Wednesday, October 18, 2017

Dürer for St. Luke’s Day

Filed under: G-Notes,General,Geometry — Tags: — m759 @ 1:00 PM

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.

Thursday, December 6, 2012

Magic Square

Filed under: General — m759 @ 10:18 PM

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.

IMAGE- Chess grandmaster and Dürer's angel with magic square

Wednesday, March 20, 2019

Secret Characters

Filed under: General — Tags: , , , , — m759 @ 2:23 PM

"Cell 461" quote from Curzio Malaparte superimposed on a scene from
the 1963 Godard film "Le Mépris " ("Contempt") —

"The architecture… beomes closely linked to the script…."

Malaparte's cell number , 461, is somewhat less closely  linked
to the phrase "eternal blazon" —

Irving was quoted here on Dec. 22, 2008

The Tale of
the Eternal Blazon

by Washington Irving

Blazon  meant originally a shield , and then
the heraldic bearings on a shield .
Later it was applied to the art of describing
or depicting heraldic bearings in the proper
manner; and finally the term came to signify 
ostentatious display  and also description or
record by words or other means 
. In Hamlet ,
Act I Sc. 5, the Ghost, while talking with
Prince Hamlet, says:

‘But this eternal blazon must not be
To ears of flesh and blood.’

Eternal blazon  signifies revelation or description
of things pertaining to eternity 

— Irving’s Sketch Book , p. 461

Update of 6:25 PM ET —

"Self-Blazon of Edenic Plenitude"

(The Issuu text is taken from Speaking about Godard , by Kaja Silverman
and Harun Farocki, New York University Press, 1998, page 34.)

Saturday, October 28, 2017

Just the Facts

Filed under: General,Geometry — Tags: — m759 @ 11:00 AM

The New Yorker  on the recent film "The Square"

"It’s an aesthetic that presents,
so to speak, just the facts, 
as if the facts themselves weren’t
deeply layered with living history
and crisscrossed with vectors
of divergent ideas and ideals."

— Richard Brody, Thursday, Oct. 26, 2017

For other images deeply layered  and crisscrossed ,
see Geometry of the I Ching.

Dating Harvard

Filed under: General,Geometry — Tags: — m759 @ 10:00 AM

See also this journal on today's date four years ago.

Tuesday, October 17, 2017

Plan 9 Continues

Filed under: General,Geometry — Tags: — m759 @ 9:00 PM

See also Holy Field in this journal.

Some related mathematics —

IMAGE- Herbert John Ryser, 'Combinatorial Mathematics' (1963), page 1

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

Monday, October 16, 2017

Highway 61 Revisited

Filed under: G-Notes,General,Geometry — Tags: , , — m759 @ 10:13 AM

"God said to Abraham …." — Bob Dylan, "Highway 61 Revisited"

Related material — 

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):

"Ultra Super Magic Square"

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


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

Thursday, May 11, 2017

Reopening the Tesseract

Filed under: General,Geometry — Tags: — m759 @ 11:27 AM

Dialogue from the film "Interstellar" —

Cooper: Did it work?

TARS: I think it might have.

Cooper: How do you know?

TARS: Because the bulk beings
            are closing the tesseract.

Related material — "Bulk apperception"
in this journal, and

Wednesday, May 10, 2017

In the Park with Yin and Yang

Filed under: General,Geometry — Tags: — m759 @ 10:35 PM

In memory of an art dealer who 
reportedly died on Sunday, May 7—

Decorations for a Cartoon Graveyard

Tuesday, May 9, 2017

Text and Context

Filed under: General,Geometry — Tags: — m759 @ 2:11 PM

Some context for the previous post, which was about
a new Art Space  Pinterest board

Monday, May 8, 2017

New Pinterest Board

Filed under: General,Geometry — Tags: — m759 @ 9:29 PM


Sunday, May 7, 2017

Art Space

Filed under: General,Geometry — Tags: — m759 @ 1:00 PM

Detail of an image in the previous post

This suggests a review of a post on a work of art by fashion photographer
Peter Lindbergh, made when he was younger and known as "Sultan."

The balls in the foreground relate Sultan's work to my own.

Linguistic backstory —

The art space where the pieces by Talman and by Lindbergh
were displayed is Museum Tinguely in Basel.

As the previous post notes, the etymology of "glamour" (as in
fashion photography) has been linked to "grammar" (as in 
George Steiner's Grammars of Creation ). A sculpture by 
Tinguely (fancifully representing Heidegger) adorns one edition
of Grammars .

