Monday, December 31, 2012

6 PM ET is…

Filed under: General — m759 @ 6:00 PM

  Midnight in Paris

IMAGE- 'Vin Mariani' poster illustrating the Belle Époque

   See also Mariani  in this journal.


Filed under: General — m759 @ 1:26 PM

IMAGE- The Art World: Shapes of Things- The Birth of the Abstract [New Yorker title]

"What possessed a generation of young European artists,
and a few Americans, to suddenly suppress recognizable imagery
in pictures and sculptures? Unthinkable at one moment, the strategy
became practically compulsory in the next." 

Peter Schjeldahl in the current New Yorker

The following remarks may or may not be relevant.

Paul Valéry, "Introduction to the Method of Leonardo da Vinci,"
La Nouvelle Revue , Paris, Vol. 95 (1895)—

"Regarded thus, the ornamental conception is to the individual arts
what mathematics is to the other sciences. …the objects chosen
and arranged with a view to a particular effect seem as if disengaged
from most of their properties and only reassume them in the effect,
in, that is to say, the mind of the detached spectator. It is thus
by means of an abstraction that the work of art can be constructed,
and is more or less easy to define according as the elements borrowed
from reality for it are more or less complex. Inversely it is by a sort of
induction, by the production of mental images, that all works of art are
appreciated, and this production must equally be more or less active,
more or less tiring, according as it is set in motion by a simple
interlacing on a vase or a broken phrase by Pascal."

— Translated by Thomas McGreevy (Valéry's Selected Writings,
     New Directions, 1950)

Paul Valéry, "Introduction a la Méthode de Léonard de Vinci,"
La Nouvelle Revue , Paris, Tome 95 (1895), p. 762

"De ce point de vue, la conception ornementale est aux arts
particuliers ce que la mathématique est aux autres sciences. De
même que les notions physiques de temps, longueur, densité,
masse, etc., ne sont dans les calculs que des quantités homo-
gènes et ne retrouvent leur individualité que dans l'interprétation
des résultats, de même les objets choisis et ordonnés en vue d'un
effet sont comme détachés de la plupart de leurs propriétés et
ne les reprennent que dans cet effet, dans l'esprit non prévenu
du spectateur. C'est donc par une abstraction que l'œuvre d'art
peut se construire, et cette abstraction est plus ou moins éner-
gique, plus ou moins facile à y découvrir, que les éléments em-
pruntés à la réalité en sont des portions plus ou moins com-
plexes. Inversement, c'est par une sorte d'induction, par la
production d'images mentales que toute œuvre d'art s'apprécie;
et cette production doit être également plus ou moins énergique,
plus ou moins fatigante  selon qu'un simple entrelacs sur un
vase ou une phrase brisée de Pascal la sollicite." 

Holiday Philosophy

Filed under: General,Geometry — m759 @ 4:56 AM

Obituaries for New Year's Eve—

A link from Christmas Day—

Easter meditations—

See also

Sunday, December 30, 2012

Rien de Rien (continued)

Filed under: General — m759 @ 5:01 PM


"I never had any hesitation or regrets…."

— Rita Levi-Montalcini, who died today at 103

See also today's previous post and

Turning the Page:

The Kick

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

George Steiner, Real Presences , first published in 1989—

The inception of critical thought, of a philosophic anthropology,
is contained in the archaic Greek definition of man as a

Richard Powers, The Gold Bug Variations , first published in 1991—

Botkin, whatever her gifts as a conversationist, is almost as old
as the rediscovery of Mendel. The other extreme in age,
Joe Lovering, beat a time-honored path out of pure math
into muddy population statistics. Ressler has seen the guy
potting about in the lab, although exactly what the excitable kid
does is anybody's guess. He looks decidedly gumfooted holding
any equipment more corporeal than a chi-square. Stuart takes
him to the Y for lunch, part of a court-your-resources campaign.
He has the sub, Levering the congealed mac and cheese.
Hardly are they seated when Joe whips out a napkin and begins
sketching proofs. He argues that the genetic code, as an
algorithmic formal system, is subject to Gödel's Incompleteness
Theorem. "That would mean the symbolic language of the code
can't be both consistent and complete. Wouldn't that be a kick
in the head?"

Kid talk, competitive showing off, intellectual fantasy.
But Ressler knows what Joe is driving at. He's toyed with similar
ideas, cast in less abstruse terms. We are the by-product of the
mechanism in there. So it must be more ingenious than us.
Anything complex enough to create consciousness may be too
complex for consciousness to understand. Yet the ultimate paradox
is Lovering, crouched over his table napkin, using proofs to
demonstrate proof's limits. Lovering laughs off recursion and takes
up another tack: the key is to find some formal symmetry folded
in this four-base chaos
. Stuart distrusts this approach even more.
He picks up the tab for their two untouched lunches, thanking
Lovering politely for the insight.

Edith Piaf—

Non, rien de rien

See last midnight's post and Theme and Variations.

"The key is to find some formal symmetry…."

IMAGE- Valéry on ornament in 'Method of Leonardo,' with Valéry's serpent-and-key emblem

Not Often

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

"It is not often anyone will hear the phrase 'Galois field' and 'DNA' together…."

