Log24

Saturday, February 16, 2019

Melancholy for Dürer

Filed under: General — Tags: — m759 @ 11:53 AM

The title refers to a 1514 engraving.

See also Angel Particle in this  journal.

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.

Tuesday, February 20, 2018

A Sharper Image

Filed under: General — m759 @ 5:15 AM

Diamond-shaped face of Durer's 'Melencolia I' solid, with  four colored pencils from Diane Robertson Design

Click for some related posts.

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 —

Monday, November 3, 2014

Wisconsin Death Trip*

Filed under: General — Tags: — m759 @ 8:59 PM

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.

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.

Structure

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.)

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.

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
 

Sunday, December 2, 2012

Realism

Filed under: General — Tags: , — m759 @ 4:30 PM

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…."

more »

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

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.

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

http://www.log24.com/log/pix12/120110-AMS_page-Detail.jpg

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—

http://www.log24.com/log/pix12/120110-DiamondPuzzleFigure.jpg

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

http://www.log24.com/log/pix11A/110625-CubeHypostases.gif

Ousia

Click for further details:

http://www.log24.com/log/pix11A/110625-ProjectiveTrinitySm.jpg

 

Monday, October 3, 2011

Realism in Plato’s Cave

Filed under: General — m759 @ 8:08 PM

In memory of the late combinatorialist-philosopher Gian-Carlo Rota

Excerpts from the introduction to Allan Casebier's

Film and Phenomenology: Towards a Realist Theory of Cinematic Representation
(Cambridge Studies in Film, Cambridge University Press, 1991) —

Pages 1-2,  pages 3-4,  pages 5-6.

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

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

Monday, August 8, 2011

Diamond Theory vs. Story Theory (continued)

Filed under: General,Geometry — m759 @ 5:01 PM

Some background

Richard J. Trudeau, a mathematics professor and Unitarian minister, published in 1987 a book, The Non-Euclidean Revolution , that opposes what he calls the Story Theory of truth [i.e., Quine, nominalism, postmodernism] to what he calls the traditional Diamond Theory of truth [i.e., Plato, realism, the Roman Catholic Church]. This opposition goes back to the medieval "problem of universals" debated by scholastic philosophers.

(Trudeau may never have heard of, and at any rate did not mention, an earlier 1976 monograph on geometry, "Diamond Theory," whose subject and title are relevant.)

From yesterday's Sunday morning New York Times

"Stories were the primary way our ancestors transmitted knowledge and values. Today we seek movies, novels and 'news stories' that put the events of the day in a form that our brains evolved to find compelling and memorable. Children crave bedtime stories…."

Drew Westen, professor at Emory University

From May 22, 2009

Poster for 'Diamonds' miniseries on ABC starting May 24, 2009

The above ad is by
  Diane Robertson Design—

Credit for 'Diamonds' miniseries poster: Diane Robertson Design, London

Diamond from last night’s
Log24 entry, with
four colored pencils from
Diane Robertson Design:

Diamond-shaped face of Durer's 'Melencolia I' solid, with  four colored pencils from Diane Robertson Design
 
See also
A Four-Color Theorem.

For further details, see Saturday's correspondences
and a diamond-related story from this afternoon's
online New York Times.

Thursday, May 27, 2010

A Gathering for Gardner

Filed under: General,Geometry — Tags: — m759 @ 6:00 AM

"You ain't been blue; no, no, no.
 You ain't been blue,
 Till you've had that mood indigo."
 — Song lyrics, authorship disputed

 

Indigo (web color) (#4B0082)

"Pigment indigo (web color indigo) represents
 the way the color indigo was always reproduced
 in pigments, paints, or colored pencils in the 1950s."

Related mythology:

Indigo Children and the classic
1964 film Children of the Damned

Image-- Children of the Damned take sanctuary in St. Dunstan's Church.

