From a post, now private, in this journal on June 17, 2024 —
James Hillman
EGALITARIAN TYPOLOGIES
VERSUS THE PERCEPTION OF THE UNIQUE
“The kind of movement Olson urges is an inward deepening of the image,
an in-sighting of the superimposed levels of significance within it.
This is the very mode that Jung suggested for grasping dreams —
not as a sequence in time, but as revolving around a nodal complex.”
See as well "True Grids" (Log24, August 9, 2018).
Today's host for a special political edition of CBS Sunday Morning
is Ted Koppel. Vocabulary review:
Koppel's appearance today was backed by the usual CBS Sunday Morning
sun-disk Apollo symbol. An Apollo symbol that some may prefer —
The Ninefold Square
|
Rosalind Krauss "If we open any tract– Plastic Art and Pure Plastic Art or The Non-Objective World , for instance– we will find that Mondrian and Malevich are not discussing canvas or pigment or graphite or any other form of matter. They are talking about Being or Mind or Spirit. From their point of view, the grid is a staircase to the Universal, and they are not interested in what happens below in the Concrete. Or, to take a more up-to-date example…."
"He was looking at the nine engravings and at the circle,
"And it's whispered that soon if we all call the tune
The nine engravings of The Club Dumas
An example of the universal— or, according to Krauss,
"This is the garden of Apollo, the field of Reason…." |
"That in which space itself is contained"
— Wallace Stevens
In the ninefold square,
projective-perspectivity duality
corresponds to
projective-correlation duality.
Illustrations —
The ninefold square in the previous post suggests a review
of posts now tagged Plane of Time.
See as well . . .
Hat tip to Peter Woit for quoting the above yesterday.
* See Crichton in this weblog and the solar Apollo symbols
of CBS Sunday Morning, which this week featured Crichton.
As an Apollo symbol, I prefer the Chinese "holy field" —
a ninefold square.
From some Canadian legal boilerplate —
E. Be able to provide complete, clean, unencumbered
chain of title for the Project, must have all the rights,
releases and clearances necessary to produce, own and
exploit the Project and for deployment of the Project . . . .
Weak Links in the Chain of Title —
A 2006 biography of geometer H.S.M. Coxeter:
The Aleph (implicit in a 1950 article by Coxeter):
Click on images
for further details.
Updates from later the same day —
Related affine structures —
See also "Square+Triangles" in this journal.
The fishlike shapes within three of the above
ninefold colored triangles suggest some . . .
Related Entertainment —
This post was suggested by Chapter Two, "The Sprite and the Synergist,"
of Alfred Bester's The Deceivers , and by . . .
Fanciful version —
Less fanciful versions . . .
|
Unmagic Squares Consecutive positive integers:
1 2 3 Consecutive nonnegative integers:
0 1 2
Consecutive nonnegative integers
00 01 02
This last square may be viewed as
Note that the ninefold square so viewed
As does, similarly, the ancient Chinese
These squares are therefore equivalent under This method generalizes. — Steven H. Cullinane, Nov. 20, 2021 |
.jpg)
|
Unmagic Squares Consecutive positive integers:
1 2 3 Consecutive nonnegative integers:
0 1 2
Consecutive nonnegative integers
00 01 02
This last square may be viewed as
Note that the ninefold square so viewed
As does, similarly, the ancient Chinese
These squares are therefore equivalent under This method generalizes. — Steven H. Cullinane, Nov. 20, 2021 |
"… thinking starts with a problem, a difficulty, a contradiction."
— The late Arthur Mattuck of MIT in last night's post Gestalt .
"The 'technical support' is an underlying ground for aesthetic practice
that supports the work of art as canvas supported oil paint."
— MIT Press on the book by Rosalind Krauss titled Under Blue Cup .
From Grid View and List View (Log24, 15 April 2021) —
|
Margaret Atwood on Lewis Hyde's "Trickster is among other things the gatekeeper who opens the door into the next world; those who mistake him for a psychopath never even know such a door exists." (159) What is "the next world"? It might be the Underworld…. The pleasures of fabulation, the charming and playful lie– this line of thought leads Hyde to the last link in his subtitle, the connection of the trickster to art. Hyde reminds us that the wall between the artist and that American favourite son, the con-artist, can be a thin one indeed; that craft and crafty rub shoulders; and that the words artifice, artifact, articulation and art all come from the same ancient root, a word meaning "to join," "to fit," and "to make." (254) If it’s a seamless whole you want, pray to Apollo, who sets the limits within which such a work can exist. Tricksters, however, stand where the door swings open on its hinges and the horizon expands: they operate where things are joined together, and thus can also come apart. |
"As a Chinese jar . . . ."
