Log24

Sunday, June 19, 2005

Sunday June 19, 2005

Filed under: General,Geometry — Tags: , , — m759 @ 4:00 am
ART WARS:
Darkness Visible
"No light, but rather darkness visible
 Serv'd only to discover sights of woe"
John Milton, Paradise Lost,
Book I,  lines 63-64
 
From the cover article (pdf) in the
June/July 2005 Notices of the
American Mathematical Society–

Martin Gardner

A famed vulgarizer, Martin Gardner,
summarizes the art of Ad Reinhardt
(Adolph Dietrich Friedrich Reinhardt,
  Dec. 24, 1913 – Aug. 30, 1967):
 
"Ed Rinehart [sic] made a fortune painting canvases that were just one solid color.  He had his black period in which the canvas was totally black.  And then he had a blue period in which he was painting the canvas blue.  He was exhibited in top shows in New York, and his pictures wound up in museums.  I did a column in Scientific American on minimal art, and I reproduced one of Ed Rinehart's black paintings.  Of course, it was just a solid square of pure black.  The publisher insisted on getting permission from the gallery to reproduce it."
 
Related material
from Log24.net,
Nov. 9-12, 2004:
 

Fade to Black

"…that ineffable constellation of talents that makes the player of rank: a gift for conceiving abstract schematic possibilities; a sense of mathematical poetry in the light of which the infinite chaos of probability and permutation is crystallized under the pressure of intense concentration into geometric blossoms; the ruthless focus of force on the subtlest weakness of an opponent."

— Trevanian, Shibumi

"'Haven't there been splendidly elegant colors in Japan since ancient times?'

'Even black has various subtle shades,' Sosuke nodded."

— Yasunari Kawabata, The Old Capital

An Ad Reinhardt painting
described in the entry of
noon, November 9, 2004
is illustrated below.

Ad Reinhardt,  Greek Cross

Ad Reinhardt,
Abstract Painting, 1960-66.
Oil on canvas, 60 x 60 inches.
Solomon R. Guggenheim Museum

The viewer may need to tilt
the screen to see that this
painting is not uniformly black,
but is instead a picture of a
Greek cross, as described below.

"The grid is a staircase to the Universal…. We could think about Ad Reinhardt, who, despite his repeated insistence that 'Art is art,' ended up by painting a series of… nine-square grids in which the motif that inescapably emerges is a Greek cross.

 

Greek Cross

There is no painter in the West who can be unaware of the symbolic power of the cruciform shape and the Pandora's box of spiritual reference that is opened once one uses it."

— Rosalind Krauss,
Meyer Schapiro Professor
of Modern Art and Theory
at Columbia University

(Ph.D., Harvard U., 1969),
in "Grids"

The image “http://www.log24.com/log/pix04B/041109-Krauss.jpg” cannot be displayed, because it contains errors.

Krauss

 In memory of
St. William Golding
(Sept. 19, 1911 – June 19, 1993)

Wednesday, June 15, 2005

Wednesday June 15, 2005

Filed under: General — m759 @ 1:44 am

Cross-Referenced

From today’s New York Times,
a review of a Werner Herzog film,
“Wheel of Time,” that opens
today in Manhattan:

The image “http://www.log24.com/log/pix05A/050615-Mandala.jpg” cannot be displayed, because it contains errors.

From the June 13-14
midnight Log24 entry:

The image “http://www.log24.com/log/pix05A/050614-DarkCity.jpg” cannot be displayed, because it contains errors.

“With a little effort, anything can be
shown to connect with anything else:
existence is infinitely cross-referenced.”

— Opening sentence of
Martha Cooley’s The Archivist

These images suggest
a Google search on the phrase
crucified on the wheel of time,”

which yields the following.

Click to go to DARK CITYDARK CITY (1998)
Crucified on the Wheel Of Time.
A visual feast.

From Dark City: A Hollywood Jesus Movie Review

“There is something mesmerizing about this important film. It flows in the same vein as The Truman Show, The Game, and Pleasantville.  Something isn’t real with the world around John Murdoch. Some group is trying to control things and it isn’t God.”

Amen.

Related material:
Skewed Mirrors and
The Graces of Paranoia

Tuesday, June 14, 2005

Tuesday June 14, 2005

Filed under: General — m759 @ 12:00 am

ART WARS:
Dark City

Jennifer Connelly at
premiere of “Cinderella Man” —

The image “http://www.log24.com/log/pix05A/050614-Connelly.jpg” cannot be displayed, because it contains errors.

In memory of Martin Buber,
author of Good and Evil,
who died on June 13, 1965:

The image “http://www.log24.com/log/pix05A/050614-DarkCity.jpg” cannot be displayed, because it contains errors.

“With a little effort, anything can be
shown to connect with anything else:
existence is infinitely cross-referenced.”

— Opening sentence of
Martha Cooley’s The Archivist

Woe unto
them that
call evil
good, and
good evil;
that put
darkness
for light,
and light
for darkness

Isaiah 5:20

 

As she spoke
about the Trees
of Life and Death,
I watched her…. 
The Archivist

The world
has gone
mad today
And good’s
bad today,

And black’s
white today,
And day’s
night today


Cole Porter

Jennifer Connelly in “Dark City”

(from journal note of June 19, 2002) —

The image “http://www.log24.com/log/pix05A/050613-DarkCity.jpg” cannot be displayed, because it contains errors.

And, one might add for Flag Day,
“you sons of bitches.”

The image “http://www.log24.com/log/pix05A/050614-Flag.jpg” cannot be displayed, because it contains errors.
 

Saturday, June 4, 2005

Saturday June 4, 2005

Filed under: General,Geometry — Tags: — m759 @ 7:00 pm
  Drama of the Diagonal
  
   The 4×4 Square:
  French Perspectives

Earendil_Silmarils:
The image “http://www.log24.com/log/pix05A/050604-Fuite1.jpg” cannot be displayed, because it contains errors.
  
   Les Anamorphoses:
 
   The image “http://www.log24.com/log/pix05A/050604-DesertSquare.jpg” cannot be displayed, because it contains errors.
 
