Log24

Friday, May 6, 2022

Interality and the I Ching

Filed under: General — Tags: , , — m759 @ 12:57 am

See "Flusser and the I Ching," by Peter Zhang.

Zhang has written extensively on the concept of "interality,"
a term coined by his colleague Geling Shang.

For interality as the mathematics underlying the natural
automorphism group of the I Ching, see my own work.

Friday, September 3, 2021

“The Home Cube, Where the Couple Reside”

Filed under: General — Tags: , — m759 @ 7:12 pm

From the post "Games" of Jan. 31, 2021 —

“Once Knecht confessed to his teacher that he wished to
learn enough to be able to incorporate the system of the
I Ching  into the Glass Bead Game. Elder Brother laughed.
‘Go ahead and try’, he exclaimed. ‘You’ll see how it turns out.
Anyone can create a pretty little bamboo garden in the world.
But I doubt the gardener would succeed in incorporating
the world in his bamboo grove’ ” (P. 139).

— Hermann Hesse, The Glass Bead Game (Magister Ludi) .
Translated by Richard and Clara Winston ( London, Vintage, 2000).

Tuesday, December 29, 2020

I Ching  Geometry

Filed under: General — Tags: , , — m759 @ 11:04 am

"Before time began, there was the Cube."
Hassenfeld Brothers cinematic merchandising slogan

Sunday, September 30, 2018

Iconology of the Eightfold Cube

Filed under: General,Geometry — Tags: , — m759 @ 11:13 am

Found today in an Internet image search, from the website of
an anonymous amateur mathematics enthusiast

Forming Gray codes in the eightfold cube with the eight
I Ching  trigrams (bagua ) —

Forming Gray codes in the eightfold cube with the eight I Ching trigrams (bagua)

This  journal on Nov. 7, 2016

A different sort of cube, from the makers of the recent
Netflix miniseries "Maniac" —

See also Rubik in this  journal.

Thursday, September 27, 2012

Kummer and the Cube

Filed under: General,Geometry — Tags: , , — m759 @ 7:11 pm

Denote the d-dimensional hypercube by  γd .

"… after coloring the sixty-four vertices of  γ6
alternately red and blue, we can say that
the sixteen pairs of opposite red vertices represent
the sixteen nodes of Kummer's surface, while
the sixteen pairs of opposite blue vertices
represent the sixteen tropes."

— From "Kummer's 16," section 12 of Coxeter's 1950
    "Self-dual Configurations and Regular Graphs"

Just as the 4×4 square represents the 4-dimensional
hypercube  γ4  over the two-element Galois field GF(2),
so the 4x4x4 cube represents the 6-dimensional
hypercube  γ6  over GF(2).

For religious interpretations, see
Nanavira Thera (Indian) and
I Ching  geometry (Chinese).

See also two professors in The New York Times
discussing images of the sacred in an op-ed piece
dated Sept. 26 (Yom Kippur).

Thursday, October 22, 2009

Chinese Cubes

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

From the Bulletin of the American Mathematical Society, Jan. 26, 2005:

What is known about unit cubes
by Chuanming Zong, Peking University

Abstract: Unit cubes, from any point of view, are among the simplest and the most important objects in n-dimensional Euclidean space. In fact, as one will see from this survey, they are not simple at all….

From Log24, now:

What is known about the 4×4×4 cube
by Steven H. Cullinane, unaffiliated

Abstract: The 4×4×4 cube, from one point of view, is among the simplest and the most important objects in n-dimensional binary space. In fact, as one will see from the links below, it is not simple at all.

Solomon's Cube

The Klein Correspondence, Penrose Space-Time, and a Finite Model

Non-Euclidean Blocks

Geometry of the I Ching

Related material:

Monday's entry Just Say NO and a poem by Stevens,

"The Well Dressed Man with a Beard."

Saturday, September 3, 2022

1984 Revisited

Filed under: General — m759 @ 2:46 pm

Cube Bricks 1984 —

An Approach to Symmetric Generation of the Simple Group of Order 168

Related material

Note the three quadruplets of parallel edges  in the 1984 figure above.

Further Reading

The above Gates article appeared earlier, in the June 2010 issue of
Physics World , with bigger illustrations. For instance —

Exercise: Describe, without seeing the rest of the article,
the rule used for connecting the balls above.

Wikipedia offers a much clearer picture of a (non-adinkra) tesseract —

      And then, more simply, there is the Galois tesseract

For parts of my own  world in June 2010, see this journal for that month.

The above Galois tesseract appears there as follows:

Image-- The Dream of the Expanded Field

See also the Klein correspondence in a paper from 1968
in yesterday's 2:54 PM ET post

Monday, August 1, 2022

Review

Filed under: General — Tags: , — m759 @ 5:12 am

From Log24 posts tagged Art Space —

From a paper on Kummer varieties,
arXiv:1208.1229v3 [math.AG] 12 Jun 2013,
The Universal Kummer Threefold,” by
Qingchun Ren, Steven V Sam, Gus Schrader,
and Bernd Sturmfels —

IMAGE- 'Consider the 6-dimensional vector space over the 2-element field,' from 'The Universal Kummer Threefold'

Two such considerations —

IMAGE- 'American Hustle' and Art Cube

IMAGE- Cube for study of I Ching group actions, with Jackie Chan and Nicole Kidman 

Wednesday, March 10, 2021

“Always with a little humor.” — Dr. Yen Lo

Filed under: General — m759 @ 12:03 pm

David Carradine displays a yellow book-- the Princeton I Ching.

Click on the Yellow Book.

Tuesday, December 22, 2020

Small Venues

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

“… her art was rarely exhibited until the 1970s,
and then only sporadically and in small venues . . . .”

— New York Times  obituary suggested by
today’s review,

https://www.nytimes.com/2020/12/22/
arts/artists-who-died-2020.html

“No ordinary venue.” — Song lyric

Related material now linked to in the previous post

David Carradine displays a yellow book-- the Princeton I Ching.

Click on the Yellow Book.

Tuesday, January 28, 2020

Very Stable Kool-Aid

Filed under: General — Tags: , , — m759 @ 2:16 pm

Two of the thumbnail previews
from yesterday's 1 AM  post

"Hum a few bars"

"For 6 Prescott Street"

Further down in the "6 Prescott St." post, the link 5 Divinity Avenue
leads to

A Letter from Timothy Leary, Ph.D., July 17, 1961

Harvard University
Department of Social Relations
Center for Research in Personality
Morton Prince House
5 Divinity Avenue
Cambridge 38, Massachusetts

July 17, 1961

Dr. Thomas S. Szasz
c/o Upstate Medical School
Irving Avenue
Syracuse 10, New York

Dear Dr. Szasz:

Your book arrived several days ago. I've spent eight hours on it and realize the task (and joy) of reading it has just begun.

The Myth of Mental Illness is the most important book in the history of psychiatry.

I know it is rash and premature to make this earlier judgment. I reserve the right later to revise and perhaps suggest it is the most important book published in the twentieth century.

It is great in so many ways–scholarship, clinical insight, political savvy, common sense, historical sweep, human concern– and most of all for its compassionate, shattering honesty.

. . . .

The small Morton Prince House in the above letter might, according to
the above-quoted remarks by Corinna S. Rohse, be called a "jewel box."
Harvard moved it in 1978 from Divinity Avenue to its current location at
6 Prescott Street.