— Four Quartets
|
Rosalind Krauss "If we open any tract– Plastic Art and Pure Plastic Art or The Non-Objective World , for instance– we will find that Mondrian and Malevich are not discussing canvas or pigment or graphite or any other form of matter. They are talking about Being or Mind or Spirit. From their point of view, the grid is a staircase to the Universal, and they are not interested in what happens below in the Concrete. Or, to take a more up-to-date example…."
"He was looking at the nine engravings and at the circle,
"And it's whispered that soon if we all call the tune
The nine engravings of The Club Dumas
An example of the universal— or, according to Krauss,
"This is the garden of Apollo, |
The "Katz" of the August 7 post Art Angles
is a product of Princeton's
Department of Art and Archaeology.
ART —
ARCHAEOLOGY —
"This pattern is a square divided into nine equal parts.
It has been called the 'Holy Field' division and
was used throughout Chinese history for many
different purposes, most of which were connected
with things religious, political, or philosophical."
– The Magic Square: Cities in Ancient China,
by Alfred Schinz, Edition Axel Menges, 1996, p. 71
Robert A. Wilson on symmetries of the ninefold square —
|
"All of these ideas have shown promise at some time or other, and some are still under active investigation. But my conclusion after all this work is that the part of algebra that shows the most promise for genuinely useful applications to fundamental physics is the representation theory, real, complex, integral and modular, of the group GL(2, 3). There is, of course, no guarantee that a viable theory can be built on this foundation. But it appears to be the only part of algebra that both has a reasonable chance of success and has not already been exhaustively explored in the physics literature. It is therefore worth serious consideration." — "Potential applications of modular representation theory to quantum mechanics," arXiv, May 28, 2021, revised June 7, 2021. |
See as well GL(2,3) in this journal .
Related material: Christmas Eve 2012.
Robert A. Wilson on symmetries of the ninefold square —
|
"All of these ideas have shown promise at some time or other, and some are still under active investigation. But my conclusion after all this work is that the part of algebra that shows the most promise for genuinely useful applications to fundamental physics is the representation theory, real, complex, integral and modular, of the group GL(2, 3). There is, of course, no guarantee that a viable theory can be built on this foundation. But it appears to be the only part of algebra that both has a reasonable chance of success and has not already been exhaustively explored in the physics literature. It is therefore worth serious consideration." — "Potential applications of modular representation theory to quantum mechanics," arXiv, May 28, 2021, revised June 7, 2021. |
See as well GL(2,3) in this journal .
Some may consider more relevant the remarks of a different Robert A. Wilson —
The above phrase "the intersection of storytelling and visual arts"
suggests a review . . .
Storytelling —
Visual arts —
"This pattern is a square divided into nine equal parts.
It has been called the 'Holy Field' division and
was used throughout Chinese history for many
different purposes, most of which were connected
with things religious, political, or philosophical."
– The Magic Square: Cities in Ancient China,
by Alfred Schinz, Edition Axel Menges, 1996, p. 71
A Midrash for Michener —
For a connection of the above "Holy Field"
with pure mathematics, see Coxeter's Aleph.
From AntiChristmas 2010 —
Above: Art Theorist Rosalind Krauss and The Ninefold Square
For Krauss:
Grid View List View
See also "True Grid " in this journal.
|
Rosalind Krauss "If we open any tract– Plastic Art and Pure Plastic Art or The Non-Objective World , for instance– we will find that Mondrian and Malevich are not discussing canvas or pigment or graphite or any other form of matter. They are talking about Being or Mind or Spirit. From their point of view, the grid is a staircase to the Universal, and they are not interested in what happens below in the Concrete. Or, to take a more up-to-date example…."
"He was looking at the nine engravings and at the circle,
"And it's whispered that soon if we all call the tune
The nine engravings of The Club Dumas
An example of the universal— or, according to Krauss,
"This is the garden of Apollo, the field of Reason…." |
See as well . . .
The previous post, on the 3×3 square in ancient China,
suggests a review of group actions on that square
that include the quaternion group.
Click to enlarge —
Three links from the above finitegeometry.org webpage on the
quaternion group —
Related material —
See as well the two Log24 posts of December 1st, 2018 —
Character and In Memoriam.
The elementary shapes at the top of the figure below mirror
the looking-glass property of the classical Lo Shu square.
The nine shapes at top left* and their looking-glass reflection
illustrate the looking-glass reflection relating two orthogonal
Latin squares over the three digits of modulo-three arithmetic.
Combining these two orthogonal Latin squares,** we have a
representation in base three of the numbers from 0 to 8.
Adding 1 to each of these numbers yields the Lo Shu square.
* The array at top left is from the cover of
Wonder Years:
Werkplaats Typografie 1998-2008.
** A well-known construction.
*** For other instances of what might be
called "design grammar" in combinatorics,
see a slide presentation by Robin Wilson.
No reference to the work of Chomsky is
intended.