Ewan Birney at his weblog on July 3, 2012.

Try a Google search. (And see such a search as of Dec. 30, 2012.)

See also "Context Part III" in a Log24 post of Sept. 17, 2012.

Saturday, December 29, 2012

Mapping Problem

Filed under: General,Geometry — m759 @ 1:06 AM

A mapping problem posed (informally) in 1985
and solved 27 years later,  in 2012:

See also Finite Relativity and Finite Relativity: The Triangular Version.

(A note for fans of the recent film Looper  (see previous post)—

Hunter S. Thompson in this journal on February 22, 2005 

IMAGE- Hunter S. Thompson, the old and the young
           Hunter S. Thompson, photos from The New York Times

and on March 3, 2009.)

Friday, December 28, 2012

Big Time

Filed under: General — m759 @ 12:06 PM

Joseph Gordon-Levitt and Bruce Willis in Looper :

IMAGE- Diner scene from 'Looper'

Bruce Willis in 12 Monkeys :

IMAGE- Examining a spider web in '12 Monkeys'

See also Big Time  in this journal.

Cube Koan

Filed under: General,Geometry — Tags: , , , , — m759 @ 4:56 AM

From Don DeLillo's novel Point Omega —

I knew what he was, or what he was supposed to be, a defense intellectual, without the usual credentials, and when I used the term it made him tense his jaw with a proud longing for the early weeks and months, before he began to understand that he was occupying an empty seat. "There were times when no map existed to match the reality we were trying to create."

"What reality?"

"This is something we do with every eyeblink. Human perception is a saga of created reality. But we were devising entities beyond the agreed-upon limits of recognition or interpretation. Lying is necessary. The state has to lie. There is no lie in war or in preparation for war that can't be defended. We went beyond this. We tried to create new realities overnight, careful sets of words that resemble advertising slogans in memorability and repeatability. These were words that would yield pictures eventually and then become three-dimensional. The reality stands, it walks, it squats. Except when it doesn't."

He didn't smoke but his voice had a sandlike texture, maybe just raspy with age, sometimes slipping inward, becoming nearly inaudible. We sat for some time. He was slouched in the middle of the sofa, looking off toward some point in a high corner of the room. He had scotch and water in a coffee mug secured to his midsection. Finally he said, "Haiku."

I nodded thoughtfully, idiotically, a slow series of gestures meant to indicate that I understood completely.

"Haiku means nothing beyond what it is. A pond in summer, a leaf in the wind. It's human consciousness located in nature. It's the answer to everything in a set number of lines, a prescribed syllable count. I wanted a haiku war," he said. "I wanted a war in three lines. This was not a matter of force levels or logistics. What I wanted was a set of ideas linked to transient things. This is the soul of haiku. Bare everything to plain sight. See what's there. Things in war are transient. See what's there and then be prepared to watch it disappear."

What's there—

This view of a die's faces 3, 6, and 5, in counter-
clockwise order (see previous post) suggests a way
of labeling the eight corners  of a die (or cube):

123, 135, 142, 154, 246, 263, 365, 456.

Here opposite faces of the die sum to 7, and the
three faces meeting at each corner are listed
in counter-clockwise order. (This corresponds
to a labeling of one of MacMahon's* 30 colored cubes.)
A similar vertex-labeling may be used in describing 
the automorphisms of the order-8 quaternion group.

For a more literary approach to quaternions, see
Pynchon's novel Against the Day .

* From Peter J. Cameron's weblog:

  "The big name associated with this is Major MacMahon,
   an associate of Hardy, Littlewood and Ramanujan,
   of whom Robert Kanigel said,

His expertise lay in combinatorics, a sort of
glorified dice-throwing, and in it he had made
contributions original enough to be named
a Fellow of the Royal Society.

   Glorified dice-throwing, indeed…"

Thursday, December 27, 2012


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

IMAGE- 'Inception' totems: red die and chess bishop, with Inception 'Point Man' poster

Note that the visible faces of the die, in counter -clockwise
order, are 3 6 5. See also this journal 365 days ago and,
since 2012 is a leap year, also today's date last year.

Object Lesson

Yesterday's post on the current Museum of Modern Art exhibition
"Inventing Abstraction: 1910-1925" suggests a renewed look at
abstraction and a fundamental building block: the cube.

From a recent Harvard University Press philosophical treatise on symmetry—

The treatise corrects Nozick's error of not crediting Weyl's 1952 remarks
on objectivity and symmetry, but repeats Weyl's error of not crediting
Cassirer's extensive 1910 (and later) remarks on this subject.

For greater depth see Cassirer's 1910 passage on Vorstellung :

IMAGE- Ernst Cassirer on 'representation' or 'Vorstellung' in 'Substance and Function' as 'the riddle of knowledge'

This of course echoes Schopenhauer, as do discussions of "Will and Idea" in this journal.

For the relationship of all this to MoMA and abstraction, see Cube Space and Inside the White Cube.

"The sacramental nature of the space becomes clear…." — Brian O'Doherty

Wednesday, December 26, 2012


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

This journal on June 24, 2006—

Robert M. Pirsig, Zen and the Art of Motorcycle Maintenance , 1974:

"But what's happening is that each year our old flat earth of
conventional reason becomes less and less adequate to handle
the experiences we have and this is creating widespread feelings
of topsy-turviness. As a result we're getting more and more people
in irrational areas of thought… occultism, mysticism, drug changes
and the like… because they feel the inadequacy of classical reason
to handle what they know are real experiences."

