Log24

Friday, August 30, 2019

The Coxeter Aleph

Filed under: General — Tags: , , , — m759 @ 4:21 am

(Continued)

The previous post displayed part of a page from
a newspaper published the day Olivia Newton-John
turned 21 — Friday, September 26, 1969.

A meditation, with apologies to Coleridge:

In Xanadu did Newton-John
A stately pleasure-square decree
Where Aleph the sacred symbol ran
Through subsquares measureless to man.

A related video —

Beware, beware, her flashing eyes, her floating hair:

Set design —

As opposed to block design

Friday, January 7, 2011

Coxeter and the Aleph

Filed under: General,Geometry — Tags: , , , — m759 @ 10:31 am

In a nutshell —

Epigraph to "The Aleph," a 1945 story by Borges:

O God! I could be bounded in a nutshell,
and count myself a King of infinite space…
— Hamlet, II, 2

http://www.log24.com/log/pix11/110107-BorgesElAleph.jpg

The story in book form, 1949

A 2006 biography of geometer H.S.M. Coxeter:

http://www.log24.com/log/pix11/110107-KingOfInfiniteSpace-Sm.jpg

The Aleph (implicit in a 1950 article by Coxeter):

http://www.log24.com/log/pix11/110107-The1950Aleph-Sm.jpg

The details:

(Click to enlarge)

http://www.log24.com/log/pix11/110107-Aleph-Sm.jpg

Related material: Group Actions, 1984-2009.

Tuesday, May 14, 2013

Raiders of the Lost Aleph

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

See Coxeter + Aleph in this journal.

Epigraph to "The Aleph," a 1945 story by Borges:

"O God! I could be bounded in a nutshell,
and count myself a King of infinite space…"
– Hamlet, II, 2

Monday, February 20, 2012

Coxeter and the Relativity Problem

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

In the Beginning…

"As is well known, the Aleph is the first letter of the Hebrew alphabet."
– Borges, "The Aleph" (1945)

From some 1949 remarks of Weyl—

"The relativity problem is one of central significance throughout geometry and algebra and has been recognized as such by the mathematicians at an early time."

Hermann Weyl, "Relativity Theory as a Stimulus in Mathematical Research," Proceedings of the American Philosophical Society , Vol. 93, No. 7, Theory of Relativity in Contemporary Science: Papers Read at the Celebration of the Seventieth Birthday of Professor Albert Einstein in Princeton, March 19, 1949  (Dec. 30, 1949), pp. 535-541

Weyl in 1946—:

"This is the relativity problem: to fix objectively a class of equivalent coordinatizations and to ascertain the group of transformations S mediating between them."

— Hermann Weyl, The Classical Groups , Princeton University Press, 1946, p. 16

Coxeter in 1950 described the elements of the Galois field GF(9) as powers of a primitive root and as ordered pairs of the field of residue-classes modulo 3—

"… the successive powers of  the primitive root λ or 10 are

λ = 10,  λ2 = 21,  λ3 = 22,  λ4 = 02,
λ5 = 20,  λ6 = 12,  λ7 = 11,  λ8 = 01.

These are the proper coordinate symbols….

(See Fig. 10, where the points are represented in the Euclidean plane as if the coordinate residue 2 were the ordinary number -1. This representation naturally obscures the collinearity of such points as λ4, λ5, λ7.)"

http://www.log24.com/log/pix12/120220-CoxeterFig10.jpg

Coxeter's Figure 10 yields...

http://www.log24.com/log/pix11/110107-The1950Aleph-Sm.jpg

The Aleph

The details:

(Click to enlarge)

http://www.log24.com/log/pix11/110107-Aleph-Sm.jpg

Coxeter's phrase "in the Euclidean plane" obscures the noncontinuous nature of the transformations that are automorphisms of the above linear 2-space over GF(3).

Thursday, May 19, 2011

The Aleph, the Lottery, and the Eightfold Way

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

Three links with a Borges flavor—

Related material

The 236 in yesterday evening's NY lottery may be
 viewed as the 236 in March 18's Defining Configurations.
For some background, see Configurations and Squares.

A new illustration for that topic—

http://www.log24.com/log/pix11A/110519-8-3-Configuration.jpg

This shows a reconcilation of the triples described by Sloane
 in Defining Configurations with the square geometric
arrangement described by Coxeter in the Aleph link above.