"At the still point of the turning world. Neither flesh nor fleshless;
Neither from nor towards; at the still point, there the dance is,
But neither arrest nor movement. And do not call it fixity,
Where past and future are gathered. Neither movement from nor towards,
Neither ascent nor decline. Except for the point, the still point,
There would be no dance, and there is only the dance."
— T. S. Eliot, Four Quartets
See also a recurrent image
from this journal —
.
The contraction of the title is from group actions on
the ninefold square (with the center subsquare fixed)
to group actions on the eightfold cube.
From a post of June 4, 2014 …
At math.stackexchange.com on March 1-12, 2013:
“Is there a geometric realization of the Quaternion group?” —
The above illustration, though neatly drawn, appeared under the
cloak of anonymity. No source was given for the illustrated group actions.
Possibly they stem from my Log24 posts or notes such as the Jan. 4, 2012,
note on quaternion actions at finitegeometry.org/sc (hence ultimately
from my note “GL(2,3) actions on a cube” of April 5, 1985).
Ben Brantley in tonight's online review of a show that
reportedly opened off-Broadway on Dec. 10, 2015 —
" 'Mattress' has its charms, but they do wear thin. "
See also The New York Times on Martin Gardner Nov. 30:
A companion image from this journal
on the "Mattress" opening date —
Midrash:
Vonnegut Asterisk
A post of July 7, Haiku for DeLillo, had a link to posts tagged "Holy Field GF(3)."
As the smallest Galois field based on an odd prime, this structure
clearly is of fundamental importance.
It is, however, perhaps too small to be visually impressive.
A larger, closely related, field, GF(9), may be pictured as a 3×3 array…
|
… hence as the traditional Chinese Holy Field.
Marketing the Holy Field
The above illustration of China's Holy Field occurred in the context of
Log24 posts on Child Buyers. For more on child buyers, see an excellent
condemnation today by Diane Ravitch of the U. S. Secretary of Education.
Notes on space for day 13 of May, 2015 —
The 13 symmetry axes of the cube may be viewed as
the 13 points of the Galois projective space PG(2,3).
This space (a plane) may also be viewed as the nine points
of the Galois affine space AG(2,3) plus the four points on
an added "line at infinity."
Related poetic material:
The ninefold square and Apollo, as well as …
Barron's Educational Series (click to enlarge):
The Tablet of Ahkmenrah:
"With the Tablet of Ahkmenrah and the Cube of Rubik,
my power will know no bounds!"
— Kahmunrah in a novelization of Night at the Museum:
Battle of the Smithsonian , Barron's Educational Series
Another educational series (this journal):
Art theorist Rosalind Krauss and The Ninefold Square
On Walter Isaacson’s The Innovators :
“Yet as the book’s five hundred–plus pages unwind, Isaacson interrupts himself to present small bromides about what it means to innovate and what we might learn from these innovators, our presumed betters. “Innovation requires articulation,” he tells us, after explaining how the main strength of Grace Hopper, a trailblazing computer scientist for the US Navy, was her ability to speak in the languages of mathematicians, engineers, programmers, and soldiers alike. ‘One useful leadership talent is knowing when to push ahead against doubters and when to heed them,’ he offers later.
The book is peppered with these kinds of passages, which often intrude on the narrative, depriving us of moments of real emotional power.”
— Jacob Silverman in Bookforum , Sept/Oct/Nov 2014
From Isaacson’s book:
Related material:
In memory of T. S. Eliot…
… and in memory of Stanley Chase, producer of Colossus: The Forbin Project
and of Threepenny Opera :
Ninefold square from Colossus
(“There is another system”) —
Fourfold square introducing Brecht
in Dreigroschen Trifft Vierfarben —
The ninefold square, the eightfold cube, and monkeys.
For posts on the models above, see quaternion
in this journal. For the monkeys, see
"Nothing Is More Fun than a Hypercube of Monkeys,"
Evelyn Lamb's Scientific American weblog, May 19, 2014:
The Scientific American item is about the preprint
"The Quaternion Group as a Symmetry Group,"
by Vi Hart and Henry Segerman (April 26, 2014):
See also Finite Geometry and Physical Space.
Review of Joseph Campbell's The Inner Reaches of Outer Space—
The reviewer compares Campbell to "one of those guys who
builds his own church out of hub caps."
A simple hub cap — see ninefold in this journal.
“… the object sets up a kind of
frame or space or field
within which there can be epiphany.”
Related material —
Star Wars (January 11, 2014),
The Lyche Gate Asterisk , from 10:31 AM ET on May 22, 2010,
the date of Martin Gardner's death —
— and the March 2014 issue of the
Notices of the American Mathematical Society —
See as well Epiphany 2014 (Jan. 6) in this journal and the
March Notices on the Shaw prize —
"Established under the auspices of Run Run Shaw
in November 2002, the prize is managed and
administered by the Shaw Prize Foundation
based in Hong Kong."