"I'm not sure what you mean by classical reason."

"Analytic reason, dialectic reason. Reason which at the University
is sometimes considered to be the whole of understanding. You've
never had  to understand it really. It's always been completely
bankrupt with regard to abstract art. Nonrepresentative art is one of
the root experiences I'm talking about. Some people still condemn it
because it doesn’t make 'sense.' But what's really wrong is not
the art but the 'sense,' the classical reason, which can't grasp it.
People keep looking for branch extensions of reason that will cover
art's more recent occurrences, but the answers aren't in  the
branches, they're at the roots."

See also an exhibition at the Museum of Modern Art that opened Dec. 23—

— and an exhibition in this journal of the image "Root Circle."

Tuesday, December 25, 2012

Mot Juste

Filed under: General — m759 @ 8:00 PM
IMAGE- Anne Hathaway in 'Les Miserables'  Phantasmagoria.

At Heaven’s Gate

Filed under: General — m759 @ 12:00 PM

A meditation on the performance of the late Charles Durning
in the film The Best Little Whorehouse in Texas .

Monday, December 24, 2012

Post-Mortem for Quincy

Filed under: General,Geometry — m759 @ 11:59 PM

Raiders of the Lost Trunk, or:

Stars in the Attic

More »

See also A Glass for Klugman :

Context: Poetry and Truth,  Eternal Recreation,  
              Solid Symmetry, and Stevens's Rock.


Filed under: General — m759 @ 6:32 PM

Where Kirkland meets Quincy

IMAGE- Adolphus Busch Hall, 29 Kirkland St., Cambridge, MA

Click image for some context.

All Over Again

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

Octavio Paz —

"… the movement of analogy
begins all over once again."

See A Reappearing Number in this journal.


Figure 1 —

Background: MOG in this journal.

Figure 2 —

Image-- 'Then a miracle occurs' cartoon
Cartoon by S.Harris

Background —

Image-- Google search on 'miracle octad'-- top 3 results

Eternal Recreation

Filed under: General,Geometry — Tags: , , , , , — m759 @ 3:17 AM

Memories, Dreams, Reflections
by C. G. Jung

Recorded and edited By Aniela Jaffé, translated from the German
by Richard and Clara Winston, Vintage Books edition of April 1989

From pages 195-196:

“Only gradually did I discover what the mandala really is:
‘Formation, Transformation, Eternal Mind’s eternal recreation.’*
And that is the self, the wholeness of the personality, which if all
goes well is harmonious, but which cannot tolerate self-deceptions.”

* Faust , Part Two, trans. by Philip Wayne (Harmondsworth,
England, Penguin Books Ltd., 1959), p. 79. The original:

                   … Gestaltung, Umgestaltung, 
  Des ewigen Sinnes ewige Unterhaltung….

Jung’s “Formation, Transformation” quote is from the realm of
the Mothers (Faust Part Two, Act 1, Scene 5: A Dark Gallery).
The speaker is Mephistopheles.

See also Prof. Bruce J. MacLennan on this realm
in a Web page from his Spring 2005 seminar on Faust:

“In alchemical terms, F is descending into the dark, formless
primary matter from which all things are born. Psychologically
he is descending into the deepest regions of the
collective unconscious, to the source of life and all creation.
Mater (mother), matrix (womb, generative substance), and matter
all come from the same root. This is Faust’s next encounter with
the feminine, but it’s obviously of a very different kind than his
relationship with Gretchen.”

The phrase “Gestaltung, Umgestaltung ” suggests a more mathematical
approach to the Unterhaltung . Hence

Part I: Mothers

“The ultimate, deep symbol of motherhood raised to
the universal and the cosmic, of the birth, sending forth,
death, and return of all things in an eternal cycle,
is expressed in the Mothers, the matrices of all forms,
at the timeless, placeless originating womb or hearth
where chaos is transmuted into cosmos and whence
the forms of creation issue forth into the world of
place and time.”

— Harold Stein Jantz, The Mothers in Faust:
The Myth of Time and Creativity 
Johns Hopkins Press, 1969, page 37

Part II: Matrices


Part III: Spaces and Hypercubes

Click image for some background.

Part IV: Forms

Forms from the I Ching :

Click image for some background.

Forms from Diamond Theory :

Click image for some background.

Sunday, December 23, 2012

Digital Recreation

Filed under: General — m759 @ 2:29 AM

Denzel Washington in Deja Vu  (2006), directed by Tony Scott—

Click to enlarge.

IMAGE- Denzel Washington in 'Deja Vu' (2006)

See also Tony Scott and four and a half days ago* —

Japanese character
       for "field"

Related material from five  days ago

"At the point of convergence
the play of similarities and differences
cancels itself out in order that 
identity alone may shine forth. 
The illusion of motionlessness,
the play of mirrors of the one: 
identity is completely empty;
it is a crystallization and
in its transparent core
the movement of analogy 
begins all over once again."