Note that  the 56 from yesterday's midday NY lottery
describes the triples that appear both in the Eightfold Way
link above and also in a possible source for
the eight triples of  Sloane's 83 configuration—

http://www.log24.com/log/pix11A/110519-SloaneDesign.jpg

The geometric square arrangement discussed in the Aleph link
above appears in a different, but still rather Borgesian, context
in yesterday morning's Minimalist Icon.

Wednesday, August 9, 2023

“The Seed Crystal” — Plan 9 from Science Direct

Filed under: General — Tags: — m759 @ 2:48 am

From a 1990 novel —
http://www.log24.com/log/pix11A/110424-StoneJunction.jpg

Related remarks — Posts tagged The Coxeter Aleph.

Friday, July 30, 2021

The Secret Subterranean River

Filed under: General — Tags: — m759 @ 5:23 am

The "secret, subterranean river" of Shulevitz is
a flow of thought favorable to the cause of feminism,
but not necessarily to other "revolutionary" ideas.

Compare and contrast:

"Where Alph, the sacred river, ran"
— Coleridge, Kubla Khan

"Where Aleph the sacred symbol ran"
— Cullinane, "The Coxeter Aleph"

For group discussion:

How (if at all) is the "finitude" of Heidegger related to
mathematical  finitude and The King of Infinite Space ?

Monday, May 1, 2023

Chain of Title:  Complete, Clean, Unencumbered!

Filed under: General — Tags: — m759 @ 1:25 pm

From some Canadian legal boilerplate

E. Be able to provide complete, clean, unencumbered
chain of title for the Project, must have all the rights,
releases and clearances necessary to produce, own and
exploit the Project and for deployment of the Project . . . .

Weak Links in the Chain of Title —

http://www.log24.com/log/pix11C/111010-CollegeOfTheDesert-Seal.gif

A 2006 biography of geometer H.S.M. Coxeter:

http://www.log24.com/log/pix11/110107-KingOfInfiniteSpace-Sm.jpg

The Aleph (implicit in a 1950 article by Coxeter):

http://www.log24.com/log/pix11/110107-The1950Aleph-Sm.jpg

Click on images
for further details.

Sunday, April 10, 2022

Plan 9 Continues . . .

Filed under: General — Tags: , , — m759 @ 10:26 pm

A meditation on Coxeter's Aleph

'The 3x3 Magic Square as an Affine Transformation'

Saturday, July 3, 2021

Here, There, and Chicago

Filed under: General — Tags: , — m759 @ 9:07 pm

The above phrase "the intersection of storytelling and visual arts"
suggests a review . . .

Storytelling —

Visual arts —

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

IMAGE- The 3x3 ('ninefold') square as Chinese 'Holy Field'

"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

A Midrash for Michener —

IMAGE- Marie-Louise von Franz on the 'field' that represents 'the structural outlines of the collective unconscious'

For a connection of the above "Holy Field"
with pure mathematics, see Coxeter's Aleph.

Friday, July 5, 2019

The Motive for Metaphor

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

"János Bolyai was a nineteenth-century mathematician who
set the stage for the field  of non-Euclidean geometry."

Transylvania Now , October 26, 2018

 

From  Coxeter and the Relativity Problem

http://www.log24.com/log/pix11/110107-Aleph-Sm.jpg

Desiring the exhilarations of changes:
The motive for metaphor, shrinking from
The weight of primary noon,
The A B C of being,

The ruddy temper, the hammer
Of red and blue, the hard sound—
Steel against intimation—the sharp flash,
The vital, arrogant, fatal, dominant X.

Wallace Stevens, "The Motive for Metaphor"

Wednesday, August 9, 2017

Implosion

Filed under: General,Geometry — Tags: , — m759 @ 9:48 pm

For those whose only interest in mathematics
is as a path to the occult —

See also Coxeter's Aleph.

Monday, July 17, 2017

Athens Meets Jerusalem . . .

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

At the Googleplex .

For those whose only interest in higher mathematics
is as a path to the occult

Plato's Diamond and the Hebrew letter Aleph —

          

and some related (if only graphically) mathematics —

Click the above image for some related purely mathematical  remarks.