Yes. See …
The 48 actions of GL(2,3) on a 3×3 coordinate-array A,
when matrices of that group right-multiply the elements of A,
with A =
| (1,1) (1,0) (1,2) (0,1) (0,0) (0,2) (2,1) (2,0) (2,2) |
Actions of GL(2,p) on a pxp coordinate-array have the
same sorts of symmetries, where p is any odd prime.
Note that A, regarded in the Sallows manner as a magic square,
has the constant sum (0,0) in rows, columns, both diagonals, and
all four broken diagonals (with arithmetic modulo 3).
For a more sophisticated approach to the structure of the
ninefold square, see Coxeter + Aleph.
This post was suggested by…
1. Movie reviews links on the NY Times online front page—
2. "When Death tells a story…" in this journal
3. A search for "Recognition" in this journal
(suggested by Terence Cave's Recognitions ,
mentioned in a post of October 31.)
4. The play "Magic," by G. K. Chesterton
5. The following posts —
The post itself consists simply of the title,
together with the above items that suggested
the title.
The title refers to a classic 1960 novel by John Hersey.
“How do you get young people excited about space?”
— Megan Garber in The Atlantic , Aug. 16, 2012
(Italics added.) (See previous four posts.)
Allyn Jackson on “Simplicity, in Mathematics and in Art,”
in the new August 2013 issue of Notices of the American
Mathematical Society—
“As conventions evolve, so do notions of simplicity.
Franks mentioned Gauss’s 1831 paper that
established the respectability of complex numbers.”
This suggests a related image by Gauss, with a
remark on simplicity—
Here Gauss’s diagram is not, as may appear at first glance,
a 3×3 array of squares, but is rather a 4×4 array of discrete
points (part of an infinite plane array).
Related material that does feature the somewhat simpler 3×3 array
of squares, not seen as part of an infinite array—
Marketing the Holy Field
Click image for the original post.
For a purely mathematical view of the holy field, see Visualizing GL(2,p).
Some webpages at finitegeometry.org discuss
group actions on Sylvester’s duads and synthemes.
Those pages are based on the square model of
PG(3,2) described in the 1980’s by Steven H. Cullinane.
A rival tetrahedral model of PG(3,2) was described
in the 1990’s by Burkard Polster.
Polster’s tetrahedral model appears, notably, in
a Mathematics Magazine article from April 2009—
Click for a pdf of the article.
Related material:
“The Religion of Cubism” (May 9, 2003) and “Art and Lies”
(Nov. 16, 2008).
This post was suggested by following the link in yesterday’s
Sunday School post to High White Noon, and the link from
there to A Study in Art Education, which mentions the date of
Rudolf Arnheim‘s death, June 9, 2007. This journal
on that date—
The Fink-Guy article was announced in a Mathematical
Association of America newsletter dated April 15, 2009.
Those who prefer narrative to mathematics may consult
a Log24 post from a few days earlier, “Where Entertainment is God”
(April 12, 2009), and, for some backstory, The Judas Seat
(February 16, 2007).
From AntiChristmas 2010—
Art theorist Rosalind Krauss and The Ninefold Square
Krauss is a co-founder of the art journal October .
For some backgound, see the overlapping searches
Krauss October and Matrix Architect.
Last night’s post discussed ways of draining the world of meaning.
For some tastes, poets like Dante do the opposite, supplying too much meaning.
See a New Republic review, dated Oct. 5, in which Harvard atheist Helen Vendler discusses Dante’s
“… assertion that Beatrice herself ‘was this number [nine],’ since nine is the square of three, the number belonging to the Trinity. Dante’s fantastic reasoning requires pages of annotation, which Frisardi, drawing on a number of commentators, furnishes to the bewildered reader. The theological elaboration of the number nine— merely one instance of how far from our own* are Dante’s habits of thought— will convince any doubting reader that the Vita Nuova requires annotation far beyond what its pages might seem to demand.”
Related material— Ninefold in this journal, and remarks by Joseph Campbell in a post, Plan 9, from Sept. 5.
John Baez wrote in 1996 ("Week 91") that
"I've never quite seen anyone come right out
and admit that triality arises from the
permutations of the unit vectors i, j, and k
in 3d Euclidean space."
Baez seems to come close to doing this with a
somewhat different i , j , and k — Hurwitz
quaternions— in his 2005 book review
quoted here yesterday.
See also the Log24 post of Jan. 4 on quaternions,
and the following figures. The actions on cubes
in the lower figure may be viewed as illustrating
(rather indirectly) the relationship of the quaternion
group's 24 automorphisms to the 24 rotational
symmetries of the cube.
A physics quote relayed at Peter Woit's weblog today—
"The relation between 4D N=4 SYM and the 6D (2, 0) theory
is just like that between Darth Vader and the Emperor.
You see Darth Vader and you think 'Isn’t he just great?
How can anyone be greater than that? No way.'