— The Monkey Grammarian 

by Octavio Paz, translated by 
Helen Lane (Kindle edition of 
2011-11-07, Kindle locations 

* More precisely, what will be 4.5 days ago at 3:09 AM ET.

In a Nutshell…

Filed under: General — m759 @ 1:00 AM

The Kernel of the Concept of the Object

according to the New York Lottery yesterday—

From 4/27

From 11/24

IMAGE- Agent Smith from 'The Matrix,' 1999

A page numbered 176

A page numbered 187

Saturday, December 22, 2012

Cezanne’s Greetings

Filed under: General — Tags: — m759 @ 5:55 PM

IMAGE- Arts & Letters Daily, feed from Dec. 21, 2012: Charles Rosen, Cezanne, and Roger Scruton on high culture

Click anywhere in the above image for the high culture essay.

Here are the links to Rosen and Cézanne.

See also the December 11th Log24 post Conclusion.

Web Links:

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

Spidey Goes to Church

More realistically

  1. "Nick Bostrom is a Swedish philosopher at 
    St. Cross College, University of Oxford…."
  2. "The early location of St Cross was on a site in
    St Cross Road, immediately south of St Cross Church."
  3. "The church building is located on St Cross Road
    just south of Holywell Manor."
  4. "Balliol College has had a presence in the area since
    the purchase by Benjamin Jowett, the Master, in the 1870s
    of the open area which is the Balliol sports ground
    'The Master's Field.' "
  5. Leaving Wikipedia, we find a Balliol field at Log24:
  6. A different view of the same field, from 1950—
  7. A view from 1974, thanks to J. J. Seidel — 
  8. Yesterday's Analogies.

For Larry L. King…

Filed under: General — m759 @ 3:26 AM

… author of "The Best Little Whorehouse in Texas,"
who reportedly died on Thursday

IMAGE- Harper's magazine, Oct. 1970, pp. 94-95: John Berryman poem and Larry L. King Harvard article

For George Steiner…

Filed under: General — m759 @ 2:00 AM

… In gratitude for his book Real Presences

Will and Idea:

A related shell game:

Ad for a talk at Harvard by Nick Bostrom in April 2010—

Click ad for background on the April 10 , 2010, symposium.

See also Bostrom on the The Simulation Argument
and the Log24 April 12, 2010, Shell Game post above.

Note the black diamond logo of Bostrom's Oxford institute.

Friday, December 21, 2012


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

The Moore correspondence may be regarded
as an analogy between the 35 partitions of an
8-set into two 4-sets and the 35 lines in the
finite projective space PG(3,2).

Closely related to the Moore correspondence
is a correspondence (or analogy) between the
15 2-subsets of a 6-set and the 15 points of PG(3,2).

An analogy between  the two above analogies
is supplied by the exceptional outer automorphism of S6.

The 2-subsets of a 6-set are the points of a PG(3,2),
Picturing outer automorphisms of  S6, and
A linear complex related to M24.

(Background: InscapesInscapes III: PG(2,4) from PG(3,2),
and Picturing the smallest projective 3-space.)

* For some context, see Analogies and
  "Smallest Perfect Universe" in this journal.

Thursday, December 20, 2012


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

IMAGE- 'Inception' totems including the chess bishop of Ariadne (Ellen Page, who also starred in 'Hard Candy') and Google Doodle with Little Red Riding Hood

Click image for some background.

For Grimm Day*

Filed under: General — m759 @ 2:00 AM

* See Google Doodle for today, the
   200th anniversary of the Grimms'
   Kinder- und Hausmärchen .

   See also Ay Que Bonito Es Volar
    and The Broom Factory.


Filed under: General,Geometry — m759 @ 1:00 AM

See reference AR in "Binary Coordinate Systems."

See also Prewar Berlin in this journal.

Wednesday, December 19, 2012

Identity Shining Forth

Filed under: General,Geometry — Tags: — m759 @ 1:14 PM

From William M. Kantor's 1978 review of Peter J. Cameron's
1976 book Parallelisms of Complete Designs

"There are three ways an area of mathematics
can be surveyed: by a vast, comprehensive treatise;
by a monograph on a small corner of the field; or by
a monograph on a cross section."

An area of mathematics—

A small corner of the field—

A cross section—


The area— Four.
The corner— Identity.
The cross section— Window.

The three ways— December 8  ten years ago.

Tuesday, December 18, 2012


Filed under: General — m759 @ 3:09 PM

Today's New York Times  on a collector of Japanese art who died on December 8th—

In 1954, she made her first trip to Japan. The visit had been suggested by the architect Walter Gropius, whose disciple Benjamin Thompson was designing a modernist house for her in Oyster Bay, on Long Island.

Gropius, a titan of the Bauhaus school, was deeply influenced by the Japanese aesthetic and wanted her to experience its clean, spare lines firsthand.

See Dec. 8th in this  journal for the following clean, spare lines:

Japanese character
        for "field"

Related material: Looking Deeply and Field.