  "Pour construire un dessin en perspective,
   le peintre trace sur sa toile des repères:
   la ligne d'horizon (1),
   le point de fuite principal (2)
   où se rencontre les lignes de fuite (3)
   et le point de fuite des diagonales (4)."
   _______________________________
  
  Serge Mehl,
   Perspective &
  Géométrie Projective:
  
   "… la géométrie projective était souvent
   synonyme de géométrie supérieure.
   Elle s'opposait à la géométrie
   euclidienne: élémentaire
  
  La géométrie projective, certes supérieure
   car assez ardue, permet d'établir
   de façon élégante des résultats de
   la géométrie élémentaire."
  
  Similarly…
  
  Finite projective geometry
  (in particular, Galois geometry)
   is certainly superior to
   the elementary geometry of
  quilt-pattern symmetry
  and allows us to establish
   de façon élégante
   some results of that
   elementary geometry.
  
  Other Related Material…
  
   from algebra rather than
   geometry, and from a German
   rather than from the French:  

"This is the relativity problem:
to fix objectively a class of
equivalent coordinatizations
and to ascertain
the group of transformations S
mediating between them."
— Hermann Weyl,
The Classical Groups,
Princeton U. Press, 1946

The image “http://www.log24.com/log/pix05/050124-galois12s.jpg” cannot be displayed, because it contains errors.

Evariste Galois

 Weyl also says that the profound branch
of mathematics known as Galois theory

   "… is nothing else but the
   relativity theory for the set Sigma,
   a set which, by its discrete and
    finite character, is conceptually
   so much simpler than the
   infinite set of points in space
   or space-time dealt with
   by ordinary relativity theory."
  — Weyl, Symmetry,
   Princeton U. Press, 1952
  
   Metaphor and Algebra…  

"Perhaps every science must
start with metaphor
and end with algebra;
and perhaps without metaphor
there would never have been
any algebra." 

   — attributed, in varying forms, to
   Max Black, Models and Metaphors, 1962

For metaphor and
algebra combined, see  

  "Symmetry invariance
  in a diamond ring,"

  A.M.S. abstract 79T-A37,
Notices of the
American Mathematical Society,
February 1979, pages A-193, 194 —
the original version of the 4×4 case
of the diamond theorem.

  
More on Max Black…

"When approaching unfamiliar territory, we often, as observed earlier, try to describe or frame the novel situation using metaphors based on relations perceived in a familiar domain, and by using our powers of association, and our ability to exploit the structural similarity, we go on to conjecture new features for consideration, often not noticed at the outset. The metaphor works, according to Max Black, by transferring the associated ideas and implications of the secondary to the primary system, and by selecting, emphasising and suppressing features of the primary in such a way that new slants on it are illuminated."

— Paul Thompson, University College, Oxford,
    The Nature and Role of Intuition
     in Mathematical Epistemology

  A New Slant…  

That intuition, metaphor (i.e., analogy), and association may lead us astray is well known.  The examples of French perspective above show what might happen if someone ignorant of finite geometry were to associate the phrase "4×4 square" with the phrase "projective geometry."  The results are ridiculously inappropriate, but at least the second example does, literally, illuminate "new slants"– i.e., diagonals– within the perspective drawing of the 4×4 square.

Similarly, analogy led the ancient Greeks to believe that the diagonal of a square is commensurate with the side… until someone gave them a new slant on the subject.

Wednesday, May 11, 2005

Wednesday May 11, 2005

Filed under: General — m759 @ 11:00 am
Art History

Reuters – "Joe Grant, a legendary Disney artist who designed the Queen/Witch in 'Snow White and the Seven Dwarfs,' died of a heart attack while doing what he loved most, drawing, the Walt Disney Co. said Monday.

 

Grant, 96, died at his home in the Los Angeles suburb of Glendale last Friday while sitting at his drawing board."
 

"With a little effort, anything can be
shown to connect with anything else:
existence is infinitely cross-referenced."

— Opening sentence of
Martha Cooley's The Archivist

From Log24 last Friday,
a Greek cross:

Pandora's box, according to Rosalind Krauss

Click on picture for details.
 
And from Sunday, May 1
(Orthodox Easter)
:

Rosalind Krauss,

The image “http://www.log24.com/log/pix05/050501-Krauss.jpg” cannot be displayed, because it contains errors.

Columbia University's
Meyer Schapiro Professor
of Modern Art and Theory:

"There is no painter in the West
who can be unaware of
the symbolic power of
the cruciform shape1
and the Pandora's box

The Wicked Queen's Box

of spiritual reference2
that is opened
once one uses it."

Click on pictures for details.
Related material:
Nine is a Vine3.
 

1, 2, 3 Today's birthdays:

1 Natasha Richardson, born 11 May 1963,
   Jedi wife and costar of Nell
2 Martha Quinn, born 11 May 1959,
   MTV wit
3 Frances Fisher, born 11 May 1952,
   dazzling redhead

Friday, November 19, 2004

Friday November 19, 2004

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

From Tate to Plato
In honor of Allen Tate's birthday (today)
and of the MoMA re-opening (tomorrow)

"For Allen Tate the concept of tension was the most useful formal tool at the critic’s disposal, as irony and paradox were for Brooks. The principle of tension sustains the whole structure of meaning, and, as Tate declares in Tension in Poetry (1938), he derives it from lopping the prefixes off the logical terms extension and intension (which define the abstract and denotative aspect of the poetic language and, respectively, the concrete and connotative one). The meaning of the poem is 'the full organized body of all the extension and intension that we can find in it.'  There is an infinite line between extreme extension and extreme intension and the readers select the meaning at the point they wish along that line, according to their personal drives, interests or approaches. Thus the Platonist will tend to stay near the extension end, for he is more interested in deriving an abstraction of the object into a universal…."

— from Form, Structure, and Structurality,
   by Radu Surdulescu

"Eliot, in a conception comparable to Wallace Stevens' 'Anecdote of the Jar,' has suggested how art conquers time:

        Only by the form, the pattern,
Can words or music reach
The stillness, as a Chinese jar still
Moves perpetually in its stillness."