Then you meet the Emperor."
Some related material from this weblog—
(See Big Apple and Columbia Film Theory)
The Meno Embedding:
Some related material from the Web—
See also uses of the word triality in mathematics. For instance…
A discussion of triality by Edward Witten—
Triality is in some sense the last of the exceptional isomorphisms,
and the role of triality for n = 6 thus makes it plausible that n = 6
is the maximum dimension for superconformal symmetry,
though I will not give a proof here.
— "Conformal Field Theory in Four and Six Dimensions"
and a discussion by Peter J. Cameron—
There are exactly two non-isomorphic ways
to partition the 4-subsets of a 9-set
into nine copies of AG( 3,2).
Both admit 2-transitive groups.
— "The Klein Quadric and Triality"
Exercise: Is Witten's triality related to Cameron's?
(For some historical background, see the triality link from above
and Cameron's Klein Correspondence and Triality.)
Cameron applies his triality to the pure geometry of a 9-set.
For a 9-set viewed in the context of physics, see A Beginning—
|
From MIT Commencement Day, 2011— A symbol related to Apollo, to nine, and to "nothing"— A minimalist favicon—
This miniature 3×3 square— |
Happy April 1.
"At the still point…" — T. S. Eliot
In memory of David L. Waltz, artificial-intelligence pioneer,
who died Thursday, March 22, 2012—
The following from the First of May, 2010—
Some context–
"This pattern is a square divided into nine equal parts.
It has been called the 'Holy Field' division and
was used throughout Chinese history for many
different purposes, most of which were connected
with things religious, political, or philosophical."
– The Magic Square: Cities in Ancient China,
by Alfred Schinz, Edition Axel Menges, 1996, p. 71
From "The Talented," a post of April 26, 2011—
And for Josefine Lyche—
Unity in Multiplicity —
Pink in Wikipedia
"The 'one' with whom the reader has identified himself
has now become 'the listener, who listens in the snow';
he has become the snow man, and he knows winter
with a mind of winter, knows it in its strictest reality,
stripped of all imagination and human feeling.
But at that point when he sees the winter scene
reduced to absolute fact, as the object not of the mind,
but of the perfect perceptual eye that sees
'nothing that is not there,' then the scene,
devoid of its imaginative correspondences,
has become 'the nothing that is.'"
—Robert Pack, Wallace Stevens:
An Approach to His Poetry and Thought.
New Brunswick: Rutgers UP, 1958.
"… myths are stories, and like all narratives
they unravel through time, whereas grids
are not only spatial to start with,
they are visual structures that explicitly reject
a narrative or sequential reading of any kind."
— Rosalind Krauss in "Grids,"
October (Summer 1979), 9: 50-64.
Counterexample—
The Ninefold Square
See Coxeter and the Aleph and Ayn Sof—
| Mathematics and Narrative, Illustrated |
|
Mathematics |
 Narrative |
A reader comments on yesterday afternoon's New York Times
"The Stone" column by Justin E.H. Smith—
"I did indeed appreciate Mr. Smith’s essay.
And I’m curious as to what future contributions of his,
to the Stoner series, that we can look forward to."
From August 24, 2010—
|
|
Happy day 23 of Mental Health Month.
Another topic from today's newspaper —
Commentary —
"We're gonna need more holy water." — "Season of the Witch," a film that opened Friday
See also —
This morning's post Inception and the following site:
Note the ninefold favicon at the above site. Some background—
The Ninth Gate in yesterday's post and this image from last September—
"Rosetta Stone" as a Metaphor
in Mathematical Narratives
For some backgound, see Mathematics and Narrative from 2005.
Yesterday's posts on mathematics and narrative discussed some properties
of the 3×3 grid (also known as the ninefold square ).
For some other properties, see (at the college-undergraduate, or MAA, level)–
Ezra Brown, 2001, "Magic Squares, Finite Planes, and Points of Inflection on Elliptic Curves."
His conclusion:
When you are done, you will be able to arrange the points into [a] 3×3 magic square,
which resembles the one in the book [5] I was reading on elliptic curves….
This result ties together threads from finite geometry, recreational mathematics,
combinatorics, calculus, algebra, and number theory. Quite a feat!
5. Viktor Prasolov and Yuri Solvyev, Elliptic Functions and Elliptic Integrals ,
American Mathematical Society, 1997.
Brown fails to give an important clue to the historical background of this topic —
the word Hessian . (See, however, this word in the book on elliptic functions that he cites.)
Investigation of this word yields a related essay at the graduate-student, or AMS, level–
Igor Dolgachev and Michela Artebani, 2009, "The Hesse Pencil of Plane Cubic Curves ."
From the Dolgachev-Artebani introduction–
In this paper we discuss some old and new results about the widely known Hesse
configuration of 9 points and 12 lines in the projective plane P2(k ): each point lies
on 4 lines and each line contains 3 points, giving an abstract configuration (123, 94).