Monkey Grammar

Filed under: General,Geometry — Tags: , — m759 @ 2:29 AM

For a modern Adam and Eve—

W. Tecumseh Fitch and Gesche Westphal Fitch,
editors of a new four-volume collection titled
Language Evolution  (Feb. 2, 2012, $1,360)—

Related material—

"At the point of convergence
the play of similarities and differences
cancels itself out in order that 
identity alone may shine forth. 
The illusion of motionlessness,
the play of mirrors of the one: 
identity is completely empty;
it is a crystallization and
in its transparent core
the movement of analogy 
begins all over once again."

— The Monkey Grammarian 

by Octavio Paz, translated by
Helen Lane (Kindle edition of
2011-11-07, Kindle locations

The "play of mirrors" link above is my own.

Click on W. Tecumseh Fitch for links to some
examples of mirror-play in graphic design—
from, say, my own work in a version of 1977, not from
the Fitches' related work published online last June—

See also Log24 posts from the publication date
of the Fitches' Language Evolution

Groundhog Day, 2012.

Happy birthday to the late Alfred Bester.

Monday, December 17, 2012

The Sun Also Sets

Filed under: General — m759 @ 7:42 PM

Image adapted from a 1960 Grove Press
(Evergreen paperback E-219) cover
illustration, apparently by Roy Kuhlman,
for The Spirit of Zen , by Alan Watts

Nonlyric Stupidity

Filed under: General — m759 @ 5:35 PM

Or: Being There

(A sequel to last night's Lyric Intelligence )

IMAGE- Book reviews page of William Deresiewicz, showing reviews titled 'Be Here Now' and ''I Was There.''

William Deresiewicz reviews Kurt Vonnegut's 1952 novel Player Piano :

The novel’s prescience is chilling. Six years before the left-wing English
sociologist Michael Young published The Rise of the Meritocracy ,
a dystopian satire that coined that now-ubiquitous final word,
Vonnegut was already there.

Related material:

Intelligence Test , Gombrich,  and, more generally, Stupidity.

Sunday, December 16, 2012

Lyric Intelligence

Filed under: General — Tags: — m759 @ 8:00 PM

Тут Аполлон — идеал, там Ниобея — печаль….

The source: The link from St. Lucia's Day to "Congregated Light,"
and the link from there to the Pushkin verse.

See also St. Lucia's Day ten years ago in posts tagged Meadow-down.

Thursday, December 13, 2012


Filed under: General,Geometry — Tags: — m759 @ 8:00 PM


Cubic models of finite geometries
display an interplay between
Euclidean and Galois  geometry.

Related literary remarks:  Congregated Light.


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

The sequel to Vibrations

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.”

Or place.

See  A Prince of Darkness
and "A Clean, Well-Lighted Place."


Filed under: General,Geometry — m759 @ 11:18 AM

 Or:  A Funny Thing Happened
     on the Way to the Embedding

This journal on the morning of Saturday, Dec. 8, 2012:

Plato's Diamond embedded in The Matrix

Marilyn Monroe and her music coach in 1954,
from last night's online New York Times :

" 'We were very close to making love; I don’t remember
the stage we were at, but I would say half-dressed,'
Mr. Schaefer recalled. He added: 'And all of a sudden
for some reason, Marilyn got these vibrations, and
we went over to the window….' "  more »

"Mr. Schaefer died on Saturday at 87 at his home in
Fort Lauderdale, Fla. ….

He [had] coached Monroe through 'Diamonds Are
a Girl’s Best Friend,' her signature number in the
1953 movie 'Gentlemen Prefer Blondes' (he arranged
the music as well)…."

Perhaps on Saturday she returned the favor.

Wednesday, December 12, 2012

Chromatic Plenitude

Filed under: General,Geometry — Tags: , — m759 @ 2:00 PM

(Continued from 2 PM ET Tuesday)

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

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

"The absolute consonance is a state of chromatic plenitude."

Charles Rosen

"… the nearest precedent might be found in Becky Sharp .
The opening of the Duchess of Richmond's ball,
with its organization of strong contrasts and
display of chromatic plenitude, presents a schema…."

— Scott Higgins, Harnessing the Technicolor Rainbow:
Color Design in The 1930s 
, University of Texas Press,
2007, page 142

Schema I    (Click to enlarge.)

Note the pattern on the dance floor.

(Click for wider image.)

Schema II 

"At the still point…" — Four Quartets

Tuesday, December 11, 2012


Filed under: General — m759 @ 9:00 PM

"If a man does not keep pace with his companions,
perhaps it is because he hears a different drummer.
Let him step to the music which he hears, however
measured or far away." — Thoreau

"I got a line on you." — Spirit

See also today's previous post and Eve's Menorah.


Filed under: General,Geometry — Tags: — m759 @ 2:00 PM

In memory of Charles Rosen:

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

Related material:

The Magic Square in Doctor Faustus  (October 10th, 2012)

Elementary Finite Geometry (August 1st, 2012)

The Space of Horizons (August 7th, 2012)

Chromatic Plenitude (Rosen on Schoenberg)

IMAGE- Charles Rosen on 'a final demarcation of form'


Filed under: General — Tags: — m759 @ 12:24 AM

"The study of musical form is an attempt to understand
the way in which musical gestures can be prolonged
and brought to a conclusion. The work of a great critic
like Charles Rosen can entirely change our
understanding of form— not merely by destroying the
neat categories of traditional musicology, but by
showing that form is a large-scale working-out of forces
that are brought into being by the musical material itself."