F. O. Matthiessen
   in The Achievement of T.S. Eliot,
   Oxford University Press, 1958

From Writing Chinese Characters:

"It is practical to think of a character centered within an imaginary square grid…. The grid can… be… subdivided, usually to 9 or 16 squares…."

The image “http://www.log24.com/log/pix04B/041119-ZhongGuo.jpg” cannot be displayed, because it contains errors.

These "Chinese jars"
(as opposed to their contents)
are as follows:

The image “http://www.log24.com/log/pix04B/041119-Grids.gif” cannot be displayed, because it contains errors.

Various previous Log24.net entries have
dealt with the 3×3 "form" or "pattern"
(to use the terms of T. S. Eliot).

For the 4×4 form, see Poetry's Bones
and Geometry of the 4×4 Square.

Friday, November 12, 2004

Friday November 12, 2004

Filed under: General — Tags: , , — m759 @ 2:56 am

Dark Zen

The above link is in memory of
Iris Chang,
who ended her life at 36
on Nov. 9, 2004.

A central concept of Zen
is satori, or "awakening."
For a rude awakening, see
Satori at Pearl Harbor.

Fade to Black

See, too, my entries of
Aug. 1-7, 2003,
from which the following is taken:

"…that ineffable constellation of talents that makes the player of rank: a gift for conceiving abstract schematic possibilities; a sense of mathematical poetry in the light of which the infinite chaos of probability and permutation is crystallized under the pressure of intense concentration into geometric blossoms; the ruthless focus of force on the subtlest weakness of an opponent."

— Trevanian, Shibumi

 

" 'Haven't there been splendidly elegant colors in Japan since ancient times?'

'Even black has various subtle shades,' Sosuke nodded.' "

— Yasunari Kawabata, The Old Capital

An Ad Reinhardt painting
described in the entry of
noon, November 9, 2004 —
the date given
as that of Chang's death —
is illustrated below.

The image “http://www.log24.com/log/pix04B/041112-Reinhardt.jpg” cannot be displayed, because it contains errors.

Ad Reinhardt,
Abstract Painting, 1960–66.
Oil on canvas, 60 x 60 inches.
Solomon R. Guggenheim Museum

 

 

Thursday, September 30, 2004

Thursday September 30, 2004

Filed under: General — m759 @ 12:00 am

Midnight in the Garden

“With a little effort,
anything can be shown
to connect with anything else:
existence is infinitely

cross-referenced.”

— Opening sentence
of Martha Cooley’s
The Archivist

Woe unto
them that
call evil
good, and
good evil;
that put
darkness
for light,
and light
for darkness

Isaiah 5:20

As she spoke
about the Trees
of Life and Death,
I watched her…

The Archivist

The world
has gone
mad today
And good’s
bad today,
And black’s
white today,
And day’s
night today

Cole Porter

 Example:
Mozart’s K 265,
the page number 265,
and a story by George MacDonald.

Monday, April 19, 2004

Monday April 19, 2004

Filed under: General — m759 @ 7:59 pm

Cartesian Theatre

From aldaily.com today:

"If my mind is a tiny theatre I watch in my brain, then there is a tinier mind and theatre inside that mind to see it, and so on forever… more»"

This leads to the dream (or nightmare) of the Cartesian theatre, as pictured by Daniel Dennett.

From websurfing yesterday and today…

The tiny theatre of Ivor Grattan-Guinness:

"… mathematicians often treat history with contempt (unsullied by any practice or even knowledge of it, of course)."

The Rainbow of Mathematics

The contempt for history of the Harvard mathematics department (see previous entry) suggests a phrase….

A search on "Harvard sneer" yields, as the first page found, a memorial to an expert practitioner of the Harvard sneer… Robert Harris Chapman, Professor of English Literature, playwright, theatrical consultant, and founding Director of the Loeb Drama Center from 1960 to 1980.

Continuing the Grattan-Guinness rainbow theme in a tinier theatre, we may picture Chapman's reaction to the current Irish Repertory Theatre production of Finian's Rainbow.  Let us hope it is not a Harvard sneer.

In a yet tinier theatre, we may envision a mathematical version of Finian's Rainbow, with Og the leprechaun played by Andrew P. Ogg.  Ogg would, of course, perform a musical version of his remarks on the Jugendtraum:

"Follow the fellow who follows a dream."

Melissa Errico
in Finian's Rainbow

"Give her a song like…. 'Look to the Rainbow,' and her gleaming soprano effortlessly flies it into the stratosphere where such numbers belong. This is the voice of enchantment…."

Ben Brantley, today's NY Times

For related philosophical remarks on rainbows, infinite regress, and redheads, see

Loretta's Rainbow and

The Leonardo Code.

Wednesday, April 7, 2004

Wednesday April 7, 2004

Filed under: General — m759 @ 3:30 am

ART WARS:
Mother of Beauty

In memory of architect Pierre Koenig

Mother of Beauty: A Note on Modernism.

“… Case Study House #22 … was high drama — one in which the entire city becomes part of the architect’s composition. Approached along a winding street set high in the Hollywood Hills, the house first appears as a blank concrete screen. From here, the visitor steps out onto a concrete deck that overlooks a swimming pool. Just beyond it, the house’s living room — enclosed in a glass-and steel-frame — cantilevers out from the edge of the hill toward the horizon.

The house was immortalized in a now famous image taken by the architectural photographer Julius Shulman. In it, two women, clad in immaculate white cocktail dresses, are perched on the edge of their seats in the glass-enclosed living room, their pose suggesting a kind of sanitized suburban bliss. A night view of the city spreads out beneath them, an endless grid of twinkling lights that perfectly captures the infinite hopes of the postwar American dream….

    “My blue dream…”  
— F. Scott Fitzgerald

Perhaps no house, in fact, better sums up the mix of outward confidence and psychic unease that defined Cold War America….”

Los Angeles Times, Nicolai Ouroussoff

Tuesday, April 6, 2004

Tuesday April 6, 2004

Filed under: General — Tags: , , , — m759 @ 10:00 pm

Ideas and Art, Part III

The first idea was not our own.  Adam
In Eden was the father of Descartes…

— Wallace Stevens, from
Notes Toward a Supreme Fiction

"Quaedam ex his tanquam rerum imagines sunt, quibus solis proprie convenit ideae nomen: ut cùm hominem, vel Chimaeram, vel Coelum, vel Angelum, vel Deum cogito."

