From http://m759.net/wordpress/?s=Faustus —
Design from 1514
"One of those bells that now
and then rings" — Song lyric
From http://m759.net/wordpress/?tag=zauberberg —
From the Web today —
Wednesday, January 21, 2026
Thomas Mann’s High Concept:
Magic Square Meets Magic Mountain
Comments Off on Thomas Mann’s High Concept:
Magic Square Meets Magic Mountain
Magic Square Meets Magic Mountain
Wednesday, December 21, 2022
The Unmagic Square
Last year on this date:
A Riddler Wannabe —
Related material — The Krauss passage quoted as above
by Shechtman in The New Yorker in December 2021 appears
also in a Log24 post of October 18, 2017: "Three Small Grids."
Comments Off on The Unmagic Square
Thursday, December 6, 2012
Magic Square
This post was suggested by the December 4th death
of modernist composer Jonathan Harvey, 73,
and by Harvey's reflections on his 2007 opera
Wagner Dream .
For related reflections, see the Oct. 10 post on
the Dürer magic square in Mann's Doctor Faustus .
See also a December 2nd post on the Nov. 18 death of
chess grandmaster Elena Akhmilovskaya Donaldson.
Comments Off on Magic Square
Thursday, October 14, 2010
Diamond Theory and Magic Squares
"A world of made
is not a world of born— pity poor flesh
and trees, poor stars and stones, but never this
fine specimen of hypermagical
ultraomnipotence."
— e. e. cummings, 1944
For one such specimen, see The Matrix of Abraham—
a 5×5 square that is hypermagical… indeed, diabolical.
Related material on the algebra and geometry underlying some smaller structures
that have also, unfortunately, become associated with the word "magic"—
- Finite Geometry of the Square and Cube
- Clifford Pickover on a 4×4 square
-
Christopher J. Henrich on the geometry of 4×4 magic squares
(without any mention of [1] above or related work dating back to 1976)
" … listen: there's a hell
of a good universe next door; let's go"
— e. e. cummings
Happy birthday, e. e.
Comments Off on Diamond Theory and Magic Squares
Sunday, August 16, 2026
“Magic” Affine Transformations
Exercise: Show that Dürer's 1514 "magic" square is an affine automorphism.
Comments Off on “Magic” Affine Transformations
Monday, July 27, 2026
Note for a Jain Temple:
Magic Objects in Every Dimension
From ICM Invited Lectures & Panels for July 27, 2026:
|
Bhargava — "Inscribed on a wall near the entrance of the majestic Parshvanath Jain Temple in Khajuraho, India (built around 960 CE) is a 4 × 4 magic square with remarkable arithmetic properties. We will explain why this magic square is the unique one of its kind, up to certain transformations. This answers a question posed by the legendary geometer HSM Coxeter in 1938. We will also explain why there is a unique such magic object in every dimension." |
Related reading
-
Bhargava in this journal
-
How are algebraic, combinatorial, and geometric
structures unified within the Brick Space model?
- Coxeter, Symmetry, and Diabolic Magic Squares —
Comments Off on Note for a Jain Temple:
Magic Objects in Every Dimension
Magic Objects in Every Dimension
Wednesday, January 1, 2025
Wednesday, August 21, 2024
For a Fictional “Magic Child” *
Comments Off on For a Fictional “Magic Child” *
Saturday, January 20, 2024
Needful Things: Faustus at the Magic Mountain
Image from a Sunday, January 7th, 2024, post now tagged "A Seventh Seal" —
Related image from a "Mathematics for Davos" post of
Thursday, January 18, 2024 —
Comments Off on Needful Things: Faustus at the Magic Mountain
Saturday, December 9, 2023
How Many Magic Beans, Jack?
Two notes from this journal on December 2, 2023 —
A flashback for The November Man —
As for " my own 'diamond theorem' " —
|
Question — "What is mathematician Steven H. Cullinane known for?" From the Copilot response (shorn of footnotes) — "Steven H. Cullinane is known for his work in finite geometry, particularly his Diamond Theorem. This theorem explains the surprising symmetry properties of some simple graphic designs, such as those found in quilts. It also provides a model for the 15 points, 35 lines, and 15 hyperplanes of the finite projective space PG (3,2). Cullinane’s work underlies the structure of the 35 square patterns in R. T. Curtis’s Miracle Octad Generator. His theorem has applications to sporadic simple groups, the connection between orthogonal Latin squares and projective spreads, and the symmetry of Walsh functions." |
Comments Off on How Many Magic Beans, Jack?