The Aesthetics of Music , by Roger Scruton,
     Oxford University Press, 1997 (according to
     a free online text version of the book)

The link to Rosen has been added. He died on Sunday.

Monday, December 10, 2012

Valery on Ornament

Filed under: General — m759 @ 3:33 PM

The following passage is suggested by last night's "Eve's Menorah"
and by today's "Review of Leonardo  Article" —

IMAGE- Valéry on ornament in 'Method of Leonardo,' with Valéry's serpent-and-key emblem

Click image to enlarge.

Review of Leonardo Article

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

Review of an often-cited Leonardo  article that is
now available for purchase online

The Tiling Patterns of Sebastien Truchet 
and the Topology of Structural Hierarchy

Authors: Cyril Stanley Smith and Pauline Boucher

Source: Leonardo , Vol. 20, No. 4,
20th Anniversary Special Issue:
Art of the Future: The Future of Art (1987),
pp. 373-385

Published by: The MIT Press

Stable URL: http://www.jstor.org/stable/1578535 .

Smith and Boucher give a well-illustrated account of
the early history of Truchet tiles, but their further remarks
on the mathematics underlying patterns made with
these tiles (see the diamond theorem* of 1976) are

For instance

Excerpt from pages 383-384—

"A detailed analysis of Truchet's
patterns touches upon the most fundamental
questions of the relation between
mathematical formalism and the structure
of the material world. Separations
between regions differing in density
require that nothing  be as important as
something  and that large and small cells of
both must coexist. The aggregation of
unitary choice of directional distinction
at interfaces lies at the root of all being
and becoming."

* This result is about Truchet-tile patterns, but the
    underlying mathematics was first discovered by
    investigating superimposed patterns of half-circles .
    See Half-Circle Patterns at finitegeometry.org.

Sunday, December 9, 2012

Eve’s Menorah

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

"Now the serpent was more subtle
than any beast of the field…."
Genesis 3:1

"“The serpent’s eyes shine
As he wraps around the vine….”
Don Henley

"Nine is a vine."
Folk rhyme

Part I

Part II

Part III

Halloween 2005

The image “http://log24.com/log/pix03/030109-gridsmall.gif” cannot be displayed, because it contains errors.

Click images for some background.

Adam in Eden

Filed under: General,Geometry — Tags: , — m759 @ 7:00 PM


"… we have taken the first steps
in decoding the uniquely human
fascination with visual patterns…."

W. Tecumseh Fitch et al. ,  July 2012

Fitch cites the following as a reference:

IMAGE- Washburn and Crowe, 'Symmetries of Culture' (1988)

Washburn and Crowe discuss symmetries in general, but
not the Galois geometry underlying patterns like some of
those shown on their book's cover.

Deep Structure

Filed under: General,Geometry — Tags: , — m759 @ 10:18 AM

The concept of "deep structure," once a popular meme,
has long been abandoned by Chomskians.

It still applies, however, to the 1976 mathematics, diamond theory  ,
underlying the formal patterns discussed in a Royal Society paper
this year.

A review of deep structure, from the Wikipedia article Cartesian linguistics

[Numbers in parentheses refer to pages in the original 1966 Harper edition of Chomsky's book Cartesian Linguistics .]

Deep structure vs. surface structure

"Pursuing the fundamental distinction between body and mind, Cartesian linguistics characteristically assumes that language has two aspects" (32). These are namely the sound/character of a linguistic sign and its significance (32). Semantic interpretation or phonetic interpretation may not be identical in Cartesian linguistics (32). Deep structures are often only represented in the mind (a mirror of thought), as opposed to surface structures, which are not.

Deep structures vary less between languages than surface structures. For instance, the transformational operations to derive surface forms of Latin and French may obscure common features of their deep structures (39). Chomsky proposes, "In many respects, it seems to me quite accurate, then, to regard the theory of transformational generative grammar, as it is developing in current work, as essentially a modern and more explicit version of the Port-Royal theory" (39).

Summary of Port Royal Grammar

The Port Royal Grammar is an often cited reference in Cartesian Linguistics  and is considered by Chomsky to be a more than suitable example of Cartesian linguistic philosophy. "A sentence has an inner mental aspect (a deep structure that conveys its meaning) and an outer, physical aspect as a sound sequence"***** This theory of deep and surface structures, developed in Port Royal linguistics, meets the formal requirements of language theory. Chomsky describes it in modern terms as "a base system that generates deep structures and a transformational system that maps these into surface structures", essentially a form of transformational grammar akin to modern studies (42).

The corresponding concepts from diamond theory are

"Deep structure"— The line diagrams indicating the underlying
structure of varying patterns

"A base system that generates deep structures"—
Group actions on square arrays for instance, on the 4×4 square

"A transformational system"— The decomposition theorem 
that maps deep structure into surface structure (and vice-versa)

Saturday, December 8, 2012

It’s 10 PM

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

Do you know where the mushrooms are?

IMAGE- Cover image for a free mixtape, 'Lawrence Class - The Diamond Theory,' that contains images from Steven H. Cullinane's 'Diamond Theory.'

Above: Image from Log24 on Dec. 4th, 2012, at 4:23 PM ET.