Descartes, Meditationes III, 5

"Of my thoughts some are, as it were, images of things, and to these alone properly belongs the name idea; as when I think [represent to my mind] a man, a chimera, the sky, an angel or God."

Descartes, Meditations III, 5

Begin, ephebe, by perceiving the idea
Of this invention, this invented world,
The inconceivable idea of the sun.

You must become an ignorant man again
And see the sun again with an ignorant eye
And see it clearly in the idea of it.

— Wallace Stevens, from
Notes Toward a Supreme Fiction

"… Quinimo in multis saepe magnum discrimen videor deprehendisse: ut, exempli causâ, duas diversas solis ideas apud me invenio, unam tanquam a sensibus haustam, & quae maxime inter illas quas adventitias existimo est recensenda, per quam mihi valde parvus apparet, aliam verò ex rationibus Astronomiae desumptam, hoc est ex notionibus quibusdam mihi innatis elicitam, vel quocumque alio modo a me factam, per quam aliquoties major quàm terra exhibetur; utraque profecto similis eidem soli extra me existenti esse non potest, & ratio persuadet illam ei maxime esse dissimilem, quae quàm proxime ab ipso videtur emanasse."

Descartes, Meditationes III, 11

"… I have observed, in a number of instances, that there was a great difference between the object and its idea. Thus, for example, I find in my mind two wholly diverse ideas of the sun; the one, by which it appears to me extremely small draws its origin from the senses, and should be placed in the class of adventitious ideas; the other, by which it seems to be many times larger than the whole earth, is taken up on astronomical grounds, that is, elicited from certain notions born with me, or is framed by myself in some other manner. These two ideas cannot certainly both resemble the same sun; and reason teaches me that the one which seems to have immediately emanated from it is the most unlike."

Descartes, Meditations III, 11

"Et quamvis forte una idea ex aliâ nasci possit, non tamen hîc datur progressus in infinitum, sed tandem ad aliquam primam debet deveniri, cujus causa sit in star archetypi, in quo omnis realitas formaliter contineatur, quae est in ideâ tantùm objective."

Descartes, Meditationes III, 15

"And although an idea may give rise to another idea, this regress cannot, nevertheless, be infinite; we must in the end reach a first idea, the cause of which is, as it were, the archetype in which all the reality [or perfection] that is found objectively [or by representation] in these ideas is contained formally [and in act]."

Descartes, Meditations III, 15

Michael Bryson in an essay on Stevens's "Notes Toward a Supreme Fiction,"

The Quest for the Fiction of the Absolute:

"Canto nine considers the movement of the poem between the particular and the general, the immanent and the transcendent: "The poem goes from the poet's gibberish to / The gibberish of the vulgate and back again. / Does it move to and fro or is it of both / At once?" The poet, the creator-figure, the shadowy god-figure, is elided, evading us, "as in a senseless element."  The poet seeks to find the transcendent in the immanent, the general in the particular, trying "by a peculiar speech to speak / The peculiar potency of the general." In playing on the senses of "peculiar" as particular and strange or uncanny, these lines play on the mystical relation of one and many, of concrete and abstract."

Brian Cronin in Foundations of Philosophy:

"The insight is constituted precisely by 'seeing' the idea in the image, the intelligible in the sensible, the universal in the particular, the abstract in the concrete. We pivot back and forth between images and ideas as we search for the correct insight."

— From Ch. 2, Identifying Direct Insights

Michael Bryson in an essay on Stevens's "Notes Toward a Supreme Fiction":

"The fourth canto returns to the theme of opposites. 'Two things of opposite natures seem to depend / On one another . . . . / This is the origin of change.'  Change resulting from a meeting of opposities is at the root of Taoism: 'Tao produced the One. / The One produced the two. / The two produced the three. / And the three produced the ten thousand things' (Tao Te Ching 42) …."

From an entry of March 7, 2004

 

From the web page

Introduction to the I Ching–
By Richard Wilhelm
:

"He who has perceived the meaning of change fixes his attention no longer on transitory individual things but on the immutable, eternal law at work in all change. This law is the tao of Lao-tse, the course of things, the principle of the one in the many. That it may become manifest, a decision, a postulate, is necessary. This fundamental postulate is the 'great primal beginning' of all that exists, t'ai chi — in its original meaning, the 'ridgepole.' Later Chinese philosophers devoted much thought to this idea of a primal beginning. A still earlier beginning, wu chi, was represented by the symbol of a circle. Under this conception, t'ai chi was represented by the circle divided into the light and the dark, yang and yin,

.

This symbol has also played a significant part in India and Europe. However, speculations of a gnostic-dualistic character are foreign to the original thought of the I Ching; what it posits is simply the ridgepole, the line. With this line, which in itself represents oneness, duality comes into the world, for the line at the same time posits an above and a below, a right and left, front and back-in a word, the world of the opposites."

The t'ai chi symbol is also illustrated on the web page Cognitive Iconology, which says that

"W.J.T. Mitchell calls 'iconology'
a study of the 'logos'
(the words, ideas, discourse, or 'science')
of 'icons' (images, pictures, or likenesses).
It is thus a 'rhetoric of images'
(Iconology: Image, Text, Ideology, p. 1)."

A variation on the t'ai chi symbol appears in a log24.net entry for March 5:

The Line,
by S. H. Cullinane

See too my web page Logos and Logic, which has the following:

"The beautiful in mathematics resides in contradiction. Incommensurability, logoi alogoi, was the first splendor in mathematics."

— Simone Weil, Oeuvres Choisies, ed. Quarto, Gallimard, 1999, p. 100

 Logos Alogos,
by S. H. Cullinane 

In the conclusion of Section 3, Canto X, of "Notes," Stevens says

"They will get it straight one day
at the Sorbonne.
We shall return at twilight
from the lecture
Pleased that
the irrational is rational…."

This is the logoi alogoi of Simone Weil.

