Exercise: Show that Dürer's 1514 "magic" square is an affine automorphism.
For a solution, see other posts now tagged Affine Squares.
Sunday, June 9, 2024
Thursday, August 25, 2022
Monday, May 30, 2022
Friday, December 10, 2021
Dance of the Lo Shu
The ancient Chinese matrix known as the Lo Shu
is one of 432 matrices equivalent under the action of . . .
The Lo Shu Group:
For related material, see (for instance) AGL(2,3) in . . .
"Let be be finale of seem.
The only emperor is the emperor of ice-cream."
— Wallace Stevens
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Thursday, December 9, 2021
Lo Shu Space . . .
. . . is now at loshu.space. (Update on 10 Dec. — See also loshu.group.)
See as well GL(2,3) in this journal.
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Saturday, July 20, 2024
Annals of Western Philosophy:
Blue Opera Meets Blue Magic
For the "Blue Opera" part, see other posts so tagged.
The above may, if one likes, be regarded as ekphrasis for . . .
(For the part played by "six" and "nine" above, see Lo Shu.)
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Blue Opera Meets Blue Magic
Blue Opera Meets Blue Magic
Tuesday, May 9, 2023
Monday, November 21, 2022
Deep Pool
"Nietzsche in Basel studied the deep pool
Of these discolorations, mastering
The moving and the moving of their forms
In the much-mottled motion of blank time."
— Wallace Stevens, "Description Without Place"
Also in Basel, a mathematics professor contemplated the Lo Shu —
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Thursday, September 22, 2022
Affine Dürer
The previous post's image illustrating the
ancient Lo Shu square as an affine transformation
suggests a similar view of Dürer's square.
That view illustrates the structural principle
underlying the diamond theorem —
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Saturday, September 10, 2022
Orthogonal Latin Triangles
From a 1964 recreational-mathematics essay —
Note that the first two triangle-dissections above are analogous to
mutually orthogonal Latin squares . This implies a connection to
affine transformations within Galois geometry. See triangle graphics
in this journal.
Update of 4:40 AM ET —
Other mystical figures —
"Before time began, there was the Cube."
— Optimus Prime in "Transformers" (Paramount, 2007)
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Wednesday, June 22, 2022
Occupy Space Continues.
Alternate Title —
Types of Ambiguity:
The Circle in the Triangle,
the Singer in the Song.
From an excellent June 17 Wall Street Journal review of a new
Isaac Bashevis Singer book from Princeton University Press —
" 'Old Truths and New Clichés,' a collection of 19
prose articles, most appearing in English for the
first time, reveals that Singer was as consummate
an essayist as he was a teller of tales." — Benjamin Balint
From a search in this journal for Singer —
Related material —
From a post of June 2, "Self-Enclosing" —
|
"… the self-enclosing processes by which late 20th-century
— Colin Burrow in the June 9, 2022 issue |
From the December 14, 2021, post Notes on Lines —
The triangle, a percussion instrument that was
featured prominently in the Tom Stoppard play
"Every Good Boy Deserves Favour."
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Tuesday, June 14, 2022
A Triangle of Sadness
The title refers to this year's
Cannes Film Festival winner.
Related material:
From a post of June 2, "Self-Enclosing" —
|
"… the self-enclosing processes by which late 20th-century
— Colin Burrow in the June 9, 2022 issue |
From a post of June 13, "The Theater Game" —
From a post of June 12, "Triangle.graphics, 2012-2022" —
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Sunday, June 12, 2022
Saturday, June 4, 2022
Thursday, May 26, 2022
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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Thursday, December 2, 2021
Something Old, Something New
Night at the Museum:
Fictional Version —
Not So Fictional —
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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, 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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Monday, July 12, 2021
Educational Series
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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
Do Hillbillies Dream of Dinner Parties?
The title was suggested by a New Yorker photo caption
about Yale on June 19, 2021 —
"Amy Chua, a celebrity professor at the top-ranked
law school in the country, is at the center of a
campus-wide fracas known as 'Dinner Party-gate.' "
Other recent Yale material —
Remarks related to New Haven and geometry —
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Friday, November 13, 2020
Raiders of the Lost Dorm Room
“That really is, really, I think, the Island of the Misfit Toys at that point.
You have crossed the Rubicon, you jumped on the crazy train and
you’re headed into the cliffs that guard the flat earth at that time, brother,”
said Rep. Denver Riggleman, a Republican congressman from Virginia,
in an interview."
— Jon Ward, political correspondent, Yahoo News , Nov. 12, 2020
The instinct for heaven had its counterpart:
The instinct for earth, for New Haven, for his room,
The gay tournamonde as of a single world
In which he is and as and is are one.
— Wallace Stevens, "An Ordinary Evening in New Haven"
Related material for comedians —
See as well Sallows in this journal.
“There exists a considerable literature
devoted to the Lo shu , much of it infected
with the kind of crypto-mystic twaddle
met with in Feng Shui.”
— Lee C. F. Sallows, Geometric Magic Squares ,
Dover Publications, 2013, page 121
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Thursday, October 19, 2017
Design Grammar***
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.
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Tuesday, October 17, 2017
Plan 9 Continues
See also Holy Field in this journal.
Some related mathematics —
Analysis of the Lo Shu structure —
Structure of the 3×3 magic square:
4 9 2
3 5 7 decreased by 1 is …
8 1 6
3 8 1
2 4 6
7 0 5
In base 3 —
10 22 01
02 11 20
21 00 12
As orthogonal Latin squares
(a well-known construction) —
1 2 0 0 2 1
0 1 2 2 1 0
2 0 1 1 0 2 .
— Steven H. Cullinane,
October 17, 2017
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Sunday, November 9, 2014
Twaddle
“There exists a considerable literature
devoted to the Lo shu , much of it infected
with the kind of crypto-mystic twaddle
met with in Feng Shui.”
— Lee C. F. Sallows, Geometric Magic Squares ,
Dover Publications, 2013, page 121
Cf. Raiders of the Lost Theorem, Oct. 13, 2014.
See also tonight’s previous post and
“Feng Shui” in this journal.
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Tuesday, October 21, 2014
Art as a Tool
Two news items on art as a tool:
Two Log24 posts related to the 3×3 grid, the underlying structure for China’s
ancient Lo Shu “magic” square:
Finally, leftist art theorist Rosalind Krauss in this journal
on AntiChristmas, 2010:
Which is the tool here, the grid or Krauss?
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