See also… on that date at that time …
The American College of Neuropsychopharmacology… (click to enlarge)

Defining the Contest…

Filed under: General,Geometry — Tags: , , , , — m759 @ 5:48 AM

Chomsky vs. Santa

From a New Yorker  weblog yesterday—

"Happy Birthday, Noam Chomsky." by Gary Marcus—

"… two titans facing off, with Chomsky, as ever,
defining the contest"

"Chomsky sees himself, correctly, as continuing
a conversation that goes back to Plato, especially
the Meno dialogue, in which a slave boy is
revealed by Socrates to know truths about
geometry that he hadn’t realized he knew."

See Meno Diamond in this journal. For instance, from 
the Feast of Saint Nicholas (Dec. 6th) this year—

The Meno Embedding

Plato's Diamond embedded in The Matrix

For related truths about geometry, see the diamond theorem.

For a related contest of language theory vs. geometry,
see pattern theory (Sept. 11, 16, and 17, 2012).

See esp. the Sept. 11 post,  on a Royal Society paper from July 2012
claiming that

"With the results presented here, we have taken the first steps
in decoding the uniquely human  fascination with visual patterns,
what Gombrich* termed our ‘sense of order.’ "

The sorts of patterns discussed in the 2012 paper —

IMAGE- Diamond Theory patterns found in a 2012 Royal Society paper

"First steps"?  The mathematics underlying such patterns
was presented 35 years earlier, in Diamond Theory.

* See Gombrich-Douat in this journal.

Friday, December 7, 2012

Bach Gate

Filed under: General — m759 @ 8:00 PM

In today's New York Times :

The late film executive Steven Bach on
the late literary agent Robert Lescher —

Mr. Bach’s 1985 book about film, “Final Cut:
Dreams and Disaster in the Making of ‘Heaven’s Gate’ ”—
a crow’s-nest view of the cultural and personal tensions
on the set— became a best seller and a Hollywood-insider
classic. In his acknowledgments, Mr. Bach said Mr. Lescher
“was the spiritual father to the book and to its author as well.”

See also "The rest is the madness of art" and Eight Gate.

The Embedding

Filed under: General,Geometry — m759 @ 7:00 PM

( Continued from December 6th —  and from Duelle here and in Pynchon )

Part I

The Galois Embedding

Part II

IMAGE- Anne Hathaway views diamond-quilt bed, captioned 'Don't go for second best, baby - Madonna'

In Nomine

Filed under: General — m759 @ 12:00 PM

The last name of the author Camelia Elias matches
the first name of the father of Robert "Slowly" Lescher.

Related material: December Days and  All Saints.

For Jim Holt

Filed under: General — m759 @ 11:01 AM



Filed under: General — m759 @ 10:18 AM

"Fail better." — Samuel Beckett, "Worstward Ho"

"West. And slowly." — Stage direction for a Christmas carol

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

The Interpreter

Filed under: General — m759 @ 7:11 PM

IMAGE- Obituaries featuring an 'intrepid interpreter of Beckett' who died Nov. 27, 2012

Related material: The Revisiting (December 3rd, 2012)

The Embedding

Filed under: General — m759 @ 6:29 PM

Part I

Embedding the Stone (March 23, 2012) —

The Meno Embedding

Plato's Diamond embedded in The Matrix

Part II

ReverbNation.com — Lawrence Class —

Ten Years Later

Filed under: General — m759 @ 1:06 PM

A review of a post from December 6, 2002,
suggests a review of more recent remarks.

See June 26, 2012: Looking Deeply.

A later post supplies some mathematical
background. See Counterexample (Nov. 27, 2012).

Happy Feast of Saint Nicholas.

Wednesday, December 5, 2012

Finis Coronat

Filed under: General — m759 @ 10:25 PM

Brazilian architect Oscar Niemeyer died today at 104.

His "Crown of Thorns" cathedral
appeared here in
some posts of August 2003.

Arte Programmata*

Filed under: General,Geometry — m759 @ 9:30 PM

The 1976 monograph "Diamond Theory" was an example
of "programmed art" in the sense established by, for
instance, Karl Gerstner. The images were produced 
according to strict rules, and were in this sense 
"programmed," but were drawn by hand.

Now an actual computer program has been written,
based on the Diamond Theory excerpts published
in the Feb. 1977 issue of Computer Graphics and Art
(Vol. 2, No. 1, pp. 5-7), that produces copies of some of
these images (and a few malformed images not  in
Diamond Theory).

See Isaac Gierard's program at GitHub


As the suffix indicates, this program is in the
Processing Development Environment language.

It produces the following sketch:

IMAGE- Sketch programmed by Isaac Gierard to mimic some of the images of 'Diamond Theory' (© 1976 by Steven H. Cullinane).

The rationale for selecting and arranging these particular images is not clear,
and some of the images suffer from defects (exercise: which ones?), but the 
overall effect of the sketch is pleasing.

For some background for the program, see The ReCode Project.

It is good to learn that the Processing language is well-adapted to making the 
images in such sketches. The overall structure of the sketch gives, however,
no clue to the underlying theory  in "Diamond Theory."

For some related remarks, see Theory (Sept. 30, 2012).

