"I could a tale unfold . . ." — Hamlet's father's ghost
Guarini's Four-Knight Problem of 1512 …
Sunday, January 5, 2025
Precursor to Dürer’s 1514 Square
Comments Off on Precursor to Dürer’s 1514 Square
Friday, January 13, 2023
Dürer, Music, and Doktor Faustus
The previous post and Mann's Doktor Faustus in this journal
suggest a look at . . .
Comments Off on Dürer, Music, and Doktor Faustus
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 —
Comments Off on Affine Dürer
Saturday, February 16, 2019
Melancholy for Dürer
Comments Off on Melancholy for Dürer
Wednesday, October 18, 2017
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
Wednesday, January 21, 2026
Thomas Mann’s High Concept:
Magic Square Meets Magic Mountain
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 —
Comments Off on Thomas Mann’s High Concept:
Magic Square Meets Magic Mountain
Magic Square Meets Magic Mountain
Thursday, November 20, 2025
For Doctor Faustus*
Design from 1514
"One of those bells that now
and then rings" — Song lyric
* And Royal Holloway University.
Comments Off on For Doctor Faustus*
Wednesday, November 5, 2025
Baggage Claim
Comments Off on Baggage Claim
Saturday, March 15, 2025
Monday, January 13, 2025
From the Dies Natalis of
a Tolkien Mapmaker
This post was suggested by a New York Times
"Overlooked No More" obituary today.
See also Durer Knight in this journal.
Comments Off on From the Dies Natalis of
a Tolkien Mapmaker
a Tolkien Mapmaker
Friday, January 3, 2025
Wednesday, January 1, 2025
Thursday, September 19, 2024
Embedding a Language Witch
Continuing yesterday's "Bell, Book and Candle" theme . . .
[Such lines and planes have not been, in mathematical language,
"translated."]
— Paul R. Halmos, Finite Dimensional Vector Spaces,
Princeton University Press, 1948, page 14
Candle from Sense8, Season 1, Episode 1: “Limbic Resonance”
Comments Off on Embedding a Language Witch
Thursday, July 11, 2024
Saturday, May 18, 2024
The Godfather’s Art
Cover illustration:
Spies returning from the land of
Canaan with a cluster of grapes.
Colored woodcut from
Biblia Sacra Germanica ,
Nuremberg, Anton Koberger, 1483.
Victoria and Albert Museum, London.
Related material —
The Faustus Square :
Design from 1514
Comments Off on The Godfather’s Art
Thursday, January 18, 2024
But Seriously: Mathematics for Davos
Click image to enlarge.
Comments Off on But Seriously: Mathematics for Davos
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
Monday, January 17, 2022
Finest Trick
"The Magician’s finest trick was to
dismantle the pretensions of genius
while preserving his own lofty stature."
— Alex Ross in The New Yorker , Jan. 17, 2022
Related material —
Meanwhile . . .
Comments Off on Finest Trick
Monday, March 1, 2021
Choice of Viewpoint . . .
Comments Off on Choice of Viewpoint . . .
Sunday, August 2, 2020
Tuesday, February 20, 2018
Sunday, August 7, 2016
A Talisman for Finkelstein
The late physicist David Ritz Finkelstein on the magic square
in Dürer's "Melencolia I" —
"As a child I wondered why such a square was called magic.
The Occult Philosophy [of Agrippa] answers this question
at least. They were used as magical talismans."
The correspondence in the previous post between
Figures A and B may serve as a devotional talisman
in memory of Finkelstein, a physicist who, in the sort of
magical thinking enjoyed by traditional Catholics, might
still be lingering in Purgatory.
See also this journal on the date of Finkelstein's death —
Comments Off on A Talisman for Finkelstein
Saturday, March 7, 2015
Film and Phenomenology
Continued from All Hallows' Eve, 2014.
Last year's Halloween post displayed the
Dürer print Knight, Death, and the Devil
(illustrated below on the cover of the book
Film and Phenomenology by Allan Casebier).
Cover illustration: Knight, Death, and the Devil,
by Albrecht Dürer
Some mathematics related to a different Dürer print —
Comments Off on Film and Phenomenology
Monday, November 3, 2014
Wisconsin Death Trip*
Courtesy of Mira Sorvino.
Enter Madison :
From “Intruders,” BBC America, Season 1, Episode 2, at 1:07 of 43:31.
“You sure know how to show a girl a good time.“
* The title is a reference to a Wisconsin-related Halloween post.
Comments Off on Wisconsin Death Trip*
Friday, October 31, 2014
For the Late Hans Schneider
See a University of Wisconsin obituary for Schneider,
a leading expert on linear algebra who reportedly died
at 87 on Tuesday, October 28, 2014.
Some background on linear algebra and “magic” squares:
tonight’s 3 AM (ET) post and a search in this
journal for Knight, Death, and the Devil.
Click image to enlarge.
Comments Off on For the Late Hans Schneider
Structure
Introducing a group of 322,560 affine transformations of Dürer’s ‘Magic’ Square
The four vector-space substructures of digits in 1st, 2nd, 3rd, 4th place,
together with the diamond theorem, indicate that Dürer’s square “minus one”
can be transformed by permutations of rows, columns, and quadrants to a
square with (decimal) digits in the usual numerical order, increasing from
top left to bottom right. Such permutations form a group of order 322,560.
(Continued from Vector Addition in a Finite Field, Twelfth Night, 2013.)
Comments Off on Structure
Tuesday, April 9, 2013
Four Quartets
For the cruelest month
Click for a much larger version of the photo below.
These four Kountry Korn quartets are from the Fox Valleyaires
Men's Barbershop Chorus of Appleton, Wisconsin.
See also the fine arts here on Saturday, April 6, 2013—
The New York Times Magazine cover story
a decade ago, on Sunday, April 6, 2003:
"The artists demanded space
in tune with their aesthetic."
— "The Dia Generation,"
by Michael Kimmelman
Related material:
See Wikipedia for the difference between binary numbers
and binary coordinates from the finite Galois field GF(2).
For some background, see the relativity problem.
See also the chapter on vector spaces in Korn & Korn
(originally published by McGraw-Hill)—
.
Comments Off on Four Quartets
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
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
Sunday, December 2, 2012
Realism
From the Los Angeles Times yesterday—
"Chess player Elena Akhmilovskaya Donaldson sits
in deep concentration at the U.S. chess championship
in Seattle in 2002. (Greg Gilbert / Seattle Times /
January 5, 2002)"
Linda Shaw, Seattle Times :
"Elena Akhmilovskaya Donaldson, who was once the world's
second-ranked women's chess player and eloped in 1988
with the captain of the U.S. chess team when they were both
playing at a tournament in Greece, has died. She was 55.
Donaldson, who earned the title of international women's
grandmaster, died Nov. 18 in her adopted hometown of Seattle…."
From the Log24 post "Sermon" on the date of Donaldson's death,
Sunday, Nov. 18, 2012—
"You must allow us to play every conceivable combination of chess."
— Marie-Louise von Franz in Number and Time
An October 2011 post titled Realism in Plato's Cave displays
the following image:
Cover illustration: Knight, Death, and the Devil,
by Albrecht Dürer
George Steiner and myself in Closing the Circle, a Log24 post
of Sept. 4, 2009:
“Allegoric associations of death with chess are perennial….”
"Yes, they are."
For related remarks on knight moves and the devil, see
today's previous two posts, Knight's Labyrinth and The Rite.
Comments Off on Realism