Yale University Press, 2001:

Tinguely, "Martin Heidegger,
Philosopher," sculpture, 1988

Sunday, August 7, 2016

A Talisman for Finkelstein

Filed under: General,Geometry — Tags: — m759 @ 12:00 AM

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 —

Saturday, March 7, 2015

Film and Phenomenology

Filed under: General,Geometry — m759 @ 1:18 PM

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: Durer's 'Knight, Death, and the Devil'

Cover illustration: Knight, Death, and the Devil
by Albrecht Dürer

Some mathematics related to a different Dürer print —

Friday, October 31, 2014

For the Late Hans Schneider

Filed under: General — Tags: — m759 @ 3:30 AM

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.


Filed under: General,Geometry — m759 @ 3:00 AM

On Devil’s Night

Introducing a group of 322,560 affine transformations of Dürer’s ‘Magic’ Square

IMAGE- Introduction to 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.)

Thursday, August 21, 2014

Remember me to…

Filed under: General,Geometry — Tags: — m759 @ 12:00 PM

Herald Square.

Friday, April 25, 2014


Filed under: General,Geometry — Tags: — m759 @ 2:45 AM

Quoted here on April 11 —

“…direct access to the godhead, which
in this case was Creativity.”
— Tom Wolfe, From Bauhaus to Our House

From “Today in History: April 25, 2014,” by The Associated Press:

“Five years ago… University of Georgia professor
George Zinkhan, 57, shot and killed his wife
and two men outside a community theater in Athens
before taking his own life.”

Related material:

A Google Scholar search for Zinkhan’s 1993 paper,
Creativity in Advertising,” Journal of Advertising  22,2: 1-3 —

Obiter Dicta:

“Dour wit” — Obituary of a Scots herald who died on Palm Sunday

“Remember me to Herald Square.” — Song lyric

“Welcome to Scotland.” — Kincade in Skyfall

Monday, October 28, 2013

Harvard Anniversary

Filed under: General,Geometry — Tags: — m759 @ 12:00 PM

From the AP Today in History  page
for October 28, 2013 —

IMAGE- Harvard founded Oct. 28.

From this journal seven years ago:

The Practical Cogitator


Wednesday, June 26, 2013


Filed under: General,Geometry — Tags: — m759 @ 10:10 AM

“I could a tale unfold whose lightest word
Would harrow up thy soul….

— Hamlet’s Father’s Ghost

The results of a search in this journal for “a tale unfold” suggest
a review of the following passage from Donna Tartt’s Secret History

A math weblog discussed this passage on January 24, 2013.
For related alliances, see this  weblog on that same date.

Monday, June 24, 2013

What Dreams

Filed under: General,Geometry — Tags: — m759 @ 10:00 PM

“For in that sleep of death what dreams may come
When we have shuffled off this mortal coil,
Must give us pause.” — Hamlet

Sleep well, Mr. Matheson.

Tuesday, April 9, 2013

Four Quartets

Filed under: General,Geometry — m759 @ 5:10 PM

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:

IMAGE- Clifford A. Pickover on symmetries in the Dürer 4x4 magic square, with a critique

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)—


Monday, April 8, 2013

Magic for Jews

Filed under: General,Geometry — Tags: , — m759 @ 12:00 PM

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.

Tuesday, January 10, 2012

Defining Form

Filed under: General,Geometry — Tags: , — m759 @ 9:00 AM

(Continued from Epiphany and from yesterday.)

Detail from the current American Mathematical Society homepage


Further detail, with a comparison to Dürer's magic square

http://www.log24.com/log/pix12/120110-Donmoyer-Still-Life-Detail.jpg http://www.log24.com/log/pix12/120110-DurerSquare.jpg

The three interpenetrating planes in the foreground of Donmoyer's picture
provide a clue to the structure of the the magic square array behind them.

Group the 16 elements of Donmoyer's array into four 4-sets corresponding to the
four rows of Dürer's square, and apply the 4-color decomposition theorem.
Note the symmetry of the set of 3 line diagrams that result.

Now consider the 4-sets 1-4, 5-8, 9-12, and 13-16, and note that these
occupy the same positions in the Donmoyer square that 4-sets of
like elements occupy in the diamond-puzzle figure below—


Thus the Donmoyer array also enjoys the structural  symmetry,
invariant under 322,560 transformations, of the diamond-puzzle figure.