* For the title, see Darko Fritz, "Notions of the Program in 1960s Art."

December Days

Filed under: General — m759 @ 12:31 PM


Living Legend Takes Five.


Filed under: General — m759 @ 12:00 AM

(Midnight in the Garden, continued)

E is for Ending 

Cullinane = Cullinan + E

And for Everlast.

(The Cullinan link above is to a woman
who apparently is the wife of writer
Alan Cowell. See The Revisiting, Dec. 3rd.)

Tuesday, December 4, 2012

McKenna Theory

Filed under: General,Geometry — m759 @ 4:23 PM

A 1976 monograph:

IMAGE- 'Diamond Theory,' © 1976 by Steven H. Cullinane

A 2012 mixtape cover:

IMAGE- Cover image for a free mixtape, 'Lawrence Class - The Diamond Theory,' that contains images from Steven H. Cullinane's 'Diamond Theory.'

A new "Diamond Theory" image found on the Web
today links my work to the "Stoned Ape Theory"
of human evolution due to Terence McKenna

This link is via a picture, apparently copied from deviantart.com,
of two apes contemplating some psychedelic mushrooms.
The picture is titled "Stoned Ape Theory." The mushrooms in
the picture are apparently taken from an image at DrugNet.net:

Actually, the mathematical work called "diamond theory"
has nothing whatever to do with psychedelic experiences,
although some of the illustrations may appeal to McKenna fans.

December Days

Filed under: General — Tags: — m759 @ 8:48 AM

For the Dec. 3rd-4th graduate conference 
at the University of Cambridge on
"Occultism, Magic, and the History of Art"—

Four novels by Charles Williams—

IMAGE- Charles Williams novels: Shadows of Ecstasy, The Greater Trumps, Many Dimensions, and The Place of the Lion

See also the life, and Dec. 1st death, of a former Chief Justice of South Africa.

Monday, December 3, 2012

The Revisiting

Filed under: General,Geometry — Tags: , , — m759 @ 5:18 PM

Alan Cowell in the The New York Times ,
October 21, 2006— 

"Mr. Pinter played the role of Krapp,
a 69-year-old man revisiting
a tape recording he had made at 39…."

See also a weblog post by a 69-year-old man
revisiting a drawing he had made at 39.

The revisiting:

On Guy Fawkes Day 2011,
a return to Guy Fawkes day 2005—
Contrapuntal Themes in a Shadowland.

The drawing:

A clearer version, from 1981, of the central object below —

For commentary on the original 1981 drawing, see
Diamond-Faceted: Transformations of the Rock.

(A link in that page to "an earlier note from 1981
leads to remarks from exactly thirty years before
the 2011 post, made on another Guy Fawkes Day.)

Krapp vs. Hash

Filed under: General — Tags: , — m759 @ 10:01 AM

For the scholars gathered at a graduate conference
today and tomorrow at Cambridge University* on
"Occultism, Magic, and the History of Art"—

Part I: Krapp 

Click image for a 2006 New York Times  story 

Part II: Hash 

Top center front page, online NY Times, Christmas 2008-- Pinter dead at 78

* See Ledger and Red Ink.

Sunday, December 2, 2012


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.

The Rite

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

From About.com —
Click image for larger, clearer original —

"A fully lit Advent wreath with a central Christmas candle
on a home altar, in front of icons of Saint Stephen, Saint Michael,
and Our Lady of Czestochowa. (Photo © Scott P. Richert)"

Related material— Baal in The Rite , and today's previous post.

Knight’s Labyrinth

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

A magic— indeed, diabolic— square:

IMAGE- 5x5 magic- in fact, diabolic- square

For the construction, see a book
by W. W. Rouse Ball, founding president
of a Cambridge University magic society.

For some related religious remarks,
see Raiders of the Lost Matrix.

Saturday, December 1, 2012

Harvard Humor

Filed under: General — m759 @ 6:00 PM

(Continued from Grim Joke )

Happy Birthday to The Master.

Through a Lens, Darkly

Filed under: General — m759 @ 12:00 PM

Image from this journal on November 11th,
darkened in honor of a Harvard professor
who died on that date—

Related material: Lincoln and Darwin at Harvard.

Lucky Star

Filed under: General — m759 @ 10:00 AM

Evening numbers for the New York lottery
yesterday were 182 and 8691.

One interpretation of the latter number is
8/6/91, birth date of the WWW—

Friday, August 13, 2010
For a Bright Star
m759 @ 2:14 AM

The Hunt for Blue August

From Wikipedia's timeline for 1991—

August 6 – Tim Berners-Lee announces
the World Wide Web project and software
on the alt.hypertext newsgroup.

For one interpretation of the former number,
see Random Reference.

Star Wars

Filed under: General — m759 @ 2:01 AM

IMAGE- Rudolf Koch's version of the 'double cross' symbol

  Source: Rudolf KochThe Book of Signs

The American Mathematical Society
(AMS) yesterday:

Lars Hörmander (1931-2012)
Friday November 30th 2012

Hörmander, who received a Fields Medal in 1962,
died November 25 at the age of 81. …

more »

Some related material:

See also posts on Damnation Morning and, from the
date of Hörmander's death,

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