In "Notes toward a Supreme Fiction,"
Wallace Stevens lists three criteria
for a work of the imagination:

It Must Be Abstract

The Line,
by S.H. Cullinane 

It Must Change

 The 24,
by S. H. Cullinane

It Must Give Pleasure

Puzzle,
by S. H. Cullinane

Related material:

Logos and Logic.

 

Friday, February 20, 2004

Friday February 20, 2004

Filed under: General,Geometry — Tags: — m759 @ 3:24 pm

Finite Relativity

Today is the 18th birthday of my note

The Relativity Problem in Finite Geometry.”

That note begins with a quotation from Weyl:

“This is the relativity problem: to fix objectively a class of equivalent coordinatizations and to ascertain the group of transformations S mediating between them.”

— Hermann Weyl, The Classical Groups, Princeton University Press, 1946, p. 16

Here is another quotation from Weyl, on the profound branch of mathematics known as Galois theory, which he says

“… is nothing else but the relativity theory for the set Sigma, a set which, by its discrete and finite character, is conceptually so much simpler than the infinite set of points in space or space-time dealt with by ordinary relativity theory.”

— Weyl, Symmetry, Princeton University Press, 1952, p. 138

This second quotation applies equally well to the much less profound, but more accessible, part of mathematics described in Diamond Theory and in my note of Feb. 20, 1986.

Monday, September 1, 2003

Monday September 1, 2003

Filed under: General — m759 @ 3:33 pm

The Unity of Mathematics,

or “Shema, Israel”

A conference to honor the 90th birthday (Sept. 2) of Israel Gelfand is currently underway in Cambridge, Massachusetts.

The following note from 2001 gives one view of the conference’s title topic, “The Unity of Mathematics.”

Reciprocity in 2001

by Steven H. Cullinane
(May 30, 2001)

From 2001: A Space Odyssey, by Arthur C. Clarke, New American Library, 1968:

The glimmering rectangular shape that had once seemed no more than a slab of crystal still floated before him….  It encapsulated yet unfathomed secrets of space and time, but some at least he now understood and was able to command.

How obvious — how necessary — was that mathematical ratio of its sides, the quadratic sequence 1: 4: 9!  And how naive to have imagined that the series ended at this point, in only three dimensions!

— Chapter 46, “Transformation”

From a review of Himmelfarb, by Michael Krüger, New York, George Braziller, 1994:

As a diffident, unsure young man, an inexperienced ethnologist, Richard was unable to travel through the Amazonian jungles unaided. His professor at Leipzig, a Nazi Party member (a bigot and a fool), suggested he recruit an experienced guide and companion, but warned him against collaborating with any Communists or Jews, since the objectivity of research would inevitably be tainted by such contact. Unfortunately, the only potential associate Richard can find in Sao Paulo is a man called Leo Himmelfarb, both a Communist (who fought in the Spanish Civil War) and a self-exiled Jew from Galicia, but someone who knows the forests intimately and can speak several of the native dialects.

“… Leo followed the principle of taking and giving, of learning and teaching, of listening and storytelling, in a word: of reciprocity, which I could not even imitate.”

… E. M. Forster famously advised his readers, “Only connect.” “Reciprocity” would be Michael Kruger’s succinct philosophy, with all that the word implies.

— William Boyd, New York Times Book Review, October 30, 1994

Reciprocity and Euler

Applying the above philosophy of reciprocity to the Arthur C. Clarke sequence

1, 4, 9, ….

we obtain the rather more interesting sequence
1/1, 1/4, 1/9, …..

This leads to the following problem (adapted from the St. Andrews biography of Euler):

Perhaps the result that brought Euler the most fame in his young days was his solution of what had become known as the Basel problem. This was to find a closed form for the sum of the infinite series

1/1 + 1/4 + 1/9 + 1/16 + 1/25 + …

— a problem which had defeated many of the top mathematicians including Jacob Bernoulli, Johann Bernoulli and Daniel Bernoulli. The problem had also been studied unsuccessfully by Leibniz, Stirling, de Moivre and others. Euler showed in 1735 that the series sums to (pi squared)/6. He generalized this series, now called zeta(2), to zeta functions of even numbers larger than two.

Related Reading

For four different proofs of Euler’s result, see the inexpensive paperback classic by Konrad Knopp, Theory and Application of Infinite Series (Dover Publications).

Related Websites

Evaluating Zeta(2), by Robin Chapman (PDF article) Fourteen proofs!

Zeta Functions for Undergraduates

The Riemann Zeta Function

Reciprocity Laws
Reciprocity Laws II

The Langlands Program

Recent Progress on the Langlands Conjectures

For more on
the theme of unity,
see

Monolithic Form
and
ART WARS.

Friday, August 1, 2003

Friday August 1, 2003

Filed under: General — Tags: , — m759 @ 4:03 pm

For All Time

"… and the Wichita lineman is still on the line…"

(Reflection on a member of the Radcliffe Class of 1964 who lived near Wichita and now has her own home page… While listening to a song on my "home on The Range – KHYI 95.3FM, Plano, Texas.")

Readings for a seminar we never really finished:

"…that ineffable constellation of talents that makes the player of rank: a gift for conceiving abstract schematic possibilities; a sense of mathematical poetry in the light of which the infinite chaos of probability and permutation is crystallized under the pressure of intense concentration into geometric blossoms; the ruthless focus of force on the subtlest weakness of an opponent."

— Trevanian, Shibumi

" 'Haven't there been splendidly elegant colors in Japan since ancient times?'

'Even black has various subtle shades,' Sosuke nodded.' "

— Yasunari Kawabata, The Old Capital 

Thursday, April 24, 2003

Thursday April 24, 2003

Filed under: General — m759 @ 3:33 am

Cross-Referenced

Shortly after midnight on the night of April 22-23, I updated my entry for Shakespeare's birthday with the following quotation: 

"With a little effort, anything can be shown to connect with anything else: existence is infinitely cross-referenced."

Opening sentence of Martha Cooley's The Archivist

About 24 hours later, I came across the following obituary in The New York Times: 

"Edgar F. Codd, a mathematician and computer scientist who laid the theoretical foundation for relational databases, the standard method by which information is organized in and retrieved from computers, died on Friday…. He was 79."