Tuesday, January 3, 2017

Cultist Space

Filed under: General,Geometry — Tags: , , — m759 @ 6:29 pm

The image of art historian Rosalind Krauss in the previous post
suggests a review of a page from her 1979 essay "Grids" —

The previous post illustrated a 3×3 grid. That  cultist space does
provide a place for a few "vestiges of the nineteenth century" —
namely, the elements of the Galois field GF(9) — to hide.
See Coxeter's Aleph in this journal.

Sunday, October 18, 2015

Coordinatization Problem

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

There are various ways to coordinatize a 3×3 array
(the Chinese "Holy Field'). Here are some —

See  Cullinane,  Coxeter,  and  Knight tour.

Saturday, November 16, 2013

Raiders of the Lost Theorem

Filed under: General,Geometry — Tags: , — m759 @ 11:30 am

IMAGE- The 'atomic square' in Lee Sallows's article 'The Lost Theorem'

Yes. See

The 48 actions of GL(2,3) on a 3×3 coordinate-array A,
when matrices of that group right-multiply the elements of A,
with A =

(1,1) (1,0) (1,2)
(0,1) (0,0) (0,2)
(2,1) (2,0) (2,2)

Actions of GL(2,p) on a pxp coordinate-array have the
same sorts of symmetries, where p is any odd prime.

Note that A, regarded in the Sallows manner as a magic square,
has the constant sum (0,0) in rows, columns, both diagonals, and  
all four broken diagonals (with arithmetic modulo 3).

For a more sophisticated approach to the structure of the
ninefold square, see Coxeter + Aleph.

Tuesday, August 13, 2013

The Story of N

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

(Continued from this morning)

http://www.log24.com/log/pix11/110107-The1950Aleph-Sm.jpg

The above stylized "N," based on
an 8-cycle in the 9-element Galois field
GF(9), may also be read as
an Aleph.

Graphic designers may prefer a simpler,
bolder version:

Sunday, December 9, 2012

Eve’s Menorah

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

"Now the serpent was more subtle
than any beast of the field…."
Genesis 3:1

"“The serpent’s eyes shine
As he wraps around the vine….”
Don Henley

"Nine is a vine."
Folk rhyme

Part I

Part II

Part III

Halloween 2005

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

Click images for some background.

Sunday, January 1, 2012

Sunday Shul

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

"… myths are stories, and like all narratives
they unravel through time, whereas grids
are not only spatial to start with,
they are visual structures that explicitly reject
a narrative or sequential reading of any kind."

— Rosalind Krauss in "Grids,"
October  (Summer 1979), 9: 50-64.

Counterexample—

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

The Ninefold Square

See Coxeter and the Aleph and Ayn Sof

Mathematics and Narrative, Illustrated
http://www.log24.com/log/pix11/110107-The1950Aleph-Sm.jpg

Mathematics
http://www.log24.com/log/pix11/110107-ScriptAlephSm.jpg
Narrative

Monday, October 10, 2011

10/10

Filed under: General — Tags: — m759 @ 10:00 am

See last year's Day of the Tetraktys.

Those who prefer Hebrew to Greek may consult Coxeter and the Aleph.

See also last midnight's The Aleph as well as Saturday morning's
An Ordinary Evening in Hartford and Saturday evening's
For Whom the Bell (with material from March 20, 2011).

For connoisseurs of synchronicity, there is …

THE LAST CONCERT

Cached from http://mrpianotoday.com/tourdates.htm —
The last concert of Roger Williams — March 20, 2011 —

   March 20

"Roger Williams" In Concert,
The Legendary Piano Man!!
Roger Williams & his Band
(Sierra Ballroom)
7:30-9:00pm

Palm Desert, CA    

Background music… Theme from "Somewhere in Time"

Wednesday, May 18, 2011

Minimalist Icon

Filed under: General,Geometry — Tags: — m759 @ 6:48 am

The source of the mysterious generic
3×3 favicon with one green cell

http://www.log24.com/log/pix11A/110518-GenericFavicon.jpg

— has been identified.

For minimalists, here is a purer 3×3 matrix favicon—

http://www.log24.com/log/pix11A/110518-3x3FaviconURL.jpg

This may, if one likes, be viewed as the "nothing"
present at the Creation.  See Jim Holt on physics.

See also Visualizing GL(2,p), Coxeter and the Aleph, and Ayn Sof.

Tuesday, May 10, 2011

Groups Acting

Filed under: General,Geometry — Tags: , , — m759 @ 10:10 am

The LA Times  on last weekend's film "Thor"—

"… the film… attempts to bridge director Kenneth Branagh's high-minded Shakespearean intentions with Marvel Entertainment's bottom-line-oriented need to crank out entertainment product."

