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
Tuesday, May 9, 2023
Thursday, August 25, 2022
Meditation on a Song Lyric
"I saw a werewolf with a Chinese menu in his hand
Walking through the streets of Soho in the rain"
See other posts now tagged Structure Character.
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Thursday, May 26, 2022
Monday, May 23, 2022
Sunday, May 22, 2022
“Church Socials”* Continues . . .
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Friday, May 20, 2022
Squares to Triangles
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Wednesday, May 18, 2022
“The form, the pattern”
An image from Slovenia missed earlier* in the search above —
"Et cetera, et cetera, et cetera." — Oscar Hammerstein
* See "Robin Wilson" in the Design Grammar post of
19 Oct. 2017. The author of the above document may
or may not be the Robin Wilson of Gresham College.
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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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Wednesday, April 27, 2022
Ennead (Pace Moon Knight)
Putting the graphic in lexicographic —
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Monday, April 18, 2022
Iconic Simplicity
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Sunday, April 10, 2022
Monday, January 3, 2022
Nine Perfect Coordinates: Ex Fano Continues.
Update of Jan. 4, 2022 —
"Nine Perfect Strangers was certainly not helped by premiering
just days after HBO aired the finale of The White Lotus,
the summer’s word-of-mouth TV hit that’s also a stacked
ensemble of strangers gathering at a lush hotel, and a much more
scathing, focused and riveting satire . . . ."
— Adrian Horton in The Guardian , Aug. 25, 2021
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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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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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Friday, January 7, 2011
Coxeter and the Aleph
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.
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Thursday, August 4, 2005
Thursday August 4, 2005
Visible Mathematics, continued
Today's mathematical birthdays:
Saunders Mac Lane, John Venn,
and Sir William Rowan Hamilton.
It is well known that the quaternion group is a subgroup of GL(2,3), the general linear group on the 2-space over GF(3), the 3-element Galois field.
The figures below illustrate this fact.
Related material: Visualizing GL(2,p)
"The typical example of a finite group is GL(n,q), the general linear group of n dimensions over the field with q elements. The student who is introduced to the subject with other examples is being completely misled."
— J. L. Alperin, book review,
Bulletin (New Series) of the American
Mathematical Society 10 (1984), 121
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Thursday, January 9, 2003
Thursday January 9, 2003
Balanchine's Birthday
Today seems an appropriate day to celebrate Apollo and the nine Muses.
From a website on Balanchine's and Stravinsky's ballet, "Apollon Musagete":
In his Poetics of Music (1942) Stravinsky says: "Summing up: What is important for the lucid ordering of the work– for its crystallization– is that all the Dionysian elements which set the imagination of the artist in motion and make the life-sap rise must be properly subjugated before they intoxicate us, and must finally be made to submit to the law: Apollo demands it." Stravinsky conceived Apollo as a ballet blanc– a "white ballet" with classical choreography and monochromatic attire. Envisioning the work in his mind's eye, he found that "the absence of many-colored hues and of all superfluities produced a wonderful freshness." Upon first hearing Apollo, Diaghilev found it "music somehow not of this world, but from somewhere else above." The ballet closes with an Apotheosis in which Apollo leads the Muses towards Parnassus. Here, the gravely beautiful music with which the work began is truly recapitulated "on high"– ceaselessly recycled, frozen in time.
— Joseph Horowitz
Another website invoking Apollo:
The icon that I use… is the nine-fold square…. The nine-fold square has centre, periphery, axes and diagonals. But all are present only in their bare essentials. It is also a sequence of eight triads. Four pass through the centre and four do not. This is the garden of Apollo, the field of Reason….
In accordance with these remarks, here is the underlying structure for a ballet blanc:
This structure may seem too simple to support movements of interest, but consider the following (click to enlarge):
As Sir Arthur Quiller-Couch, paraphrasing Horace, remarks in his Whitsun, 1939, preface to the new edition of the Oxford Book of English Verse, "tamen usque recurret Apollo."
The alert reader will note that in the above diagrams, only eight of the positions move.
Which muse remains at the center?
Consider the remark of T. S. Eliot, "At the still point, there the dance is," and the fact that on the day Eliot turned 60, Olivia Newton-John was born. How, indeed, in the words of another "sixty-year-old smiling public man," can we know the dancer from the dance?