Related "jewel box" material for those who
prefer narrative to mathematics —

"In The Electric Kool-Aid Acid Test , Tom Wolfe writes about encountering 
'a young psychologist,' 'Clifton Fadiman’s nephew, it turned out,' in the
waiting room of the San Mateo County jail. Fadiman and his wife were
'happily stuffing three I-Ching coins into some interminable dense volume*
of Oriental mysticism' that they planned to give Ken Kesey, the Prankster-
in-Chief whom the FBI had just nabbed after eight months on the lam.
Wolfe had been granted an interview with Kesey, and they wanted him to
tell their friend about the hidden coins. During this difficult time, they
explained, Kesey needed oracular advice."

— Tim Doody in The Morning News  web 'zine on July 26, 2012**

Oracular advice related to yesterday evening's
"jewel box" post …

A 4-dimensional hypercube H (a tesseract ) has 24 square
2-dimensional faces
.  In its incarnation as a Galois  tesseract
(a 4×4 square array of points for which the appropriate transformations
are those of the affine 4-space over the finite (i.e., Galois) two-element
field GF(2)), the 24 faces transform into 140 4-point "facets." The Galois 
version of H has a group of 322,560 automorphisms. Therefore, by the
orbit-stabilizer theorem, each of the 140 facets of the Galois version has
a stabilizer group of  2,304 affine transformations.

Similar remarks apply to the I Ching  In its incarnation as  
a Galois hexaract , for which the symmetry group — the group of
affine transformations of the 6-dimensional affine space over GF(2) —
has not 322,560 elements, but rather 1,290,157,424,640.

* The volume Wolfe mentions was, according to Fadiman, the I Ching.

** See also this  journal on that date — July 26, 2012.

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

Monday, March 11, 2019

Ant-Man Meets Doctor Strange

Filed under: General — m759 @ 1:22 pm

IMAGE- Concepts of Space

The 4×4 square may also be called the Galois Tesseract .
By analogy, the 4x4x4 cube may be called the Galois Hexeract .

"Think outside the tesseract.

Sunday, July 1, 2018

The Perpetual Motion of T. S. Eliot

Filed under: General — Tags: — m759 @ 10:28 am

IMAGE- Cube for study of I Ching group actions, with Jackie Chan and Nicole Kidman

Friday, June 29, 2018

For St. Stanley

Filed under: General,Geometry — m759 @ 1:26 pm

The phrase "Blue Dream" in the previous post
suggests a Web search for Traumnovelle .
That search yields an interesting weblog post
from 2014 commemorating the 1999 dies natalis 
(birth into heaven) of St. Stanley Kubrick.

Related material from March 7, 2014,
in this  journal

IMAGE- Cube for study of I Ching group actions, with Jackie Chan and Nicole Kidman 

That  2014 post was titled "Kummer Varieties." It is now tagged
"Kummerhenge." For some backstory, see other posts so tagged.

Tuesday, March 27, 2018

Compare and Contrast

Filed under: General,Geometry — Tags: , — m759 @ 4:28 pm

Weyl on symmetry, the eightfold cube, the Fano plane, and trigrams of the I Ching

Related material on automorphism groups —

The "Eightfold Cube" structure shown above with Weyl
competes rather directly with the "Eightfold Way" sculpture 
shown above with Bryant. The structure and the sculpture
each illustrate Klein's order-168 simple group.

Perhaps in part because of this competition, fans of the Mathematical
Sciences Research Institute (MSRI, pronounced "Misery') are less likely
to enjoy, and discuss, the eight-cube mathematical structure  above
than they are an eight-cube mechanical puzzle  like the one below.

Note also the earlier (2006) "Design Cube 2x2x2" webpage
illustrating graphic designs on the eightfold cube. This is visually,
if not mathematically, related to the (2010) "Expert's Cube."

Wednesday, March 7, 2018

Unite the Seven.

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


Related material —

The seven points of the Fano plane within 

The Eightfold Cube.
 

Weyl on symmetry, the eightfold cube, the Fano plane, and trigrams of the I Ching


"Before time began . . . ."

  — Optimus Prime

Wednesday, November 22, 2017

Goethe on All Souls’ Day

Filed under: General,Geometry — Tags: , — m759 @ 11:00 pm

David E. Wellbery on Goethe

From an interview published on 2 November 2017 at

http://literaturwissenschaft-berlin.de/interview-with-david-wellbery/

as later republished in 

https://thepointmag.com/2017/dialogue/
irreducible-significance-david-wellbery-literature-goethe-cavell
 —

 

The logo at left above is that of The Point .
The menu icon at right above is perhaps better
suited to illustrate Verwandlungslehre .

Weyl on symmetry, the eightfold cube, the Fano plane, and trigrams of the I Ching

Thursday, October 12, 2017

“But Back to the Action…”

Filed under: General — m759 @ 11:40 am

The title is from this morning's online New York Times  review
of a new Jackie Chan film.

Click the image below for some related posts.

IMAGE- Cube for study of I Ching group actions, with Jackie Chan and Nicole Kidman

Monday, July 24, 2017

Penguin Classics Deluxe Edition

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

The above title was suggested by a film trailer quoted here Saturday

" Jeremy Irons' dry Alfred Pennyworth:
'One misses the days when one's biggest concerns
were exploding wind-up penguins.' "

"Penguin Classics Deluxe Edition" describes, among other books,
an edition of the I Ching  published on December 1, 2015.

Excerpt from this journal on that date

Tuesday, December 1, 2015

Verhexung

Filed under: Uncategorized — m759 @ 9:00 PM 

(Continued)

"The positional meaning of a symbol derives from
its relationship to other symbols in a totality, a Gestalt,
whose elements acquire their significance from the
system as a whole."

— Victor Turner, The Forest of Symbols , Ithaca, NY,
Cornell University Press, 1967, p. 51, quoted by
Beth Barrie in "Victor Turner."

(Turner pioneered the use of the term "symbology,"
a term later applied by Dan Brown to a fictional
scholarly pursuit at Harvard.)

. . . .

Related material —

IMAGE by Cullinane- 'Solomon's Cube' with 64 identical, but variously oriented, subcubes, and six partitions of these 64 subcubes

The I Ching's underlying group has 1,290,157,424,640 permutations.

Wednesday, March 29, 2017

Art Space, Continued

Filed under: General — Tags: , , — m759 @ 4:35 am

"And as the characters in the meme twitch into the abyss
that is the sky, this meme will disappear into whatever
internet abyss swallowed MySpace."

—Staff writer Kamila Czachorowski, Harvard Crimson  today

From Log24 posts tagged Art Space

From a recent paper on Kummer varieties,
arXiv:1208.1229v3 [math.AG] 12 Jun 2013,
The Universal Kummer Threefold,” by
Qingchun Ren, Steven V Sam, Gus Schrader, and
Bernd Sturmfels —

IMAGE- 'Consider the 6-dimensional vector space over the 2-element field,' from 'The Universal Kummer Threefold'

Two such considerations —

IMAGE- 'American Hustle' and Art Cube

IMAGE- Cube for study of I Ching group actions, with Jackie Chan and Nicole Kidman 

Saturday, February 18, 2017

Verbum

Filed under: General,Geometry — Tags: , , — m759 @ 3:00 am

The Log24 version  (Nov. 9, 2005, and later posts) —

VERBUM
SAT
SAPIENTI

 

Escher's 'Verbum'

Escher's Verbum


Solomon's Cube

Solomon's Cube
 

I Ching hexagrams as parts of 4x4x4 cube

Geometry of the I Ching

The Warner Brothers version

The Paramount version

See also related material in the previous post, Transformers.