Those averse to Nordic religion may contemplate a different approach to entertainment (such as Taymor's recent approach to Spider-Man).

A high-minded— if not Shakespearean— non-Nordic approach to groups acting—

"What was wrong? I had taken almost four semesters of algebra in college. I had read every page of Herstein, tried every exercise. Somehow, a message had been lost on me. Groups act . The elements of a group do not have to just sit there, abstract and implacable; they can do  things, they can 'produce changes.' In particular, groups arise naturally as the symmetries of a set with structure. And if a group is given abstractly, such as the fundamental group of a simplical complex or a presentation in terms of generators and relators, then it might be a good idea to find something for the group to act on, such as the universal covering space or a graph."

— Thomas W. Tucker, review of Lyndon's Groups and Geometry  in The American Mathematical Monthly , Vol. 94, No. 4 (April 1987), pp. 392-394

"Groups act "… For some examples, see

Related entertainment—

High-minded— Many Dimensions

Not so high-minded— The Cosmic Cube

http://www.log24.com/log/pix11A/110509-SpideySuperStories39Sm.jpg

One way of blending high and low—

The high-minded Charles Williams tells a story
in his novel Many Dimensions about a cosmically
significant cube inscribed with the Tetragrammaton—
the name, in Hebrew, of God.

The following figure can be interpreted as
the Hebrew letter Aleph inscribed in a 3×3 square—

http://www.log24.com/log/pix11A/110510-GaloisAleph.GIF

The above illustration is from undated software by Ed Pegg Jr.

For mathematical background, see a 1985 note, "Visualizing GL(2,p)."

For entertainment purposes, that note can be generalized from square to cube
(as Pegg does with his "GL(3,3)" software button).

For the Nordic-averse, some background on the Hebrew connection—

Friday, January 7, 2011

Ayn Sof

Filed under: General — Tags: , — m759 @ 7:26 pm

(A continuation of this morning's Coxeter and the Aleph)

"You've got to pick up every stitch… Must be the season of the witch."
Donovan song at the end of Nicole Kidman's "To Die For"

Mathematics and Narrative, Illustrated
http://www.log24.com/log/pix11/110107-The1950Aleph-Sm.jpg

Mathematics

http://www.log24.com/log/pix11/110107-ScriptAlephSm.jpg
Narrative

"As is well known, the Aleph is the first letter of the Hebrew alphabet.
Its use for the strange sphere in my story may not be accidental.
For the Kabbala, the letter stands for the En Soph ,
the pure and boundless godhead; it is also said that it takes
the shape of a man pointing to both heaven and earth, in order to show
that the lower world is the map and mirror of the higher; for Cantor's
Mengenlehre , it is the symbol of transfinite numbers,
of which any part is as great as the whole."

— Borges, "The Aleph"

From WorldLingo.com

Ein Sof

Ein Soph or Ayn Sof (Hebrew  אין סוף, literally "without end", denoting "boundlessness" and/or "nothingness"), is a Kabbalistic term that usually refers to an abstract state of existence preceding God's Creation of the limited universe. This Ein Sof , typically referred to figuratively as the "light of Ein Sof " ("Or Ein Sof "), is the most fundamental emanation manifested by God. The Ein Sof  is the material basis of Creation that, when focused, restricted, and filtered through the sefirot , results in the created, dynamic universe.
….

Cultural impact

Mathematician Georg Cantor labeled different sizes of infinity using the Aleph. The smallest size of infinity is aleph-null (0), the second size is aleph-one (1), etc. One theory about why Cantor chose to use the aleph is because it is the first letter of Ein-Sof. (See Aleph number)

"Infinite Jest… now stands as the principal contender
for what serious literature can aspire to
in the late twentieth and early twenty-first centuries."

All Things Shining, a work of pop philosophy published January 4th

http://www.log24.com/log/pix10B/101231-AllThingsShining-Cover.jpg

"You're gonna need a bigger boat." — Roy Scheider in "Jaws"

"We're gonna need more holy water." — "Season of the Witch," a film opening tonight

See also, with respect to David Foster Wallace, infinity, nihilism,
and the above reading of "Ayn Sof" as "nothingness,"
the quotations compiled as "Is Nothing Sacred?"

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