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 —
Comments Off on Sketch for a Magic Triangle
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 |
Comments Off on The Unmagicking
Friday, February 26, 2021
Non-Chaos Non-Magic
For fans of “WandaVision” —
“1978 was perhaps the seminal year in the origin of chaos magic. . . .”
— Wikipedia article on Chaos Magic
Non-Chaos Non-Magic from Halloween 1978 —
Related material —
A doctoral student of a different Peter Cameron —
( Not to be confused with The Tin Man’s Hat. )
Comments Off on Non-Chaos Non-Magic
Monday, April 8, 2013
Magic for Jews
A commenter on Saturday's "Seize the Dia" has
suggested a look at the work of one Mark Collins.
Here is such a look (click to enlarge):
I find attempts to associate pure mathematics with the words
"magic" or "mystic" rather nauseating. (H. F. Baker's work
on Pascal's mystic hexagram is no exception; Baker was
stuck with Pascal's obnoxious adjective, but had no truck
with any mystic aspects of the hexagram.)
The remarks above by Clifford Pickover on Collins, Dürer, and
binary representations may interest some non-mathematicians,
who should not be encouraged to waste their time on this topic.
For the mathematics underlying the binary representation of
Dürer's square, see, for instance, my 1984 article "Binary
Coordinate Systems."
Those without the background to understand that article
may enjoy, instead of Pickover's abortive attempts above at
mathematical vulgarization, his impressively awful 2009 novel
Jews in Hyperspace .
Pickover's 2002 book on magic squares was, unfortunately,
published by the formerly reputable Princeton University Press.
Related material from today's Daily Princetonian :
See also Nash + Princeton in this journal.
Comments Off on Magic for Jews
Monday, August 17, 2026
Twaddle Waddle
“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
Comments Off on Twaddle Waddle
Thursday, November 6, 2025
On Middlemarch: “The Patterns Are Out There!”
|
The Hunt for the World’s Oldest Story From thunder gods to serpent slayers, scholars are reconstructing myths that vanished millennia ago. How much further can we go—and what might we find? By Manvir Singh in The New Yorker
October 13, 2025 The Reverend Edward Casaubon is Eliot’s grand study in futility: an aging, self-important, faintly ridiculous clergyman who has dedicated his life to an audacious quest. Casaubon is convinced that every mythic system is a decayed remnant of a single original revelation—a claim he plans to substantiate in his magnum opus, “The Key to All Mythologies.” He means to chart the world’s myths, trace their similarities, and produce a codex that, as Eliot puts it, would make “the vast field of mythical constructions . . . intelligible, nay, luminous with the reflected light of correspondences.” The ill-fated project founders between the unruly diversity of cultural traditions and the fantasy of a single source, between the expanse of his material and the impossibility of ever mastering it, between the need for theory and the distortions it introduces. These failures are deepened by Casaubon’s limitations—his pedantic love of minutiae (he “dreams footnotes”) and his refusal to engage with scholarship in languages he doesn’t know (if only he’d learned German).
Casaubon’s quest stands as both an indictment of overreach and a warning about the senselessness of such sweeping comparisons. But is this entirely fair? The patterns are out there. Floods, tricksters, battles with monsters, creation and apocalypse—sometimes the resemblances are uncanny. |
"Before time began . . ." — Optimus Prime
Comments Off on On Middlemarch: “The Patterns Are Out There!”
Thursday, September 26, 2024
Puzzle Art
From the post Belgian Puzzle Art —
Related reading . . .
— "The Devil, unlike the angels, was at home in the world of phenomena.
He knew how to combine pure concepts with empirical intuitions …
which is the basic principle of linguistic creation."
(Noah Jonathan Jacobs, Naming-Day in Eden, Macmillan, 1958 …
In Macmillan 1969 revised edition, page 21.)
The figure of 25 parts discussed in
"On Linguistic Creation"–
— "Such is the square dance of Numbers."
(Jacques Derrida, Dissemination, 1972)
— "It all adds up."
(Saul Bellow, book title, 1994)
Comments Off on Puzzle Art
Saturday, January 21, 2023
Dead-Poet Witcraft
"Death is the mother of beauty." — Wallace Stevens
From the 2020 Feast of St. Wallace Stevens,
who reportedly died in 1955 on August 2 —
Related material —
Exercise: Can each order-4 magic square be obtained by some
transformation like the one above (i.e., preserving affine hyperplanes)?
If not, why not?
Update of 31 Jan. 2023 — Spoiler: Remarks by Tilman Piesk.