Thursday, January 12, 2017

Changes

Filed under: General,Geometry — Tags: , , — m759 @ 1:00 pm

Despite a remark at ichingpsychics.com, the I Ching's underlying group actually has 1,290,157,424,640 permutations.

Sunday, January 8, 2017

A Theory of Everything

Filed under: General,Geometry — Tags: , — m759 @ 7:11 pm

The title refers to the Chinese book the I Ching ,
the Classic of Changes .

The 64 hexagrams of the I Ching  may be arranged
naturally in a 4x4x4 cube. The natural form of transformations
("changes") of this cube is given by the diamond theorem.

A related post —

The Eightfold Cube, core structure of the I Ching

Saturday, October 31, 2015

Raiders of the Lost Crucible

Filed under: General,Geometry — Tags: , , — m759 @ 10:15 am

Stanford Encyclopedia of Philosophy
on the date Friday, April 5, 2013 —

Paraconsistent Logic

“First published Tue Sep 24, 1996;
substantive revision Fri Apr 5, 2013”

This  journal on the date Friday, April 5, 2013 —

The object most closely resembling a “philosophers’ stone”
that I know of is the eightfold cube .

For some related philosophical remarks that may appeal
to a general Internet audience, see (for instance) a website
by I Ching  enthusiast Andreas Schöter that displays a labeled
eightfold cube in the form of a lattice diagram —

Related material by Schöter —

A 20-page PDF, “Boolean Algebra and the Yi Jing.”
(First published in The Oracle: The Journal of Yijing Studies ,
Vol 2, No 7, Summer 1998, pp. 19–34.)

I differ with Schöter’s emphasis on Boolean algebra.
The appropriate mathematics for I Ching  studies is,
I maintain, not Boolean algebra  but rather Galois geometry.

See last Saturday’s post Two Views of Finite Space.
Unfortunately, that post is, unlike Schöter’s work, not
suitable for a general Internet audience.

Monday, December 1, 2014

Change Arises

Filed under: General,Geometry — Tags: — m759 @ 11:00 am

Flashback to St. Andrew's Day, 2013 —

Saturday, November 30, 2013

Waiting for Ogdoad

Filed under: Uncategorized — Tags:  — m759 @ 10:30 AM 

Continued from October 30 (Devil's Night), 2013.

“In a sense, we would see that change
arises from the structure of the object.”

— Theoretical physicist quoted in a
Simons Foundation article of Sept. 17, 2013

This suggests a review of mathematics and the
"Classic of Change ," the I Ching .

If the object is a cube, change arises from the fact
that the object has six  faces…

and is the unit cell for the six -dimensional
hyperspace H over the two-element field —

Spaces as Hypercubes

A different representation of the unit cell of
the hyperspace H (and of the I Ching ) —

Wednesday, August 27, 2014

Schau der Gestalt

Filed under: General,Geometry — Tags: , , , — m759 @ 5:01 am

(Continued from Aug. 19, 2014)

“Christian contemplation is the opposite
of distanced consideration of an image:
as Paul says, it is the metamorphosis of
the beholder into the image he beholds
(2 Cor 3.18), the ‘realisation’ of what the
image expresses (Newman). This is
possible only by giving up one’s own
standards and being assimilated to the
dimensions of the image.”

— Hans Urs von Balthasar,
The Glory of the Lord:
A Theological Aesthetics,

Vol. I: Seeing the Form
[ Schau der Gestalt ],
Ignatius Press, 1982, p. 485

A Bauhaus approach to Schau der Gestalt :

I prefer the I Ching ‘s approach to the laws of cubical space.

Monday, May 19, 2014

Rubik Quote

Filed under: General — Tags: — m759 @ 12:08 pm

“The Cube was born in 1974 as a teaching tool
to help me and my students better understand
space and 3D. The Cube challenged us to find
order in chaos.”

— Professor Ernő Rubik at Chrome Cube Lab

For a Chinese approach to order and chaos,
see I Ching  Cube in this journal.

Friday, May 9, 2014

Models of Everything

Filed under: General,Geometry — Tags: , — m759 @ 11:16 am

“The About page contains detailed descriptions of the project….”

The Illustris project on constructing a model of the universe

For the mathematics of a simpler traditional Chinese model
of everything, see

Friday, March 7, 2014

Kummer Varieties

Filed under: General,Geometry — Tags: , , — m759 @ 11:20 am

The Dream of the Expanded Field continues

Image-- The Dream of the Expanded Field

From Klein's 1893 Lectures on Mathematics —

"The varieties introduced by Wirtinger may be called Kummer varieties…."
E. Spanier, 1956

From this journal on March 10, 2013 —

From a recent paper on Kummer varieties,
arXiv:1208.1229v3 [math.AG] 12 Jun 2013,
"The Universal Kummer Threefold," by
Qingchun Ren, Steven V Sam, Gus Schrader, and Bernd Sturmfels —

IMAGE- 'Consider the 6-dimensional vector space over the 2-element field,' from 'The Universal Kummer Threefold'

Two such considerations —

IMAGE- 'American Hustle' and Art Cube

IMAGE- Cube for study of I Ching group actions, with Jackie Chan and Nicole Kidman 

Update of 10 PM ET March 7, 2014 —

The following slides by one of the "Kummer Threefold" authors give
some background related to the above 64-point vector space and
to the Weyl group of type E7(E7):

The Cayley reference is to "Algorithm for the characteristics of the
triple ϑ-functions," Journal für die Reine und Angewandte
Mathematik  87 (1879): 165-169. <http://eudml.org/doc/148412>.
To read this in the context of Cayley's other work, see pp. 441-445
of Volume 10 of his Collected Mathematical Papers .

Monday, December 9, 2013

Heaven Descending

An I Ching  study quoted in Waiting for Ogdoad (St. Andrew's Day, 2013)—

(Click for clearer image.)

The author of the above I Ching  study calls his lattice "Arising Heaven."

The following lattice might, therefore, be called "Heaven Descending."

IMAGE- Construction of 'Heaven Descending' lattice

Click for the source, mentioned in Anatomy of a Cube (Sept. 18, 2011).

Saturday, November 30, 2013

Waiting for Ogdoad

Filed under: General,Geometry — Tags: , , , — m759 @ 10:30 am

Continued from October 30 (Devil’s Night), 2013.

“In a sense, we would see that change
arises from the structure of the object.”

— Theoretical physicist quoted in a
Simons Foundation article of Sept. 17, 2013

This suggests a review of mathematics and the
Classic of Change ,” the I Ching .

The physicist quoted above was discussing a rather
complicated object. His words apply to a much simpler
object, an embodiment of the eight trigrams underlying
the I Ching  as the corners of a cube.

The Eightfold Cube and its Inner Structure

See also

(Click for clearer image.)

The Cullinane image above illustrates the seven points of
the Fano plane as seven of the eight I Ching  trigrams and as
seven natural ways of slicing the cube.

For a different approach to the mathematics of cube slices,
related to Gauss’s composition law for binary quadratic forms,
see the Bhargava cube  in a post of April 9, 2012.