PlanetMath.org on the Hesse configuration—
A picture of the Hesse configuration–
(See Visualizing GL(2,p), a note from 1985).
Related notes from this journal —
From last November —
|
From the December 2010 American Mathematical Society Notices—
Related material from this journal— Consolation Prize (August 19, 2010) |
From 2006 —
|
Sunday December 10, 2006
“Function defined form, expressed in a pure geometry
– J. G. Ballard on Modernism
“The greatest obstacle to discovery is not ignorance –
— Daniel J. Boorstin, |
Also from 2006 —
|
Sunday November 26, 2006
Rosalind Krauss "If we open any tract– Plastic Art and Pure Plastic Art or The Non-Objective World , for instance– we will find that Mondrian and Malevich are not discussing canvas or pigment or graphite or any other form of matter. They are talking about Being or Mind or Spirit. From their point of view, the grid is a staircase to the Universal, and they are not interested in what happens below in the Concrete. Or, to take a more up-to-date example…."
"He was looking at the nine engravings and at the circle,
"And it's whispered that soon if we all call the tune
The nine engravings of The Club Dumas
An example of the universal*– or, according to Krauss,
"This is the garden of Apollo, the field of Reason…."
For more on the field of reason, see
A reasonable set of "strange correspondences" Unreason is, of course, more popular. * The ninefold square is perhaps a "concrete universal" in the sense of Hegel: "Two determinations found in all philosophy are the concretion of the Idea and the presence of the spirit in the same; my content must at the same time be something concrete, present. This concrete was termed Reason, and for it the more noble of those men contended with the greatest enthusiasm and warmth. Thought was raised like a standard among the nations, liberty of conviction and of conscience in me. They said to mankind, 'In this sign thou shalt conquer,' for they had before their eyes what had been done in the name of the cross alone, what had been made a matter of faith and law and religion– they saw how the sign of the cross had been degraded."
– Hegel, Lectures on the History of Philosophy ,
"For every kind of vampire, |
And from last October —
|
Friday, October 8, 2010
Starting Out in the Evening This post was suggested by last evening's post on mathematics and narrative and by Michiko Kakutani on Vargas Llosa in this morning's New York Times .
"One must proceed cautiously, for this road— of truth and falsehood in the realm of fiction— is riddled with traps and any enticing oasis is usually a mirage."
– "Is Fiction the Art of Lying?"* by Mario Vargas Llosa,
* The Web version's title has a misprint— |
In a nutshell —
Epigraph to "The Aleph," a 1945 story by Borges:
O God! I could be bounded in a nutshell,
and count myself a King of infinite space…
— Hamlet, II, 2
The story in book form, 1949
A 2006 biography of geometer H.S.M. Coxeter:
The Aleph (implicit in a 1950 article by Coxeter):
The details:
Related material: Group Actions, 1984-2009.
See today's earlier posts Ode and True Grid (continued) and, in the latter's
context of tic-tac-toe war games — Balance, from Halloween 2005 —
“An asymmetrical balance is sought since it possesses more movement.
This is achieved by the imaginary plotting of the character
upon a nine-fold square, invented by some ingenious writer of the Tang dynasty.
If the square were divided in half or in four, the result would be symmetrical,
but the nine-fold square permits balanced asymmetry."
— Paraphrase of a passage in Chiang Yee's Chinese Calligraphy
Between midnight and dawn, when the past is all deception,
The future futureless, before the morning watch
When time stops and time is never ending….
Art Theorist Rosalind Krauss and The Ninefold Square
* Title from a quotation in the Sunday New York Times today
(in a story about a June 25 death beginning on page MB1 of the New York edition)
Art Theorist Rosalind Krauss and The Ninefold Square
天鈞
Made famous by Ursula K. Le Guin
as the book title "Lathe of Heaven,"
this Chinese phrase, tianjun, apparently
means something more like "Scales of Heaven"–
an appropriate image for Law Day 2010.
An anonymous forum user says that
"…if you switch the two characters around,
you get: 鈞天, which is one of
the nine heavens, more specifically,
the middle heaven."
This is supported by a
non-anonymous source:
"I follow A.C. Graham’s translation of
Juntian as 'Level Heaven (the innermost
of the nine divisions of heaven)';
he renders Juntian guangyue as
'the mighty music of the innermost heaven.'"
— "Music in the World of Su Shi (1037-1101):
Terminology," by Stuart H. Sargent,
Colorado State University,
Journal of Sung-Yuan Studies 32 (2002), 39-81
The Nine Divisions of Heaven–
Some context–
"This pattern is a square divided into nine equal parts.
It has been called the 'Holy Field' division and
was used throughout Chinese history for many
different purposes, most of which were connected
with things religious, political, or philosophical."
