless release from the 'outer compulsions' of the world. Still another variation is the passage on the Chinese jar in the final section. Here Eliot, in a conception comparable to Wallace Stevens' 'Anecdote of the Jar,' has suggested how art conquers time:

       Only by the form, the pattern,
Can words or music reach
The stillness, as a Chinese jar still
Moves perpetually in its stillness."

— F. O. Matthiessen,
   The Achievement of T.S. Eliot,
   Oxford University Press, 1958,
   as quoted in On "Burnt Norton"

Thursday, January 9, 2003

Thursday January 9, 2003

Filed under: General — Tags: , — m759 @ 4:48 PM

Balanchine's Birthday

Today seems an appropriate day to celebrate Apollo and the nine Muses.

From a website on Balanchine's and Stravinsky's ballet, "Apollon Musagete":

In his Poetics of Music (1942) Stravinsky says: "Summing up: What is important for the lucid ordering of the work– for its crystallization– is that all the Dionysian elements which set the imagination of the artist in motion and make the life-sap rise must be properly subjugated before they intoxicate us, and must finally be made to submit to the law: Apollo demands it."  Stravinsky conceived Apollo as a ballet blanc– a "white ballet" with classical choreography and monochromatic attire. Envisioning the work in his mind's eye, he found that "the absence of many-colored hues and of all superfluities produced a wonderful freshness." Upon first hearing Apollo, Diaghilev found it "music somehow not of this world, but from somewhere else above." The ballet closes with an Apotheosis in which Apollo leads the Muses towards Parnassus. Here, the gravely beautiful music with which the work began is truly recapitulated "on high"– ceaselessly recycled, frozen in time.

— Joseph Horowitz

 

 

Another website invoking Apollo:

The icon that I use… is the nine-fold square…. The nine-fold square has centre, periphery, axes and diagonals.  But all are present only in their bare essentials.  It is also a sequence of eight triads.  Four pass through the centre and four do not.  This is the garden of Apollo, the field of Reason…. 

In accordance with these remarks, here is the underlying structure for a ballet blanc:

A version of 'grid3x3.gif.'

This structure may seem too simple to support movements of interest, but consider the following (click to enlarge):

As Sir Arthur Quiller-Couch, paraphrasing Horace, remarks in his Whitsun, 1939, preface to the new edition of the Oxford Book of English Verse, "tamen usque recurret Apollo."

The alert reader will note that in the above diagrams, only eight of the positions move.

Which muse remains at the center?

Consider the remark of T. S. Eliot, "At the still point, there the dance is," and the fact that on the day Eliot turned 60, Olivia Newton-John was born.  How, indeed, in the words of another "sixty-year-old smiling public man," can we know the dancer from the dance?
 

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