Monday, October 14, 2013

Dream of the Expanded Field

Filed under: General,Geometry — m759 @ 8:28 pm

(Continued)

IMAGE- Cube for study of I Ching group actions, with Jackie Chan and Nicole Kidman

Further context: Galois I Ching

Thursday, June 13, 2013

Gate

Filed under: General,Geometry — Tags: , , , — m759 @ 2:13 pm

"Eight is a Gate." — Mnemonic rhyme

Today's previous post, Window, showed a version
of the Chinese character for "field"—

This suggests a related image

The related image in turn suggests

Unlike linear perspective, axonometry has no vanishing point,
and hence it does not assume a fixed position by the viewer.
This makes axonometry 'scrollable'. Art historians often speak of
the 'moving' or 'shifting' perspective in Chinese paintings.

Axonometry was introduced to Europe in the 17th century by
Jesuits returning from China.

Jan Krikke

As was the I Ching.  A related structure:

Monday, December 24, 2012

Eternal Recreation

Memories, Dreams, Reflections
by C. G. Jung

Recorded and edited By Aniela Jaffé, translated from the German
by Richard and Clara Winston, Vintage Books edition of April 1989

From pages 195-196:

"Only gradually did I discover what the mandala really is:
'Formation, Transformation, Eternal Mind's eternal recreation.'*
And that is the self, the wholeness of the personality, which if all
goes well is harmonious, but which cannot tolerate self-deceptions."

* Faust , Part Two, trans. by Philip Wayne (Harmondsworth,
England, Penguin Books Ltd., 1959), p. 79. The original:

                   … Gestaltung, Umgestaltung, 
  Des ewigen Sinnes ewige Unterhaltung….

Jung's "Formation, Transformation" quote is from the realm of
the Mothers (Faust Part Two, Act 1, Scene 5: A Dark Gallery).
The speaker is Mephistopheles.

See also Prof. Bruce J. MacLennan on this realm
in a Web page from his Spring 2005 seminar on Faust:

"In alchemical terms, F is descending into the dark, formless
primary matter from which all things are born. Psychologically
he is descending into the deepest regions of the
collective unconscious, to the source of life and all creation.
Mater (mother), matrix (womb, generative substance), and matter
all come from the same root. This is Faust's next encounter with
the feminine, but it's obviously of a very different kind than his
relationship with Gretchen."

The phrase "Gestaltung, Umgestaltung " suggests a more mathematical
approach to the Unterhaltung . Hence

Part I: Mothers

"The ultimate, deep symbol of motherhood raised to
the universal and the cosmic, of the birth, sending forth,
death, and return of all things in an eternal cycle,
is expressed in the Mothers, the matrices of all forms,
at the timeless, placeless originating womb or hearth
where chaos is transmuted into cosmos and whence
the forms of creation issue forth into the world of
place and time."

— Harold Stein Jantz, The Mothers in Faust:
The Myth of Time and Creativity 
,
Johns Hopkins Press, 1969, page 37

Part II: Matrices

        

Part III: Spaces and Hypercubes

Click image for some background.

Part IV: Forms

Forms from the I Ching :

Click image for some background.

Forms from Diamond Theory :

Click image for some background.

Wednesday, November 7, 2012

Board

Filed under: General,Geometry — m759 @ 1:00 pm

In several posts now tagged Chessboard
an I Ching chessboard
image in which adjacent squares 
have the Karnaugh property

— has been replaced by a picture of
the original 1989 version in which
the Karnaugh property applies only to cells
that are adjacent in a cubic, not square,
arrangement—

I Ching chessboard (original 1989 arrangement)

Wednesday, January 19, 2011

Intermediate Cubism

Filed under: General,Geometry — Tags: , — m759 @ 2:22 pm

The following is a new illustration for Cubist Geometries

IMAGE- A Galois cube: model of the 27-point affine 3-space

(For elementary cubism, see Pilate Goes to Kindergarten and The Eightfold Cube.
 For advanced, see Solomon's Cube and Geometry of the I Ching .)

Cézanne's Greetings.

Wednesday, June 30, 2010

Field Dream

Filed under: General,Geometry — Tags: , , , , — m759 @ 10:23 am

In memory of Wu Guanzhong, Chinese artist who died in Beijing on Friday

Image-- The Dream of the Expanded Field

"Once Knecht confessed to his teacher that he wished to learn enough to be able to incorporate the system of the I Ching into the Glass Bead Game.  Elder Brother laughed.  'Go ahead and try,' he exclaimed.  'You'll see how it turns out.  Anyone can create a pretty little bamboo garden in the world.  But I doubt that the gardener would succeed in incorporating the world in his bamboo grove.'"

— Hermann Hesse, The Glass Bead Game, translated by Richard and Clara Winston

"The Chinese painter Wu Tao-tzu was famous because he could paint nature in a unique realistic way that was able to deceive all who viewed the picture. At the end of his life he painted his last work and invited all his friends and admirers to its presentation. They saw a wonderful landscape with a romantic path, starting in the foreground between flowers and moving through meadows to high mountains in the background, where it disappeared in an evening fog. He explained that this picture summed up all his life’s work and at the end of his short talk he jumped into the painting and onto the path, walked to the background and disappeared forever."

Jürgen Teichmann. Teichmann notes that "the German poet Hermann Hesse tells a variation of this anecdote, according to his own personal view, as found in his 'Kurzgefasster Lebenslauf,' 1925."

Wednesday, June 16, 2010

Brightness at Noon

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

David Levine's portrait of Arthur Koestler (see Dec. 30, 2009) —

Image-- Arthur Koestler by David Levine, NY Review of Books, Dec. 17, 1964, review of 'The Act of Creation'

Image-- Escher's 'Verbum'

Escher’s Verbum

Image-- Solomon's Cube

Solomon’s Cube

Image-- The 64 I Ching hexagrams in the 4 layers of the Cullinane cube

Geometry of the I Ching

See also this morning's post as well as
Monday's post quoting George David Birkhoff

"If I were a Leibnizian mystic… I would say that…
God thinks multi-dimensionally — that is,
uses multi-dimensional symbols beyond our grasp."

Geometry of Language

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

(Continued from April 23, 2009, and February 13, 2010.)

Paul Valéry as quoted in yesterday’s post:

“The S[elf] is invariant, origin, locus or field, it’s a functional property of consciousness” (Cahiers, 15:170 [2: 315])

The geometric example discussed here yesterday as a Self symbol may seem too small to be really impressive. Here is a larger example from the Chinese, rather than European, tradition. It may be regarded as a way of representing the Galois field GF(64). (“Field” is a rather ambiguous term; here it does not, of course, mean what it did in the Valéry quotation.)

From Geometry of the I Ching

Image-- The 64 hexagrams of the I Ching

The above 64 hexagrams may also be regarded as
the finite affine space AG(6,2)— a larger version
of the finite affine space AG(4,2) in yesterday’s post.
That smaller space has a group of 322,560 symmetries.
The larger hexagram  space has a group of
1,290,157,424,640 affine symmetries.

From a paper on GL(6,2), the symmetry group
of the corresponding projective  space PG(5,2),*
which has 1/64 as many symmetries—

(Click to enlarge.)

Image-- Classes of the Group GL(6,)

For some narrative in the European  tradition
related to this geometry, see Solomon’s Cube.

* Update of July 29, 2011: The “PG(5,2)” above is a correction from an earlier error.

