* Vide November 7, 2022.
** log24.com/log/pix25/250804-Bohr-personal-yin-yang-emblem.jpg
Monday, August 4, 2025
For the Boys in the Kitchen* —
Yin-Yang Man** at Ninefold Window of Opportunity
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Yin-Yang Man** at Ninefold Window of Opportunity
Yin-Yang Man** at Ninefold Window of Opportunity
Saturday, May 31, 2025
Plan 9* from Cold Mountain** — Ninefold.space
The URL http://ninefold.space now forwards to …
http://m759.net/wordpress/?tag=lo-shu .
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Monday, January 6, 2025
Wednesday, September 4, 2024
Autumn Ninefold Square
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).
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Tuesday, July 22, 2025
Wednesday, July 9, 2025
Plan 9 from Toronto
This 1950 use of Galois coordinates on a grid
foreshadows their later use by Cullinane to construct
the Miracle Octad Generator of R. T. Curtis and to
illustrate, as below, point-line duality in the Fano plane.
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Tuesday, July 1, 2025
Sunday, March 23, 2025
Graphic Sermon
The release date of the Paul Schrader film "First Reformed"
was May 18, 2018. This suggests the following graphic art . . .
From this journal on that date, art from The New York Times —
Ninefold-square art that is much more recent —
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Sunday, September 8, 2024
Sunday Morning Koppel
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…." |
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Sunday, August 11, 2024
Double Duals
"That in which space itself is contained"
— Wallace Stevens
In the ninefold square,
projective-perspectivity duality
corresponds to
projective-correlation duality.
Illustrations —
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Sunday, August 4, 2024
MacGuffin Variations
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Saturday, July 13, 2024
The Plane of Time
The ninefold square in the previous post suggests a review
of posts now tagged Plane of Time.
See as well . . .
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Wednesday, June 5, 2024
For a future Michael Crichton*— The Tachikawa Hint
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.
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Saturday, June 1, 2024
New Social Medium: Cara.app
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Monday, May 1, 2023
Chain of Title: Complete, Clean, Unencumbered!
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.
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Monday, May 16, 2022
Sketch for a Magic Triangle
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 —
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Tuesday, December 14, 2021
The Illustrator
This post was suggested by Chapter Two, "The Sprite and the Synergist,"
of Alfred Bester's The Deceivers , and by . . .
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Tuesday, December 7, 2021
Tortoise Variations
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)
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Saturday, November 20, 2021
The Unmagicking
|
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 |
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Tuesday, November 9, 2021
Gestalt für Schicksalstag
"… 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) —
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Tuesday, August 10, 2021
Ex Fano Apollinis
|
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
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Saturday, July 10, 2021
Plan 9 from Ancient China
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.
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Monday, July 5, 2021
For Stonehearst Asylum
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 —
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Saturday, July 3, 2021
Here, There, and Chicago
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.
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Thursday, April 15, 2021
Grid View and List View
From AntiChristmas 2010 —
Above: Art Theorist Rosalind Krauss and The Ninefold Square
For Krauss:
Grid View List View
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Monday, February 17, 2020
RIP Charles Portis
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 . . .
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Sunday, May 26, 2019
Friday, May 17, 2019
Sunday, December 9, 2018
Quaternions in a Small Space
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 —
-
Visualizing GL(2,p) — A 1985 note illustrating group actions
on the 3×3 (ninefold) square. -
Another 1985 note showing group actions on the 3×3 square
transferred to the 2x2x2 (eightfold) cube. - Quaternions in an Affine Galois Plane — A webpage from 2010.
Related material —
See as well the two Log24 posts of December 1st, 2018 —
Character and In Memoriam.
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