The Times does not mention that the Friday it refers to is Good  Friday.  God will  have his little jokes.

From Computerworld.com:

1969: Edgar F. "Ted" Codd invents the relational database.
1969: Edgar F. "Ted" Codd invents the relational database.

1969: Edgar F. “Ted” Codd invents the relational database.

1973: Cullinane, led by John J. Cullinane, ships IDMS, a network-model database for IBM mainframes.

1976: Honeywell ships Multics Relational Data Store, the first commercial relational database.

For a better (and earlier) obituary than the Times's, see The San Jose Mercury News of Easter Sunday.  For some thoughts on death and the afterlife appropriate to last weekend, see The Matthias Defense.

The Exorcist, 1973
 

Wednesday, April 23, 2003

Wednesday April 23, 2003

Filed under: General — m759 @ 12:00 am

Midnight in the Garden
of Good and Evil
on Shakespeare’s Birthday

Tony Scherman on an April 7, 1968, recording by Nina Simone:

“…nobody could telescope more emotion into a single, idiosyncratically turned syllable (listen to the way she says the word “Savannah” in her spoken intro to “Sunday in Savannah.” It breaks your heart — and she ain’t even singin’ yet!).”

See also the following entries on midnight in the garden:

Trinity, Oct. 25, 2002

Midnight in the Garden, Oct. 26, 2002

Point of No Return, Dec. 10, 2002

Culture Clash at Midnight, Dec. 11, 2002

Dead Poets Society, Dec. 13, 2002

For the Dark Lady, Dec. 18, 2002

Nightmare Alley, Dec. 21, 2002

For the Green Lady, Dec. 21, 2002

“With a little effort, anything can be shown to connect with anything else: existence is infinitely cross-referenced.”

Opening sentence of Martha Cooley’s The Archivist

Woe unto
them that
call evil
good, and
good evil;
that put
darkness
for light,
and light
for darkness

Isaiah 5:20

 

 

As she spoke about the Trees of Life and Death, I watched her…. 
The Archivist

The world
has gone
mad today
And good’s
bad today,

And black’s
white today,
And day’s
night today

Cole Porter

 

 

Tuesday, December 31, 2002

Tuesday December 31, 2002

Filed under: General — m759 @ 3:17 pm

To Sir Anthony Hopkins
on His Birthday

From “The Wardrobe Wars,” by Paul Willis:

“I was back at Wheaton for a conference just a couple of years ago. During a period of announcements, a curator from the Wade Collection invited the conference participants to visit the collection and see the many books and papers that had belonged to Lewis and his associates. At the end of her announcement, she told us, ‘We also have the wardrobe that served as the original for the one in the Narnia Chronicles.’

There it was, that definite article again. In a remarkable display of maturity I put up my hand and said, ‘Excuse me, but the wardrobe is at Westmont College in Santa Barbara.’

The woman gave me a long, hard look of the ‘we are not amused’ variety. That was all. I wasn’t able to find her after the session was over to clear things up.

Not that we could have, really. Of course, if pressed, I suspect we would both admit the wardrobe we are really concerned with exists only within the covers of a book, and that not even this wardrobe is so important as the story of which it is a part, and that the story is not so important as the sense of infinite longing that it stirs within our souls, and that this longing is not so important as the One—more real than Aslan himself—to whom it directs us. But that would be asking too much of either the curator or myself. To worship at our respective wardrobes, whether they be in Jerusalem or Samaria, is indeed to live in the shadowlands. And that is where we like it.

Lewis himself would doubtless say that the physical wardrobes in our possession are but copies of a faint copy. He might even claim, to our horror, that no single wardrobe inspired the one found in his book. Then he might add under his breath, like the professor in The Last Battle who has passed on to the next life, ‘It’s all in Plato, all in Plato: bless me, what do they teach them at these schools!'”

Tuesday, December 17, 2002

Tuesday December 17, 2002

Filed under: General — m759 @ 12:00 am

ART WARS:


Just Seventeen

'Just 17' illustration

 

Durga

Today's site music*
is in honor of
a memorable date.

 

1963
Northern Songs.

Quiet may be restored by using
the midi control box at the top right
of this page.  Please let me know
if your browser is not showing
this control box.

 

 

Veronica  

From a June/July 1997
Hadassah Magazine article:

"Plato is obviously Jewish."

— Rebecca Goldstein

Readings on the Dark Lady  

From a July 27, 1997
New York Times article
by Holland Cotter:

"The single most important and sustained model for Khmer culture was India, from which Cambodia inherited two religions, Buddhism and Hinduism, and an immensely sophisticated art. This influence announces itself early in this exhibition in a spectacular seventh-century figure of the Hindu goddess Durga, whose hip-slung pose and voluptuous torso, as plush and taut as ripe fruit, combine the naturalism and idealism of the very finest Indian work."

From The Dancing Wu Li Masters,
by Gary Zukav, Harvard '64:

"The Wu Li Masters know that physicists are doing more than 'discovering the endless diversity of nature.' They are dancing with Kali [or Durga], the Divine Mother of Hindu mythology."

"Eastern religions have nothing to say about physics, but they have a great deal to say about human experience. In Hindu mythology, Kali, the Divine Mother, is the symbol for the infinite diversity of experience. Kali represents the entire physical plane. She is the drama, tragedy, humor, and sorrow of life. She is the brother, father, sister, mother, lover, and friend. She is the fiend, monster, beast, and brute. She is the sun and the ocean. She is the grass and the dew. She is our sense of accomplishment and our sense of doing worthwhile. Our thrill of discovery is a pendant on her bracelet. Our gratification is a spot of color on her cheek. Our sense of importance is the bell on her toe.

This full and seductive, terrible and wonderful earth mother always has something to offer. Hindus know the impossibility of seducing her or conquering her and the futility of loving her or hating her; so they do the only thing that they can do. They simply honor her."

How could I dance with another….?