Sunday, March 21, 2010

Galois Field of Dreams

Filed under: General,Geometry — Tags: , — m759 @ 10:01 am

It is well known that the seven (22 + 2 +1) points of the projective plane of order 2 correspond to 2-point subspaces (lines) of the linear 3-space over the two-element field Galois field GF(2), and may be therefore be visualized as 2-cube subsets of the 2×2×2 cube.

Similarly, recent posts* have noted that the thirteen (32 + 3 + 1) points of the projective plane of order 3 may be seen as 3-cube subsets in the 3×3×3 cube.

The twenty-one (42 + 4 +1) points of the (unique) projective plane of order 4 may also be visualized as subsets of a cube– in this case, the 4×4×4 cube. This visualization is somewhat more complicated than the 3×3×3 case, since the 4×4×4 cube has no central subcube, and each projective-plane point corresponds to four, not three, subcubes.

These three cubes, with 8, 27, and 64 subcubes, thus serve as geometric models in a straightforward way– first as models of finite linear spaces, hence as models for small Galois geometries derived from the linear spaces. (The cubes with 8 and 64 subcubes also serve in a less straightforward, and new, way as finite-geometry models– see The Eightfold Cube, Block Designs, and Solomon's Cube.)

A group of collineations** of the 21-point plane is one of two nonisomorphic simple groups of order 20,160. The other is the linear group acting on the linear 4-space over the two-element Galois field  GF(2). The 1899 paper establishing the nonisomorphism notes that "the expression Galois Field is perhaps not yet in general use."

Coordinates of the 4×4×4 cube's subcubes can, of course, be regarded as elements of the Galois field GF(64).

The preceding remarks were purely mathematical. The "dreams" of this post's title are not. See…

Number and Time, by Marie-Louise von Franz

See also Geometry of the I Ching and a search in this journal for "Galois + Ching."

* February 27 and March 13

** G20160 in Mitchell 1910,  LF(3,22) in Edge 1965

— Mitchell, Ulysses Grant, "Geometry and Collineation Groups
   of the Finite Projective Plane PG(2,22),"
   Princeton Ph.D. dissertation (1910)

— Edge, W. L., "Some Implications of the Geometry of
   the 21-Point Plane," Math. Zeitschr. 87, 348-362 (1965)

Monday, September 7, 2009

Monday September 7, 2009

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

Magic Boxes

"Somehow it seems to fill my head with ideas– only I don't exactly know what they are!…. Let's have a look at the garden first!"

— A passage from Lewis Carroll's Through the Looking-Glass. The "garden" part– but not the "ideas" part– was quoted by Jacques Derrida in Dissemination in the epigraph to Chapter 7, "The Time before First."

Commentary
 on the passage:

Part I    "The Magic Box,"  shown on Turner Classic Movies earlier tonight

Part II: "Mimsy Were the Borogoves," a classic science fiction story:

"… he lifted a square, transparent crystal block, small enough to cup in his palm– much too small to contain the maze of apparatus within it. In a moment Scott had solved that problem. The crystal was a sort of magnifying glass, vastly enlarging the things inside the block. Strange things they were, too. Miniature people, for example– They moved. Like clockwork automatons, though much more smoothly. It was rather like watching a play."

Part III:  A Crystal Block

Cube, 4x4x4

Four coloring pencils, of four different colors

Image of pencils is by
Diane Robertson Design.

Related material:
"A Four-Color Theorem."

Part IV:

David Carradine displays a yellow book-- the Princeton I Ching.

"Click on the Yellow Book."

Saturday, September 5, 2009

Saturday September 5, 2009

Filed under: General,Geometry — Tags: , , — m759 @ 10:31 pm
For the
Burning Man

'The Stars My Destination,' current edition (with cover slightly changed)

(Cover slightly changed.)

 
Background —

 
SAT
 
Part I:

Sophists (August 20th)

Part II:

VERBUM
SAT
SAPIENTI

Escher's 'Verbum'

Escher's Verbum


Solomon's Cube



Part III:

From August 25th

Equilateral triangle on a cube, each side's length equal to the square root of two

"Boo, boo, boo,
  square root of two.
"

Tuesday, February 17, 2009

Tuesday February 17, 2009

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

Diamond-Faceted:
Transformations
of the Rock

A discussion of Stevens's late poem "The Rock" (1954) in Wallace Stevens: A World of Transforming Shapes, by Alan D. Perlis, Bucknell University Press, 1976, p. 120:

For Stevens, the poem "makes meanings of the rock." In the mind, "its barrenness becomes a thousand things/And so exists no more." In fact, in a peculiar irony that only a poet with Stevens's particular notion of the imagination's function could develop, the rock becomes the mind itself, shattered into such diamond-faceted brilliance that it encompasses all possibilities for human thought:

The rock is the gray particular of man's life,
The stone from which he rises, up—and—ho,
The step to the bleaker depths of his descents ...

The rock is the stern particular of the air,
The mirror of the planets, one by one,
But through man's eye, their silent rhapsodist,

Turquoise the rock, at odious evening bright
With redness that sticks fast to evil dreams;
The difficult rightness of half-risen day.

The rock is the habitation of the whole,
Its strength and measure, that which is near,
     point A
In a perspective that begins again

At B: the origin of the mango's rind.

                    (Collected Poems, 528)

A mathematical version of
this poetic concept appears
in a rather cryptic note
from 1981 written with
Stevens's poem in mind:

http://www.log24.com/log/pix09/090217-SolidSymmetry.jpg

For some explanation of the
groups of 8 and 24
motions referred to in the note,
see an earlier note from 1981.

For the Perlis "diamond facets,"
see the Diamond 16 Puzzle.

For a much larger group
of motions, see
Solomon's Cube.

As for "the mind itself"
and "possibilities for
human thought," see
Geometry of the I Ching.

Monday, December 8, 2008

Monday December 8, 2008

Filed under: General — Tags: — m759 @ 10:12 am

An Indiana Jones Xmas
continues…

 

Chalice, Grail,
Whatever

 

Last night on TNT:
The Librarian Part 3:
Curse of the Judas Chalice,
in which The Librarian
encounters the mysterious
Professor Lazlo

Related material:

An Arthur Waite quotation
from the Feast of St. Nicholas:

“It is like the lapis exilis of
the German Graal legend”

as well as
yesterday’s entry
relating Margaret Wertheim’s
Pearly Gates of Cyberspace:
A History of Space from
Dante to the Internet

 to a different sort of space–
that of the I Ching— and to
Professor Laszlo Lovasz’s
cube space

David Carradine displays a yellow book-- the Princeton I Ching.

“Click on the Yellow Book.”

Happy birthday, David Carradine.

Sunday, December 7, 2008

Sunday December 7, 2008

Filed under: General,Geometry — Tags: — m759 @ 11:00 am
Space and
 the Soul

On a book by Margaret Wertheim:

“She traces the history of space beginning with the cosmology of Dante. Her journey continues through the historical foundations of celestial space, relativistic space, hyperspace, and, finally, cyberspace.” –Joe J. Accardi, Northeastern Illinois Univ. Lib., Chicago, in Library Journal, 1999 (quoted at Amazon.com)

There are also other sorts of space.

Froebel's third gift, the eightfold cube
© 2005 The Institute for Figuring

Photo by Norman Brosterman
fom the Inventing Kindergarten
exhibit at The Institute for Figuring
(co-founded by Margaret Wertheim)

This photo may serve as an
introduction to a different
sort of space.

See The Eightfold Cube.