Just as the decomposition theorem's interpenetrating lines  explain the structure
of a 4×4 square , the foreground's interpenetrating planes  explain the structure
of a 2x2x2 cube .

For an application to theology, recall that interpenetration  is a technical term
in that field, and see the following post from last year—

Saturday, June 25, 2011


Theology for Antichristmas

— m759 @ 12:00 PM

Hypostasis (philosophy)

"… the formula 'Three Hypostases  in one Ousia '
came to be everywhere accepted as an epitome
of the orthodox doctrine of the Holy Trinity.
This consensus, however, was not achieved
without some confusion…." —Wikipedia



Click for further details:



Friday, October 8, 2010

Starting Out in the Evening

Filed under: General,Geometry — Tags: , — m759 @ 12:00 PM

… and Finishing Up at Noon

This post was suggested by last evening’s post on mathematics and narrative
and by Michiko Kakutani on Vargas Llosa in this morning’s New York Times.


Above: Frank Langella in
Starting Out in the Evening

Right: Johnny Depp in
The Ninth Gate


“One must proceed cautiously, for this road— of truth and falsehood in the realm of fiction— is riddled with traps and any enticing oasis is usually a mirage.”

— “Is Fiction the Art of Lying?”* by Mario Vargas Llosa, New York Times  essay of October 7, 1984

My own adventures in that realm— as reader, not author— may illustrate Llosa’s remark.

A nearby stack of paperbacks I haven’t touched for some months (in order from bottom to top)—

  1. Pale Rider by Alan Dean Foster
  2. Franny and Zooey by J. D. Salinger
  3. The Hobbit by J. R. R. Tolkien
  4. Le Petit Prince by Antoine de Saint Exupéry
  5. Literary Reflections by James A. Michener
  6. The Ninth Configuration by William Peter Blatty
  7. A Streetcar Named Desire by Tennessee Williams
  8. Nine Stories by J. D. Salinger
  9. A Midsummer Night’s Dream by William Shakespeare
  10. The Tempest by William Shakespeare
  11. Being There by Jerzy Kosinski
  12. What Dreams May Come by Richard Matheson
  13. Zen and the Art of Motorcycle Maintenance by Robert M. Pirsig
  14. A Gathering of Spies by John Altman
  15. Selected Poems by Robinson Jeffers
  16. Hook— Tinkerbell’s Challenge by Tristar Pictures
  17. Rising Sun by Michael Crichton
  18. Changewar by Fritz Leiber
  19. The Painted Word by Tom Wolfe
  20. The Hustler by Walter Tevis
  21. The Natural by Bernard Malamud
  22. Truly Tasteless Jokes by Blanche Knott
  23. The Man Who Was Thursday by G. K. Chesterton
  24. Under the Volcano by Malcolm Lowry

What moral Vargas Llosa might draw from the above stack I do not know.

Generally, I prefer the sorts of books in a different nearby stack. See Sisteen, from May 25. That post the fanciful reader may view as related to number 16 in the above list. The reader may also relate numbers 24 and 22 above (an odd couple) to By Chance, from Thursday, July 22.

* The Web version’s title has a misprint— “living” instead of “lying.”

Friday, April 17, 2009

Friday April 17, 2009

Filed under: General,Geometry — Tags: — m759 @ 10:31 AM

Begettings of
the Broken Bold

Thanks for the following
quotation (“Non deve…
nella testa“) go to the
weblog writer who signs
himself “Conrad H. Roth.”

of Goethe

(Vol. II, London, Bell & Daldy,
1868, at Google Books):

… Yesterday I took leave of my Captain, with a promise of visiting him at Bologna on my return. He is a true


representative of the majority of his countrymen. Here, however, I would record a peculiarity which personally distinguished him. As I often sat quiet and lost in thought he once exclaimed “Che pensa? non deve mai pensar l’uomo, pensando s’invecchia;” which being interpreted is as much as to say, “What are you thinking about: a man ought never to think; thinking makes one old.” And now for another apophthegm of his; “Non deve fermarsi l’uomo in una sola cosa, perche allora divien matto; bisogna aver mille cose, una confusione nella testa;” in plain English, “A man ought not to rivet his thoughts exclusively on any one thing, otherwise he is sure to go mad; he ought to have in his head a thousand things, a regular medley.”

Certainly the good man could not know that the very thing that made me so thoughtful was my having my head mazed by a regular confusion of things, old and new. The following anecdote will serve to elucidate still more clearly the mental character of an Italian of this class. Having soon discovered that I was a Protestant, he observed after some circumlocution, that he hoped I would allow him to ask me a few questions, for he had heard such strange things about us Protestants that he wished to know for a certainty what to think of us.