— John Lennon and Paul McCartney, 1962-1963  

Tuesday, October 15, 2002

Tuesday October 15, 2002

Filed under: General — m759 @ 2:10 pm

From the Archives:

On this date in 1971, “Rick Nelson was booed off the stage when he didn’t stick to all oldies at the seventh Annual Rock ’n’ Roll Revival show at Madison Square Garden, New York. He tried to slip in some of his new material and the crowd did not approve. The negative reaction to his performance inspired Nelson to write his last top-40 hit, ‘Garden Party,’ which hit the top-ten about a year after the Madison Square Garden debacle. ‘Garden Party,’ ironically, was Nelson’s biggest hit in years.”

“With a little effort, anything can be shown to connect with anything else: existence is infinitely cross-referenced.”

Opening sentence of Martha Cooley’s The Archivist

Woe unto
them that
call evil
good, and
good evil;
that put
darkness
for light,
and light
for darkness

Isaiah 5:20

As she spoke
about the Trees
of Life
and Death,
I watched her…
The Archivist

The world
has gone
mad today
And good’s
bad today,

And black’s
white today,
And day’s
night today

Cole Porter

Actor Pat O’Brien died on this date in 1983.

“A man in Ireland, who came in contact with a Bible colporteur, at first repulsed him. Finally he was persuaded to take a Bible and later he said: ‘I read a wee bit out of the New Testament every day, and I pray to God every night and morning.’  When asked if it helped him to read God’s Word and to pray, he answered: ‘Indade it does. When I go to do anything wrong, I just say to myself, “Pat, you’ll be talking to God tonight.” That keeps me from doing it!'”
worldmissions.org

colporteur 
… noun…
Etymology: French, alteration of Middle French comporteur, from comporter to bear, peddle….
a peddler of religious books

Sunday, September 29, 2002

Sunday September 29, 2002

Filed under: General — m759 @ 10:18 pm

New from Miracle Pictures
– IF IT’S A HIT, IT’S A MIRACLE! –

Pi in the Sky
for Michaelmas 2002

“Fear not, maiden, your prayer is heard.
Michael am I, guardian of the highest Word.”

A Michaelmas Play

Contact, by Carl Sagan:

Chapter 1 – Transcendental Numbers

  In the seventh grade they were studying “pi.” It was a Greek letter that looked like the architecture at Stonehenge, in England: two vertical pillars with a crossbar at the top. If you measured the circumference of a circle and then divided it by the diameter of the circle, that was pi. At home, Ellie took the top of a mayonnaise jar, wrapped a string around it, straightened the string out, and with a ruler measured the circle’s circumference. She did the same with the diameter, and by long division divided the one number by the other. She got 3.21. That seemed simple enough.

  The next day the teacher, Mr. Weisbrod, said that pi was about 22/7, about 3.1416. But actually, if you wanted to be exact, it was a decimal that went on and on forever without repeating the pattern of numbers. Forever, Ellie thought. She raised her hand. It was the beginning of the school year and she had not asked any questions in this class.
  “How could anybody know that the decimals go on and on forever?”
  “That’s just the way it is,” said the teacher with some asperity.
  “But why? How do you know? How can you count decimals forever?”
  “Miss Arroway” – he was consulting his class list – “this is a stupid question. You’re wasting the class’s time.”

  No one had ever called Ellie stupid before and she found herself bursting into tears….

  After school she bicycled to the library at the nearby college to look through books on mathematics. As nearly as she could figure out from what she read, her question wasn’t all that stupid. According to the Bible, the ancient Hebrews had apparently thought that pi was exactly equal to three. The Greeks and Romans, who knew lots of things about mathematics, had no idea that the digits in pi went on forever without repeating. It was a fact that had been discovered only about 250 years ago. How was she expected to know if she couldn’t ask questions? But Mr. Weisbrod had been right about the first few digits. Pi wasn’t 3.21. Maybe the mayonnaise lid had been a little squashed, not a perfect circle. Or maybe she’d been sloppy in measuring the string. Even if she’d been much more careful, though, they couldn’t expect her to measure an infinite number of decimals.

  There was another possibility, though. You could calculate pi as accurately as you wanted. If you knew something called calculus, you could prove formulas for pi that would let you calculate it to as many decimals as you had time for. The book listed formulas for pi divided by four. Some of them she couldn’t understand at all. But there were some that dazzled her: pi/4, the book said, was the same as 1 – 1/3 + 1/5 – 1/7 + …, with the fractions continuing on forever. Quickly she tried to work it out, adding and subtracting the fractions alternately. The sum would bounce from being bigger than pi/4 to being smaller than pi/4, but after a while you could see that this series of numbers was on a beeline for the right answer. You could never get there exactly, but you could get as close as you wanted if you were very patient. It seemed to her

a miracle


 Cartoon by S.Harris

that the shape of every circle in the world was connected with this series of fractions. How could circles know about fractions? She was determined to learn

calculus.

  The book said something else: pi was called a “transcendental” number. There was no equation with ordinary numbers in it that could give you pi unless it was infinitely long. She had already taught herself a little algebra and understood what this meant. And pi wasn’t the only transcendental number. In fact there was an infinity of transcendental numbers. More than that, there were infinitely more transcendental numbers that ordinary numbers, even though pi was the only one of them she had ever heard of. In more ways than one, pi was tied to infinity.

  She had caught a glimpse of something majestic.

Chapter 24 – The Artist’s Signature

  The anomaly showed up most starkly in Base 2 arithmetic, where it could be written out entirely as zeros and ones. Her program reassembled the digits into a square raster, an equal number across and down. Hiding in the alternating patterns of digits, deep inside the transcendental number, was a perfect circle, its form traced out by unities in a field of noughts.

  The universe was made on purpose, the circle said. In whatever galaxy you happen to find yourself, you take the circumference of a circle, divide it by its diameter, measure closely enough, and uncover

a miracle

— another circle, drawn kilometers downstream of the decimal point. There would be richer messages farther in. It doesn’t matter what you look like, or what you’re made of, or where you come from. As long as you live in this universe, and have a modest talent for mathematics, sooner or later you’ll find it. It’s already here. It’s inside everything. You don’t have to leave your planet to find it. In the fabric of space and in the nature of matter, as in a great work of art, there is, written small, the artist’s signature. Standing over humans, gods, and demons… there is an intelligence that antedates the universe. The circle had closed. She found what she had been searching for.