For the religious meaning
of this small space, see

Richard Wilhelm on
the eight I Ching trigrams
.

For a related larger space,
see the entry and links of
 St. Augustine’s Day, 2006.

Monday, October 6, 2008

Monday October 6, 2008

Filed under: General,Geometry — Tags: — m759 @ 1:26 pm
Leap Day of Faith

Yesterday's entry contained the following unattributed quotation:

"One must join forces with friends of like mind."

As the link to Leap Day indicated, the source of the quotation is the I Ching.

Yesterday's entry also quoted the late Terence McKenna, a confused writer on psychosis and the I Ching. Lest the reader conclude that I consider McKenna or similar authors (for instance, Timothy Leary in Cuernavaca) as "friends of like mind," I would point rather to more sober students of the I Ching (cf. my June 2002 notes on philosophy, religion, and science) and to the late Scottish theologian John Macquarrie:


The Rev. John Macquarrie, Scottish Theologian, Dies at 87

Macquarrie's connection in this journal to the I Ching is, like that book itself, purely coincidental.  For details, click on the figure below.
 

A 4x4x4 cube

The persistent reader will
find a further link that
leads to an entry titled
"Notes on the I Ching."

 

McKenna's writing was of value to me for its (garbled) reference to a thought of Alfred North Whitehead:

"A colour is eternal.  It haunts time like a spirit.  It comes and it goes.  But where it comes it is the same colour.  It neither survives nor does it live.  It appears when it is wanted."

Science and the Modern World, 1925

Friday, August 8, 2008

Friday August 8, 2008

Filed under: General,Geometry — Tags: — m759 @ 8:08 am
Weyl on symmetry, the eightfold cube, the Fano plane, and trigrams of the I Ching

Click on image for details.

Wednesday, July 9, 2008

Wednesday July 9, 2008

Filed under: General,Geometry — m759 @ 8:28 am
God, Time, Epiphany

8:28:32 AM

Anthony Hopkins, from
All Hallows' Eve
last year
:

"For me time is God,
God is time. It's an equation,
like an Einstein equation."

James Joyce, from
June 26 (the day after
AntiChristmas) this year
:

"… he glanced up at the clock
of the Ballast Office and smiled:
— It has not epiphanised yet,
he said."

Ezra Pound (from a page
linked to yesterday morning):

"It seems quite natural to me
that an artist should have
just as much pleasure in an
arrangement of planes
or in a pattern of figures,
  as in painting portraits…."

From Epiphany 2008:

An arrangement of planes:

http://www.log24.com/log/pix08/080709-Epiphany.gif

From May 10, 2008:

A pattern of figures:

 

Seven partitions of the 2x2x2 cube in 'Paradise of Childhood'

See also Richard Wilhelm on
Hexagram 32 of the I Ching:

 

"Duration is a state whose movement is not worn down by hindrances. It is not a state of rest, for mere standstill is regression. Duration is rather the self-contained and therefore self-renewing movement of an organized, firmly integrated whole, taking place in accordance with immutable laws and beginning anew at every ending. The end is reached by an inward movement, by inhalation, systole, contraction, and this movement turns into a new beginning, in which the movement is directed outward, in exhalation, diastole, expansion."

 

'The Middle-English Harrowing of Hell,' by Hulme, 1907, page 64, line 672: 'with this he gaf the gaste'

The Middle-English
    Harrowing of Hell…

    by Hulme, 1907, page 64

Monday, June 16, 2008

Monday June 16, 2008

Filed under: General,Geometry — Tags: , , — m759 @ 9:00 pm
Bloomsday for Nash:
The Revelation Game

(American Mathematical Society Feb. 2008
review of Steven Brams’s Superior Beings:
If They Exist, How Would We Know?)

(pdf, 15 megabytes)

"Brams does not attempt to prove or disprove God. He uses elementary ideas from game theory to create situations between a Person (P) and God (Supreme Being, SB) and discusses how each reacts to the other in these model scenarios….

In the 'Revelation Game,' for example, the Person (P) has two options:
1) P can believe in SB's existence
2) P can not believe in SB's existence
The Supreme Being also has two options:
1) SB can reveal Himself
2) SB can not reveal Himself

Each player also has a primary and secondary goal. For the Person, the primary goal is to have his belief (or non-belief) confirmed by evidence (or lack thereof). The secondary goal is to 'prefer to believe in SB’s existence.' For the Supreme Being, the primary goal is to have P believe in His existence, while the secondary goal is to not reveal Himself. These goals allow us to rank all the outcomes for each player from best (4) to worst (1). We end up with a matrix as follows (the first number in the parentheses represents the SB's ranking for that box; the second number represents P's ranking):

Revelation Game payoff matrix

The question we must answer is: what is the Nash equilibrium in this case?"

Analogously:

Lotteries on
Bloomsday,
June 16,
2008
Pennsylvania
(No revelation)
New York
(Revelation)
Mid-day
(No belief)
418

 

The Exorcist

No belief,
no revelation

064

 

4x4x4 cube summarizing geometry of the I Ching

Revelation
without belief

Evening
(Belief)
709

Human Conflict Number Five album by The 10,000 Maniacs

 

Belief without
revelation

198

 

(A Cheap
Epiphany)

Black disc from end of Ch. 17 of Ulysses

Belief and
revelation

The holy image

Black disc from end of Ch. 17 of Ulysses

denoting belief and revelation
may be interpreted as
a black hole or as a
symbol by James Joyce:

When?

Going to dark bed there was a square round Sinbad the Sailor roc's auk's egg in the night of the bed of all the auks of the rocs of Darkinbad the Brightdayler.

Where?

Black disc from end of Ch. 17 in Ulysses

Ulysses, conclusion of Chapter 17

Thursday, May 22, 2008

Thursday May 22, 2008

Filed under: General,Geometry — Tags: , — m759 @ 9:00 am
The Undertaking:
An Exercise in
Conceptual Art

I Ching hexagram 54: The Marrying Maiden

Hexagram 54:
THE JUDGMENT

Undertakings bring misfortune.
Nothing that would further.

The image “http://www.log24.com/log/pix08/080522-Irelandslide1.jpg” cannot be displayed, because it contains errors.

Brian O’Doherty, an Irish-born artist,
before the [Tuesday, May 20] wake
of his alter ego* ‘Patrick Ireland’
on the grounds of the
Irish Museum of Modern Art.”
New York Times, May 22, 2008    

THE IMAGE

Thus the superior man
understands the transitory
in the light of
the eternity of the end.

Another version of
the image:

Images of time and eternity in memory of Michelangelo
See 2/22/08
and  4/19/08.


Related material:

Michael Kimmelman in today’s New York Times

“An essay from the ’70s by Mr. O’Doherty, ‘Inside the White Cube,’ became famous in art circles for describing how modern art interacted with the gallery spaces in which it was shown.”

Brian O’Doherty, “Inside the White Cube,” 1976 Artforum essays on the gallery space and 20th-century art:

“The history of modernism is intimately framed by that space. Or rather the history of modern art can be correlated with changes in that space and in the way we see it. We have now reached a point where we see not the art but the space first…. An image comes to mind of a white, ideal space that, more than any single picture, may be the archetypal image of 20th-century art.”

An archetypal image

THE SPACE:

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

A non-archetypal image

THE ART:

Jack in the Box, by Natasha Wescoat

Natasha Wescoat, 2004
See also Epiphany 2008:

How the eightfold cube works

“Nothing that would further.”
— Hexagram 54

Lear’s fool:

 …. Now thou art an 0
without a figure. I am better
than thou art, now. I am a fool;
thou art nothing….