Notes for Roth:

Roth and Corleone in Havana

The title of this entry,
“Begettings of the Broken Bold,”
is from Wallace Stevens’s
“The Owl in the Sarcophagus”–

This was peace after death, the brother of sleep,
The inhuman brother so much like, so near,
Yet vested in a foreign absolute,

Adorned with cryptic stones and sliding shines,
An immaculate personage in nothingness,
With the whole spirit sparkling in its cloth,

Generations of the imagination piled
In the manner of its stitchings, of its thread,
In the weaving round the wonder of its need,

And the first flowers upon it, an alphabet
By which to spell out holy doom and end,
A bee for the remembering of happiness.

Peace stood with our last blood adorned, last mind,
Damasked in the originals of green,
A thousand begettings of the broken bold.

This is that figure stationed at our end,
Always, in brilliance, fatal, final, formed
Out of our lives to keep us in our death....

Related material:

  • Yesterday’s entry on Giordano Bruno and the Geometry of Language
  • James Joyce and Heraldry
  • “One might say that he [Joyce] invented a non-Euclidean geometry of language; and that he worked over it with doggedness and devotion….” —Unsigned notice in The New Republic, 20 January 1941
  • Joyce’s “collideorscape” (scroll down for a citation)
  • “A Hanukkah Tale” (Log24, Dec. 22, 2008)
  • Stevens’s phrase from “An Ordinary Evening in New Haven” (Canto XXV)

Some further context:

Roth’s entry of Nov. 3, 2006–
Why blog, sinners?“–
and Log24 on that date:
First to Illuminate.”

Monday, December 22, 2008

Monday December 22, 2008

Filed under: General,Geometry — Tags: , — m759 @ 9:00 PM

The Folding

Hamlet, Act 1, Scene 5


“I could a tale unfold whose lightest word
Would harrow up thy soul, freeze thy young blood,
Make thy two eyes, like stars, start from their spheres,
Thy knotted and combined locks to part
And each particular hair to stand on end,
Like quills upon the fretful porpentine:
But this eternal blazon must not be
To ears of flesh and blood. List, list, O, list!”

This recalls the title of a piece in this week’s New Yorker:”The Book of Lists:
Susan Sontag’s early journals
.” (See Log24 on Thursday, Dec. 18.)

In the rather grim holiday spirit of that piece, here are some journal notes for Sontag, whom we may imagine as the ghost of Hanukkah past.

There are at least two ways of folding a list (or tale) to fit a rectangular frame.The normal way, used in typesetting English prose and poetry, starts at the top, runs from left to right, jumps down a line, then again runs left to right, and so on until the passage is done or the bottom right corner of the frame is reached.

The boustrophedonic way again goes from top to bottom, with the first line running from left to right, the next from right to left, the next from left to right, and so on, with the lines’ directions alternating.

The word “boustrophedon” is from the Greek words describing the turning, at the end of each row, of an ox plowing (or “harrowing”) a field.

The Tale of
the Eternal Blazon

by Washington Irving

Blazon meant originally a shield, and then the heraldic bearings on a shield.
Later it was applied to the art of describing or depicting heraldic bearings
in the proper manner; and finally the term came to signify ostentatious display
and also description or record by words or other means. In Hamlet, Act I. Sc. 5,
the Ghost, while talking with Prince Hamlet, says:

‘But this eternal blazon
must not be
To ears of flesh and blood.’

Eternal blazon signifies revelation or description of things pertaining to eternity.”

Irving’s Sketch Book, p. 461

By Washington Irving and Mary Elizabeth Litchfield, Ginn & Company, 1901

Related material:

Folding (and harrowing up)
some eternal blazons —

The 16 Puzzle: transformations of a 4x4 square
These are the foldings
described above.

They are two of the 322,560
natural ways to fit
the list (or tale)
“1, 2, 3, … 15, 16”
into a 4×4 frame.

For further details, see
The Diamond 16 Puzzle.

Moral of the tale:

Cynthia Zarin in The New Yorker, issue dated April 12, 2004–

“Time, for L’Engle, is accordion-pleated. She elaborated, ‘When you bring a sheet off the line, you can’t handle it until it’s folded, and in a sense, I think, the universe can’t exist until it’s folded– or it’s a story without a book.'”

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