– The Magic Square: Cities in Ancient China,
by Alfred Schinz, Edition Axel Menges, 1996, p. 71
Some historians consider today's date, April 7, to be the date of the Crucifixion in the Roman calendar (a solar calendar, as opposed to the Jewish lunar scheme).
Since the ninefold square has been called both a symbol of Apollo and the matrix of a cross, it will serve as an icon for today–
Adapted from
Ad Reinhardt
Detail from yesterday's post—
Surrealistic Alarm Clock
_________________________________________________________
THE SEQUEL:
"…and Surrealistic Pillow became
a household name
in the house of rock ‘n’ roll."
— Denise Sullivan in Crawdaddy,
October 8, 2009
Related material:
"Which Dreamed It?"
— Title of final chapter,
Through the Looking Glass
"Go ask Alice…
I think she'll know."
— Grace Slick, 1967
The Crawdaddy date Oct. 8, 2009
leads to the Log24 post
Graphic Austerity.
Clicking on those words
in that post will lead you to…
The Logic of Dreams.
Midnight in the Garden, Autumn 2009
Review:
|
From this journal on the following day, Sept. 21:
Happy birthday, Stephen King.
Today's previous entry is based on a song, "Unthought Known,"
from the above album; the cover of the album uses the 3×3 grid
shown in Sept. 20's midnight review. For related material
on the unconscious, see June 13-15, 2005.
I know more than Apollo,
For oft when he lies sleeping
I see the stars at mortal wars
In the wounded welkin weeping.
mentioned in
this afternoon's entry
"Emblematizing the Modern"
was the object of a recent
cinematic chase sequence
(successful and inspiring)
starring Mira Sorvino
at the Metropolitan
Museum of Art.
In memory of
Dr. Hunter S. Thompson,
dead by his own hand
on this date
four years ago —
There is
another sort of object
we may associate with a
different museum and with
a modern Constantine …
See "Art Wars for MoMA"
(Dec. 14, 2008).
This object, modern
rather than medieval,
is the ninefold square:
It may suit those who,
like Rosalind Krauss
(see "Emblematizing"),
admire the grids of modern art
but view any sort of Christian
cross with fear and loathing.
For some background that
Dr. Thompson might appreciate,
see notes on Geometry and Death
in this journal, June 1-15, 2007,
and the five Log24 entries
ending at 9 AM Dec. 10. 2006,
which include this astute
observation by J. G. Ballard:
Selah.
A Sunrise
for Sunrise
“If we open any tract– Plastic Art and Pure Plastic Art or The Non-Objective World, for instance– we will find that Mondrian and Malevich are not discussing canvas or pigment or graphite or any other form of matter. They are talking about Being or Mind or Spirit. From their point of view, the grid is a staircase to the Universal, and they are not interested in what happens below in the Concrete.”
Yesterday’s entry featured a rather simple-minded example from Krauss of how the ninefold square (said to be a symbol of Apollo)
may be used to create a graphic design– a Greek cross, which appears also in crossword puzzles:
Illustration by
Paul Rand
(born Peretz Rosenbaum)
A more sophisticated example
of the ninefold square
in graphic design:
“That old Jew
gave me this here.”
— A Flag for Sunrise
(continued from
June 15, 2007)
Today is the anniversary
of the 1955 death of poet
Wallace Stevens.
Related material:
an essay on the
relationships between
poets and philosophers —
“Bad Blood,” by
Leonard Michaels —
and
“This translation plane is defined by
a spreadset in a 2-dimensional
vector space over the field GF(3),
consisting of the following matrices.”
Priv.-Doz. Dr. H. Klein,
Arbeitsgruppe Geometrie,
Mathematical Seminar of
Christian-Albrechts University
(See Log24, The Nine
and Translation Plane
for Rosh Hashanah.)
At the still point…
— Eliot
…da ist der Tanz;
Doch weder Stillstand noch Bewegung.
Und nenne es nicht Beständigkeit,
Wo Vergangenheit und Zukunft sich sammeln.
|
|
|
| to put one's back into something |
bei etwas Einsatz zeigen |
| to up the ante |
den Einsatz erhöhen |
| to debrief | den Einsatz nachher besprechen |
| to be on duty |
im Einsatz sein |
| mil.to be in action | im Einsatz sein |
| to play for high stakes |
mit hohem Einsatz spielen |
"Nine is a very
powerful Nordic number."
— Katherine Neville,
The Magic Circle
Rosalind Krauss
in "Grids," 1979:
"If we open any tract– Plastic Art and Pure Plastic Art or The Non-Objective World, for instance– we will find that Mondrian and Malevich are not discussing canvas or pigment or graphite or any other form of matter. They are talking about Being or Mind or Spirit. From their point of view, the grid is a staircase to the Universal, and they are not interested in what happens below in the Concrete.
Or, to take a more up-to-date example…."
"He was looking at
the nine engravings
and at the circle,
checking strange
correspondences
between them."