“…. in the last mystery of all the single figure of what is called the World goes joyously dancing in a state beyond moon and sun, and the number of the Trumps is done.  Save only for that which has no number and is called the Fool, because mankind finds it folly till it is known.  It is sovereign or it is nothing, and if it is nothing then man was born dead.”

The Greater Trumps,
by Charles Williams, Ch. 14

* For a different, Jungian, alter ego, see Irish Fourplay (Jan. 31, 2003) and “Outside the Box,” a New York Times review of O’Doherty’s art (featuring a St. Bridget’s Cross) by Bridget L. Goodbody dated April 25, 2007. See also Log24 on that date.

Thursday, July 5, 2007

Thursday July 5, 2007

Filed under: General,Geometry — Tags: — m759 @ 7:11 pm

In Defense of
Plato’s Realism

(vs. sophists’ nominalism–
see recent entries.)

Plato cited geometry,
notably in the Meno,
in defense of his realism.
Consideration of the
Meno’s diamond figure
leads to the following:

The Eightfold Cube and its Inner Structure

Click on image for details.

As noted in an entry,
Plato, Pegasus, and
the Evening Star,

linked to
at the end of today’s
previous entry,
the “universals”
of Platonic realism
are exemplified by
the hexagrams of
the I Ching,
which in turn are
based on the seven
trigrams above and
on the eighth trigram,
of all yin lines,
not shown above:

Trigram of K'un, the Receptive

K’un
The Receptive

_____________________________________________

Update of Nov. 30, 2013:

From  a little-known website in Kuala Lumpur:
(Click to enlarge.)

The remarks on Platonic realism are from Wikipedia.
The eightfold cube is apparently from this post.

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)
 

Monday, May 28, 2007

Monday May 28, 2007

Filed under: General,Geometry — Tags: , , , — m759 @ 5:00 pm
Space-Time

and a Finite Model

Notes by Steven H. Cullinane
May 28, 2007

Part I: A Model of Space-Time

The following paper includes a figure illustrating Penrose’s model of  “complexified, compactified Minkowski space-time as the Klein quadric in complex projective 5-space.”
 
The image “http://www.log24.com/log/pix07/070528-Twistor.jpg” cannot be displayed, because it contains errors.

Click on picture to enlarge.

For some background on the Klein quadric and space-time, see Roger Penrose, “On the Origins of Twistor Theory,” from Gravitation and Geometry: A Volume in Honor of Ivor Robinson, Bibliopolis, 1987.


Part II: A Corresponding Finite Model

 

The Klein quadric also occurs in a finite model of projective 5-space.  See a 1910 paper:

G. M. Conwell, The 3-space PG(3,2) and its group, Ann. of Math. 11, 60-76.

Conwell discusses the quadric, and the related Klein correspondence, in detail.  This is noted in a more recent paper by Philippe Cara:

The image “http://www.log24.com/log/pix07/070528-Quadric.jpg” cannot be displayed, because it contains errors.

 

As Cara goes on to explain, the Klein correspondence underlies Conwell’s discussion of eight heptads.  These play an important role in another correspondence, illustrated in the Miracle Octad Generator of R. T. Curtis, that may be used to picture actions of the large Mathieu group M24.


Related material:

The projective space PG(5,2), home of the Klein quadric in the finite model, may be viewed as the set of 64 points of the affine space AG(6,2), minus the origin.

The 64 points of this affine space may in turn be viewed as the 64 hexagrams of the Classic of Transformation, China’s I Ching.

There is a natural correspondence between the 64 hexagrams and the 64 subcubes of a 4x4x4 cube.  This correspondence leads to a natural way to generate the affine group AGL(6,2).  This may in turn be viewed as a group of over a trillion natural transformations of the 64 hexagrams.

Geometry of the I Ching.
“Once Knecht confessed to his teacher that he wished to learn enough to be able to incorporate the system of the I Ching into the Glass Bead Game.  Elder Brother laughed.  ‘Go ahead and try,’ he exclaimed.  ‘You’ll see how it turns out.  Anyone can create a pretty little bamboo garden in the world.  But I doubt that the gardener would succeed in incorporating the world in his bamboo grove.'”
— Hermann Hesse, The Glass Bead Game,
  translated by Richard and Clara Winston

Wednesday, September 20, 2006

Wednesday September 20, 2006

Filed under: General,Geometry — Tags: , — m759 @ 8:00 am

Public Space

"… the Danish cartoons crisis last March showed 'two world views colliding in public space with no common point of reference.'"

George Carey, Archbishop of Canterbury from 1991 to 2002, quoted in today's London Times.

Related material:

Geometry and Christianity
   (Google search yielding
    "about 1,540,000" results)

Geometry and Islam
   (Google search yielding
    "about 1,580,000" results)

MySpace.com/affine

The image “http://www.log24.com/log/pix06A/060828-Cube.jpg” cannot be displayed, because it contains errors.

A Public Space

 

The image “http://www.log24.com/log/pix06A/060920-Motto.jpg” cannot be displayed, because it contains errors.

— Motto of 
Plato's Academy

Background from
Log24 on Feb. 15, 2006:

Hellmut Wilhelm on the Tao
 
If we replace the Chinese word "I" (change, transformation) with the word "permutation," the relevance of Western mathematics (which some might call "the Logos") to the I Ching ("Changes Classic") beomes apparent.

For the relevance of Plato to
Islam, see David Wade's
Pattern in Islamic Art
and a Google search on
Plato and Islam
("about 1,680,000" results).

"We should let ourselves be guided by what is common to all. Yet although the Logos is common to all, most men live as if each had a private intelligence of his own."

Heraclitus of Ephesus, about 500 B.C.

Monday, August 28, 2006

Monday August 28, 2006

Filed under: General,Geometry — Tags: — m759 @ 1:00 am
Today's Sinner:

Augustine of Hippo, who is said to
have died on this date in 430 A.D.

"He is, after all, not merely taking over a Neoplatonic ontology, but he is attempting to combine it with a scriptural tradition of a rather different sort, one wherein the divine attributes most prized in the Greek tradition (e.g. necessity, immutability, and atemporal eternity) must somehow be combined with the personal attributes (e.g. will, justice, and historical purpose) of the God of Abraham, Isaac, and Jacob."

Stanford Encyclopedia of Philosophy on Augustine

Here is a rather different attempt
to combine the eternal with the temporal:

 

The Eternal

Symbol of necessity,
immutability, and
atemporal eternity:

The image “http://www.log24.com/log/pix06A/060828-Cube.jpg” cannot be displayed, because it contains errors.

For details, see
finite geometry of
the square and cube
.

The Temporal

Symbol of the
God of Abraham,
Isaac, and Jacob:

The image “http://www.log24.com/log/pix06A/060828-Cloud.jpg” cannot be displayed, because it contains errors.

For details, see
Under God
(Aug. 11, 2006)

The eternal
combined with
the temporal:

 

Singer 63-cycle in the Galois field GF(64) used to order the I Ching hexagrams

Related material:

Hitler's Still Point and
the previous entry.
 

Wednesday, February 15, 2006

Wednesday February 15, 2006

Filed under: General,Geometry — Tags: , — m759 @ 11:07 am
Anthony Hopkins
Writes Screenplay
About God, Life & Death

These topics may be illuminated
by a study of the Chinese classics.