Related non-mythology:

Colored pencils

Image-- Diamond-shaped face of Durer's 'Melencolia I' solid, with four colored pencils from Diane Robertson Design

Friday, September 4, 2009

Friday September 4, 2009

Filed under: General — m759 @ 2:02 PM
Closing the Circle

Continued from Monday

“This is a chapel 
 of mischance;
ill luck betide it, ’tis
the cursedest kirk
  that ever I came in!”

Philip Kennicott on
Kirk Varnedoe in
The Washington Post:

“Varnedoe’s lectures were
ultimately about faith,
about his faith in
the power of abstraction,
 and abstraction as a kind of
    anti-religious faith in itself….”

Kennicott’s remarks were
 on Sunday, May 18, 2003.
They were subtitled
“Closing the Circle
on Abstract Art.”

Also on Sunday, May 18, 2003:

 “Will the circle be unbroken?
  As if some southern congregation
  is praying we will come to understand.”


Princeton University Press
:

Empty canvas on cover of Varnedoe's 'Pictures of Nothing'

See also

Parmiggiani’s 
  Giordano Bruno

Parmiggiani's Bruno: empty canvas with sculpture of Durer's solid

Dürer’s Melencolia I

Durer, Melencolia I

and Log24 entries
of May 19-22, 2009,
ending with
    “Steiner System” —

Diamond-shaped face of Durer's 'Melencolia I' solid, with  four colored pencils from Diane Robertson Design

George Steiner on chess
(see yesterday morning):

“There are siren moments when quite normal creatures otherwise engaged, men such as Lenin and myself, feel like giving up everything– marriage, mortgages, careers, the Russian Revolution– in order to spend their days and nights moving little carved objects up and down a quadrate board.”

Steiner continues

“Allegoric associations of death with chess are perennial….”

Yes, they are.

April is Math Awareness Month.
This year’s theme is “mathematics and art.”

Mathematics and Art: Totentanz from Seventh Seal

Cf. both of yesterday’s entries.

Friday, May 22, 2009

Friday May 22, 2009

Filed under: General,Geometry — Tags: — m759 @ 9:29 AM
Steiner System

New York Times
banner this morning:

NYT banner, 9:21 AM Friday, May 22, 2009-- Ears are ads for HSBC.

Click to enlarge.

Related material from
July 11, 2008:

HSBC logo with framed version

The HSBC Logo Designer —

Henry Steiner

Henry Steiner, designerHe is an internationally recognized corporate identity consultant. Based in Hong Kong, his work for clients such as HongkongBank, IBM and Unilever is a major influence in Pacific Rim design.

Born in Austria and raised in New York, Steiner was educated at Yale under Paul Rand and attended the Sorbonne as a Fulbright Fellow. He is a past President of Alliance Graphique Internationale. Other professional affiliations include the American Institute of Graphic Arts, Chartered Society of Designers, Design Austria, and the New York Art Directors' Club.

His Cross-Cultural Design: Communicating in the Global Marketplace was published by Thames and Hudson (1995).

Yaneff.com

 

Charles Taylor,
"Epiphanies of Modernism,"
Chapter 24 of Sources of the Self
  (Cambridge U. Press, 1989, p. 477):

 

"… the object sets up
 a kind of frame or space or field
   within which there can be epiphany."

 

Related material suggested by
an ad last night on
ABC's Ugly Betty season finale:

Poster for 'Diamonds' miniseries on ABC starting May 24, 2009

Credit for 'Diamonds' miniseries poster: Diane Robertson Design, London

Diamond from last night's
Log24 entry, with
four colored pencils from
Diane Robertson Design:

Diamond-shaped face of Durer's 'Melencolia I' solid, with  four colored pencils from Diane Robertson Design
 
See also
A Four-Color Theorem.

Thursday, May 21, 2009

Thursday May 21, 2009

Filed under: General — m759 @ 8:14 PM
Diamonds

http://www.log24.com/log/pix09/090521-Plato-Durer.jpg

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