Song lyric not in Sagan’s book:

Will the circle be unbroken
by and by, Lord, by and by?
Is a better home a-waitin’
in the sky, Lord, in the sky?

“Contact,” the film: 

Recording:

Columbia 37669

Date Issued:

Unknown

Side:

A

Title:

Can The Circle Be Unbroken

Artist:

Carter Family

Recording Date:

May 6, 1935

Listen:

Realaudio

Music courtesy of honkingduck.com.
 
For bluegrass midi version, click here.
 

The above conclusion to Sagan’s book is perhaps the stupidest thing by an alleged scientist that I have ever read.  As a partial antidote, I offer the following.

Today’s birthday: Stanley Kramer, director of “On the Beach.”

From an introduction to a recording of the famous Joe Hill song about Pie in the Sky:

“They used a shill to build a crowd… You know, a carny shill.”


Carny

Saturday, July 20, 2002

Saturday July 20, 2002

 

ABSTRACT: Finite projective geometry explains the surprising symmetry properties of some simple graphic designs– found, for instance, in quilts. Links are provided for applications to sporadic simple groups (via the "Miracle Octad Generator" of R. T. Curtis), to the connection between orthogonal Latin squares and projective spreads, and to symmetry of Walsh functions.

We regard the four-diamond figure D above as a 4×4 array of two-color diagonally-divided square tiles.

Let G be the group of 322,560 permutations of these 16 tiles generated by arbitrarily mixing random permutations of rows and of columns with random permutations of the four 2×2 quadrants.

THEOREM: Every G-image of D (as at right, below) has some ordinary or color-interchange symmetry.

Example:


For an animated version, click here.

Remarks:

Some of the patterns resulting from the action of G on D have been known for thousands of years. (See Jablan, Symmetry and Ornament, Ch. 2.6.) It is perhaps surprising that the patterns' interrelationships and symmetries can be explained fully only by using mathematics discovered just recently (relative to the patterns' age)– in particular, the theory of automorphism groups of finite geometries.

Using this theory, we can summarize the patterns' properties by saying that G is isomorphic to the affine group A on the linear 4-space over GF(2) and that the 35 structures of the 840 = 35 x 24 G-images of D are isomorphic to the 35 lines in the 3-dimensional projective space over GF(2).

This can be seen by viewing the 35 structures as three-sets of line diagrams, based on the three partitions of the four-set of square two-color tiles into two two-sets, and indicating the locations of these two-sets of tiles within the 4×4 patterns. The lines of the line diagrams may be added in a binary fashion (i.e., 1+1=0). Each three-set of line diagrams sums to zero– i.e., each diagram in a three-set is the binary sum of the other two diagrams in the set. Thus, the 35 three-sets of line diagrams correspond to the 35 three-point lines of the finite projective 3-space PG(3,2).

For example, here are the line diagrams for the figures above:

 
Shown below are the 15 possible line diagrams resulting from row/column/quadrant permutations. These 15 diagrams may, as noted above, be regarded as the 15 points of the projective 3-space PG(3,2).


The symmetry of the line diagrams accounts for the symmetry of the two-color patterns. (A proof shows that a 2nx2n two-color triangular half-squares pattern with such line diagrams must have a 2×2 center with a symmetry, and that this symmetry must be shared by the entire pattern.)

Among the 35 structures of the 840 4×4 arrays of tiles, orthogonality (in the sense of Latin-square orthogonality) corresponds to skewness of lines in the finite projective space PG(3,2). This was stated by the author in a 1978 note. (The note apparently had little effect. A quarter-century later, P. Govaerts, D. Jungnickel, L. Storme, and J. A. Thas wrote that skew (i.e., nonintersecting) lines in a projective space seem "at first sight not at all related" to orthogonal Latin squares.)

We can define sums and products so that the G-images of D generate an ideal (1024 patterns characterized by all horizontal or vertical "cuts" being uninterrupted) of a ring of 4096 symmetric patterns. There is an infinite family of such "diamond" rings, isomorphic to rings of matrices over GF(4).

The proof uses a decomposition technique for functions into a finite field that might be of more general use.

The underlying geometry of the 4×4 patterns is closely related to the Miracle Octad Generator of R. T. Curtis– used in the construction of the Steiner system S(5,8,24)– and hence is also related to the Leech lattice, which, as Walter Feit has remarked, "is a blown up version of S(5,8,24)."

For a movable JavaScript version of these 4×4 patterns, see The Diamond 16 Puzzle.

The above is an expanded version of Abstract 79T-A37, "Symmetry invariance in a diamond ring," by Steven H. Cullinane, Notices of the American Mathematical Society, February 1979, pages A-193, 194.

For a discussion of other cases of the theorem, click here.

Related pages:

The Diamond 16 Puzzle

Diamond Theory in 1937:
A Brief Historical Note

Notes on Finite Geometry

Geometry of the 4×4 Square

Binary Coordinate Systems

The 35 Lines of PG(3,2)

Map Systems:
Function Decomposition over a Finite Field

The Diamond Theorem–
The 2×2, the 2x2x2, the 4×4, and the 4x4x4 Cases

Diamond Theory

Latin-Square Geometry

Walsh Functions

Inscapes

The Diamond Theory of Truth

Geometry of the I Ching

Solomon's Cube and The Eightfold Way

Crystal and Dragon in Diamond Theory

The Form, the Pattern

The Grid of Time

Block Designs

Finite Relativity

Theme and Variations

Models of Finite Geometries

Quilt Geometry

Pattern Groups

The Fano Plane Revisualized,
or the Eightfold Cube

The Miracle Octad Generator

Kaleidoscope

Visualizing GL(2,p)

Jung's Imago

Author's home page

AMS Mathematics Subject Classification:

20B25 (Group theory and generalizations :: Permutation groups :: Finite automorphism groups of algebraic, geometric, or combinatorial structures)

05B25 (Combinatorics :: Designs and configurations :: Finite geometries)

51E20 (Geometry :: Finite geometry and special incidence structures :: Combinatorial structures in finite projective spaces)



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Initial Xanga entry.  Updated Nov. 18, 2006.

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