— The Club Dumas,1993
"And it's whispered that soon
if we all call the tune
Then the piper will lead us
to reason."
— Robert Plant,1971
The nine engravings of
The Club Dumas
(filmed as "The Ninth Gate")
are perhaps more an example
of the concrete than of the
universal.
An example of the universal*–
or, according to Krauss, a
"staircase" to the universal–
is the ninefold square:
"This is the garden of Apollo,
the field of Reason…."
— John Outram, architect
For more on the field
of reason, see
Log24, Oct. 9, 2006.
A reasonable set of
"strange correspondences"
in the garden of Apollo
has been provided by Ezra Brown
in a mathematical essay (pdf).
Unreason is, of course,
more popular.
* The ninefold square is perhaps a "concrete universal" in the sense of Hegel:
"Two determinations found in all philosophy are the concretion of the Idea and the presence of the spirit in the same; my content must at the same time be something concrete, present. This concrete was termed Reason, and for it the more noble of those men contended with the greatest enthusiasm and warmth. Thought was raised like a standard among the nations, liberty of conviction and of conscience in me. They said to mankind, 'In this sign thou shalt conquer,' for they had before their eyes what had been done in the name of the cross alone, what had been made a matter of faith and law and religion– they saw how the sign of the cross had been degraded."
— Hegel, Lectures on the History of Philosophy, "Idea of a Concrete Universal Unity"
"For every kind of vampire,
there is a kind of cross."
— Thomas Pynchon
Pandora's Box
Part I:
The Pandora Cross
— Rosalind Krauss in "Grids"
Part II:
The Opening
Remarks by the Pope on Sept. 12,
as reported by the Vatican:
Faith, Reason, and the University:
Memories and Reflections
For the result of
the Pope's remarks, see
a transcript of
yesterday's Google News
and the following
from BBC today:
Click to enlarge the screenshot.
Part III:
Hope
"In September [2005], she had a private audience with Pope Benedict XVI at Castel Gandolfo, his summer residence outside Rome. She had criticized John Paul II for making overtures to Muslims, and for not condemning terrorism heartily enough, but she has hopes for Joseph Ratzinger."
For further details, see yesterday's Log24.
Part IV:
The Sibyl's Song
— From The Magic Circle,
a spiritual narrative
by Katherine Neville
For more on "the long-mute voice
of the past," on "darkness beneath
the volcano," and on uncorking,
see Glory Season and Harrowing.
Related material from
Log24 on Dec. 2, 2005:
Benedict XVI, before he became Pope:
and a related
Christian symbol,
the Greek Cross
(adapted from
Ad Reinhardt).
Moral of the Pandora Cross:
"Nine is a very powerful Nordic number."
— Katherine Neville in The Magic Circle…
quoted in The Nine, a Log24 entry
for Hermann Weyl's birthday,
November 9, 2004.
"Nine is a very powerful Nordic number."
— Katherine Neville, author of The Eight,
"To live is to defend a form."
("Leben, das heisst eine Form verteidigen")
— attributed to Hölderlin
In defense of the nine-square grid:
For details on the above picture, see
Translation Plane.
ART WARS:
Graphical Password
From a summary of “The Design and Analysis of Graphical Passwords“:
“Results from cognitive science show that people can remember pictures much better than words….
The 5×5 grid creates a good balance between security and memorability.”
— Ian Jermyn, New York University; Alain Mayer, Fabian Monrose, Michael K. Reiter, Bell Labs, Lucent Technologies; Aviel Rubin, AT&T Labs — Research
Illustration — Warren Beatty as
a graphical password:
“Town & Country,”
released April 27, 2001
Those who prefer the simplicity of a 3×3 grid are referred to my entry of Jan. 9, 2003, Balanchine’s Birthday. For material related to the “Town & Country” theme and to Balanchine, see Leadbelly Under the Volcano (Jan. 27, 2003). (“Sometimes I live in the country, sometimes I live in town…” – Huddie Ledbetter). Those with more sophisticated tastes may prefer the work of Stephen Ledbetter on Gershwin’s piano preludes or, in view of Warren Beatty’s architectural work in “Town & Country,” the work of Stephen R. Ledbetter on window architecture.
As noted in Balanchine’s Birthday, Apollo (of the Balanchine ballet) has been associated by an architect with the 3×3, or “ninefold” grid. The reader who wishes a deeper meditation on the number nine, related to the “Town & Country” theme and more suited to the fact that April is Poetry Month, is referred to my note of April 27 two years ago, Nine Gates to the Temple of Poetry.
Intermediate between the simplicity of the 3×3 square and the (apparent) complexity of the 5×5 square, the 4×4 square offers an introduction to geometrical concepts that appears deceptively simple, but is in reality fiendishly complex. See Geometry for Jews. The moral of this megilla?
32 + 42 = 52.
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