Comments Off on Dead-Poet Witcraft
Wednesday, April 27, 2022
Ennead (Pace Moon Knight)
Putting the graphic in lexicographic —
Comments Off on Ennead (Pace Moon Knight)
Sunday, April 10, 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)
Comments Off on Tortoise Variations
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
Comments Off on Raiders of the Lost Dorm Room
Sunday, August 2, 2020
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.
Comments Off on Design Grammar***
Wednesday, October 18, 2017
Three Small Grids
An earlier post today, now tagged "Three Small Magic Squares,"
suggests a review of a post from October 25 three years ago
that contains the following figure —
Fans of the October Revolution may enjoy a passage
by Rosalind Krauss on grids:
Comments Off on Three Small Grids
Dürer for St. Luke’s Day
Structure of the Dürer magic square
16 3 2 13
5 10 11 8 decreased by 1 is …
9 6 7 12
4 15 14 1
15 2 1 12
4 9 10 7
8 5 6 11
3 14 13 0 .
Base 4 —
33 02 01 30
10 21 22 13
20 11 12 23
03 32 31 00 .
Two-part decomposition of base-4 array
as two (non-Latin) orthogonal arrays —
3 0 0 3 3 2 1 0
1 2 2 1 0 1 2 3
2 1 1 2 0 1 2 3
0 3 3 0 3 2 1 0 .
Base 2 –
1111 0010 0001 1100
0100 1001 1010 0111
1000 0101 0110 1011
0011 1110 1101 0000 .
Four-part decomposition of base-2 array
as four affine hyperplanes over GF(2) —
1001 1001 1100 1010
0110 1001 0011 0101
1001 0110 0011 0101
0110 0110 1100 1010 .
— Steven H. Cullinane,
October 18, 2017
See also recent related analyses of
noted 3×3 and 5×5 magic squares.
Comments Off on Dürer for St. Luke’s Day
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
Comments Off on Plan 9 Continues
Monday, October 16, 2017
Highway 61 Revisited
"God said to Abraham …." — Bob Dylan, "Highway 61 Revisited"
Related material —
-
"Inner Truth" in this journal
-
"On Linguistic Creation," June 25, 1999
-
"Ultra Super Magic Squares of 5×5" at
http://mathsforeurope.digibel.be/magic.htm .
See as well Charles Small, Harvard '64,
"Magic Squares over Fields" —
— and Conway-Norton-Ryba in this journal.
Some remarks on an order-five magic square over GF(52):
on the numbers 0 to 24:
22 5 18 1 14
3 11 24 7 15
9 17 0 13 21
10 23 6 19 2
16 4 12 20 8
Base-5:
42 10 33 01 24
03 21 44 12 30
14 32 00 23 41
20 43 11 34 02
31 04 22 40 13
Regarding the above digits as representing
elements of the vector 2-space over GF(5)
(or the vector 1-space over GF(52)) …
All vector row sums = (0, 0) (or 0, over GF(52)).
All vector column sums = same.
Above array as two
orthogonal Latin squares:
4 1 3 0 2 2 0 3 1 4
0 2 4 1 3 3 1 4 2 0
1 3 0 2 4 4 2 0 3 1
2 4 1 3 0 0 3 1 4 2
3 0 2 4 1 1 4 2 0 3
— Steven H. Cullinane,
October 16, 2017
Comments Off on Highway 61 Revisited
Tuesday, September 5, 2017
Florence 2001
Or: Coordinatization for Physicists
This post was suggested by the link on the word "coordinatized"
in the previous post.
I regret that Weyl's term "coordinatization" perhaps has
too many syllables for the readers of recreational mathematics —
for example, of an article on 4×4 magic squares by Conway, Norton,
and Ryba to be published today by Princeton University Press.
Insight into the deeper properties of such squares unfortunately
requires both the ability to learn what a "Galois field" is and the
ability to comprehend seven-syllable words.
Comments Off on Florence 2001
Thursday, August 31, 2017
A Conway-Norton-Ryba Theorem
In a book to be published Sept. 5 by Princeton University Press,
John Conway, Simon Norton, and Alex Ryba present the following
result on order-four magic squares —
A monograph published in 1976, “Diamond Theory,” deals with
more general 4×4 squares containing entries from the Galois fields
GF(2), GF(4), or GF(16). These squares have remarkable, if not
“magic,” symmetry properties. See excerpts in a 1977 article.
See also Magic Square and Diamond Theorem in this journal.
Comments Off on A Conway-Norton-Ryba Theorem