The image “http://www.log24.com/theory/images/GF64-63cycleA495.gif” cannot be displayed, because it contains errors.

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

If we replace the Chinese word "I"
(change, transformation) with the
word "permutation," the relevance
of Western mathematics (which
some might call "the Logos") to
the I Ching ("Changes Classic")
beomes apparent.

Related material:

Hitler's Still Point,
Jung's Imago,
Solomon's Cube,
Geometry of the I Ching,
and Globe Award.

Yesterday's Valentine
may also have some relevance.

Wednesday, January 4, 2006

Wednesday January 4, 2006

Filed under: General,Geometry — Tags: , , — m759 @ 4:04 am
Dragon School

In memory of Humphrey Carpenter, author of The Inklings, who attended The Dragon School.  Carpenter died a year ago today.

From Log24 on Nov. 16, 2005:

 

Images

 

Adam Gopnik on C. S. Lewis in the New Yorker:

"Lewis began with a number of haunted images…."

"The best of the books are the ones… where the allegory is at a minimum and the images just flow."

"'Everything began with images,' Lewis wrote…."

The image “http://www.log24.com/log/pix05B/051116-Time.jpg” cannot be displayed, because it contains errors.

 

From Paul Preuss,
Broken Symmetries
(see previous entry):

"Mathematical relationships were enough to satisfy him, mere formal relationships which existed at all times, everywhere, at once.  It was a thin nectar, but he was convinced it was the nectar of the gods…."


From
Verbum Sat Sapienti?

The image “http://www.log24.com/theory/images/EscherVerbum2.jpg” cannot be displayed, because it contains errors.

Escher's Verbum

The image “http://www.log24.com/theory/images/DTinvar246.jpg” cannot be displayed, because it contains errors.

Solomon's Cube


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

Geometry of the I Ching

 

Wednesday, November 9, 2005

Wednesday November 9, 2005

Filed under: General,Geometry — Tags: , — m759 @ 3:09 pm
In honor of the 120th anniversary
of the birth of Hermann Weyl:

Saturday, July 20, 2002

Saturday July 20, 2002

 

ABSTRACT: Finite projective geometry explains the surprising symmetry properties of some simple graphic designs– found, for instance, in quilts. Links are provided for applications to sporadic simple groups (via the "Miracle Octad Generator" of R. T. Curtis), to the connection between orthogonal Latin squares and projective spreads, and to symmetry of Walsh functions.

We regard the four-diamond figure D above as a 4×4 array of two-color diagonally-divided square tiles.

Let G be the group of 322,560 permutations of these 16 tiles generated by arbitrarily mixing random permutations of rows and of columns with random permutations of the four 2×2 quadrants.

THEOREM: Every G-image of D (as at right, below) has some ordinary or color-interchange symmetry.

Example:


For an animated version, click here.

Remarks:

Some of the patterns resulting from the action of G on D have been known for thousands of years. (See Jablan, Symmetry and Ornament, Ch. 2.6.) It is perhaps surprising that the patterns' interrelationships and symmetries can be explained fully only by using mathematics discovered just recently (relative to the patterns' age)– in particular, the theory of automorphism groups of finite geometries.

Using this theory, we can summarize the patterns' properties by saying that G is isomorphic to the affine group A on the linear 4-space over GF(2) and that the 35 structures of the 840 = 35 x 24 G-images of D are isomorphic to the 35 lines in the 3-dimensional projective space over GF(2).

This can be seen by viewing the 35 structures as three-sets of line diagrams, based on the three partitions of the four-set of square two-color tiles into two two-sets, and indicating the locations of these two-sets of tiles within the 4×4 patterns. The lines of the line diagrams may be added in a binary fashion (i.e., 1+1=0). Each three-set of line diagrams sums to zero– i.e., each diagram in a three-set is the binary sum of the other two diagrams in the set. Thus, the 35 three-sets of line diagrams correspond to the 35 three-point lines of the finite projective 3-space PG(3,2).

For example, here are the line diagrams for the figures above:

 
Shown below are the 15 possible line diagrams resulting from row/column/quadrant permutations. These 15 diagrams may, as noted above, be regarded as the 15 points of the projective 3-space PG(3,2).


The symmetry of the line diagrams accounts for the symmetry of the two-color patterns. (A proof shows that a 2nx2n two-color triangular half-squares pattern with such line diagrams must have a 2×2 center with a symmetry, and that this symmetry must be shared by the entire pattern.)

Among the 35 structures of the 840 4×4 arrays of tiles, orthogonality (in the sense of Latin-square orthogonality) corresponds to skewness of lines in the finite projective space PG(3,2). This was stated by the author in a 1978 note. (The note apparently had little effect. A quarter-century later, P. Govaerts, D. Jungnickel, L. Storme, and J. A. Thas wrote that skew (i.e., nonintersecting) lines in a projective space seem "at first sight not at all related" to orthogonal Latin squares.)

We can define sums and products so that the G-images of D generate an ideal (1024 patterns characterized by all horizontal or vertical "cuts" being uninterrupted) of a ring of 4096 symmetric patterns. There is an infinite family of such "diamond" rings, isomorphic to rings of matrices over GF(4).

The proof uses a decomposition technique for functions into a finite field that might be of more general use.

The underlying geometry of the 4×4 patterns is closely related to the Miracle Octad Generator of R. T. Curtis– used in the construction of the Steiner system S(5,8,24)– and hence is also related to the Leech lattice, which, as Walter Feit has remarked, "is a blown up version of S(5,8,24)."

For a movable JavaScript version of these 4×4 patterns, see The Diamond 16 Puzzle.

The above is an expanded version of Abstract 79T-A37, "Symmetry invariance in a diamond ring," by Steven H. Cullinane, Notices of the American Mathematical Society, February 1979, pages A-193, 194.

For a discussion of other cases of the theorem, click here.

Related pages:

The Diamond 16 Puzzle

Diamond Theory in 1937:
A Brief Historical Note

Notes on Finite Geometry

Geometry of the 4×4 Square

Binary Coordinate Systems

The 35 Lines of PG(3,2)

Map Systems:
Function Decomposition over a Finite Field

The Diamond Theorem–
The 2×2, the 2x2x2, the 4×4, and the 4x4x4 Cases

Diamond Theory

Latin-Square Geometry

Walsh Functions

Inscapes

The Diamond Theory of Truth

Geometry of the I Ching

Solomon's Cube and The Eightfold Way

Crystal and Dragon in Diamond Theory

The Form, the Pattern

The Grid of Time

Block Designs

Finite Relativity

Theme and Variations

Models of Finite Geometries

Quilt Geometry

Pattern Groups

The Fano Plane Revisualized,
or the Eightfold Cube

The Miracle Octad Generator

Kaleidoscope

Visualizing GL(2,p)

Jung's Imago

Author's home page

AMS Mathematics Subject Classification:

20B25 (Group theory and generalizations :: Permutation groups :: Finite automorphism groups of algebraic, geometric, or combinatorial structures)

05B25 (Combinatorics :: Designs and configurations :: Finite geometries)

51E20 (Geometry :: Finite geometry and special incidence structures :: Combinatorial structures in finite projective spaces)



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Page created Jan. 6, 2006, by Steven H. Cullinane      diamondtheorem.com

 

Initial Xanga entry.  Updated Nov. 18, 2006.

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