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.
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).
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 ) —
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.
Denote the d-dimensional hypercube by
"… after coloring the sixty-four vertices of
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 166 ," section 12 of Coxeter's 1950
"Self-dual Configurations and Regular Graphs"
Just as the 4×4 square represents the 4-dimensional
hypercube
so the 4x4x4 cube represents the 6-dimensional
hypercube
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).
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.
The Klein Correspondence, Penrose Space-Time, and a Finite Model
Related material:
Monday's entry Just Say NO and a poem by Stevens,
|
Updated 8:18 PM EDT, Tue July 2, 2024 "Robert Towne, the Oscar-winning screenwriter of a number of acclaimed movies, including the classic 1974 noir thriller 'Chinatown' starring Jack Nicholson and Faye Dunaway, has died. He was 89 years old. The news was confirmed by Towne’s publicist Carri McClure, who said he died on Monday 'peacefully at home surrounded by his loving family.' No cause of death was provided. Towne won the Academy Award for best original screenplay for 'Chinatown,' which last month celebrated 50 years since being released." |
Related imagery . . .
Note the three quadruplets of parallel edges in the 1984 figure above.
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:
See also the Klein correspondence in a paper from 1968
in yesterday's 2:54 PM ET post.
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 —
Two such considerations —
“… 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 —
Two of the thumbnail previews
from yesterday's 1 AM post …
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 July 17, 1961
Dr. Thomas S. Szasz 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.
The title refers to Calabi-Yau spaces.
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 —
A related Calabi-Yau "Chinese jar" first described in detail in 1905 —
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
The 4×4 square may also be called the Galois Tesseract .
By analogy, the 4x4x4 cube may be called the Galois Hexeract .
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 —
That 2014 post was titled "Kummer Varieties." It is now tagged
"Kummerhenge." For some backstory, see other posts so tagged.
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."
Related material —
The seven points of the Fano plane within
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"Before time began . . . ."
— Optimus Prime
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
The logo at left above is that of The Point .
The menu icon at right above is perhaps better
suited to illustrate Verwandlungslehre .
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.
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 —
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Tuesday, December 1, 2015
Verhexung
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Related material —
"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 —
Two such considerations —
The Log24 version (Nov. 9, 2005, and later posts) —
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VERBUM
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See also related material in the previous post, Transformers.
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 —
Stanford Encyclopedia of Philosophy
on the date Friday, April 5, 2013 —
“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.
Flashback to St. Andrew's Day, 2013 —
Saturday, November 30, 2013
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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 —
A different representation of the unit cell of
the hyperspace H (and of the I Ching ) —
(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.
“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.
“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
The Dream of the Expanded Field continues…
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 —
Two such considerations —
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, W (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 .
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."
Click for the source, mentioned in Anatomy of a Cube (Sept. 18, 2011).
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.
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.
"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.
As was the I Ching. A related structure:
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.
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—
The following is a new illustration for Cubist Geometries—
(For elementary cubism, see Pilate Goes to Kindergarten and The Eightfold Cube.
For advanced, see Solomon's Cube and Geometry of the I Ching .)
In memory of Wu Guanzhong, Chinese artist who died in Beijing on Friday—
"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."
David Levine's portrait of Arthur Koestler (see Dec. 30, 2009) —
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."
(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—
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—
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.
It is well known that the seven
Similarly, recent posts* have noted that the thirteen
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…
See also Geometry of the I Ching and a search in this journal for
* 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)
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 —
Image of pencils is by
Diane Robertson Design.
Related material:
"A Four-Color Theorem."
Part IV:
(Cover slightly changed.)
Background —
SAT
Part I:
Part II:
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Part III:
From August 25th —
"Boo, boo, boo,
square root of two."
Diamond-Faceted:
Transformations
of the Rock
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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)
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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.
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”
“Click on the Yellow Book.”
Happy birthday, David Carradine.
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.
© 2005 The Institute for FiguringThis 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
For a related larger space,
see the entry and links of
St. Augustine’s Day, 2006.
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:
Macquarrie's connection in this journal to the I Ching is, like that book itself, purely coincidental. For details, click on the figure below.
The persistent reader will
find a further link that
leads to an entry titled
"Notes on the I Ching."
"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."
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:
From May 10, 2008:
A pattern of figures:
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See also Richard Wilhelm on
Hexagram 32 of the I Ching:
— The Middle-English
Harrowing of Hell…
by Hulme, 1907, page 64
(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….
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):
Analogously:
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Lotteries on Bloomsday, June 16, 2008 |
Pennsylvania (No revelation) |
New York (Revelation) |
|
Mid-day (No belief) |
418
No belief, |
064
Revelation |
|
Evening (Belief) |
709
Belief without |
198
|
The holy image
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?
— Ulysses, conclusion of Chapter 17 |
Undertakings bring misfortune.
Nothing that would further.
“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:
See 2/22/08
and 4/19/08.
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.”
“Nothing that would further.”
— Hexagram 54
|
…. Now thou art an 0 |
“…. 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
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:
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:
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.
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)
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.)
The "chessboard" of
Nov. 7, 2006
(as revised Nov. 7, 2012)–
(Faust, Part Two, as
quoted by Jung in
Memories, Dreams, Reflections)
and a Finite Model
Notes by Steven H. Cullinane
May 28, 2007
Part I: A Model of Space-Time
 |
Click on picture to enlarge.
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:
 |
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.
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)
A Public Space
— Motto of
Plato's Academy
|
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. |
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).
— Heraclitus of Ephesus, about 500 B.C.
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,
For details, see |
The Temporal
Symbol of the
For details, see |
|
The eternal
combined with the temporal:
|
|
Related material:
These topics may be illuminated
by a study of the Chinese classics.
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.
In memory of Humphrey Carpenter, author of The Inklings, who attended The Dragon School. Carpenter died a year ago today.
"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…."
From Paul Preuss,
Broken Symmetries
(see previous entry):
From
Verbum Sat Sapienti?
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Example:
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Initial Xanga entry. Updated Nov. 18, 2006.
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