Log24

Sunday, November 18, 2012

Sermon

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

Happy birthday to

IMAGE- Margaret Atwood, Kim Wilde, Peta Wilson

Today's sermon, by Marie-Louise von Franz

Number and Time, by Marie-Louise von Franz

For more on the modern physicist analyzed by von Franz,
see The Innermost Kernel , by Suzanne Gieser.

Another modern physicist, Niels Bohr, died
on this date in 1962

Diamond Theory version of 'The Square Inch Space' with yin-yang symbol for comparison

The circle above is marked with a version
of the classic Chinese symbol
adopted as a personal emblem
by Danish physicist Niels Bohr,
leader of the Copenhagen School.

For the square, see the diamond theorem.

"Two things of opposite natures seem to depend
On one another, as a man depends
On a woman, day on night, the imagined
On the real. This is the origin of change.
Winter and spring, cold copulars, embrace
And forth the particulars of rapture come."

— Wallace Stevens,
  "Notes Toward a Supreme Fiction,"
  Canto IV of "It Must Change"

Saturday, November 5, 2011

Shadows

Filed under: General,Geometry — Tags: — m759 @ 7:59 AM

Between the idea
And the reality
Between the motion
And the act
Falls the Shadow

— T. S. Eliot, "The Hollow Men"

A passage quoted here on this date in 2005—

Douglas Hofstadter on his magnum opus:

“… I realized that to me,
Gödel and Escher and Bach
were only shadows
cast in different directions
by some central solid essence."

This refers to Hofstadter's cover image:

IMAGE- http://www.log24.com/log/pix11C/111105-GEBshadows.jpg

Also from this date in 2005:

IMAGE- www.log24.com/theory/images/GEB.jpg
 
BackgroundYesterday's link Change Logos,
                         
and Solid Symmetry.

Midrash:         Hearts of Darkness.

Friday, November 4, 2011

Logos

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

Continued from All Hallows Eve

The Belgian Lottery logo

http://www.log24.com/log/pix11C/111104-BelgianLotteryLogo-256w.jpg

The Belgian Lottery was a sponsor of 
last month's 25th Solvay Conference —

"The Theory of the Quantum World,"
  Brussels, October 19-22, 2011.

See also this journal in October and Change Logos

http://www.log24.com/log/pix09/090309-SqInSpace2.jpg

(Physicists will recognize the kinship
with the coat of arms of Niels Bohr.)

Friday, July 1, 2011

Symmetry Review

Filed under: General,Geometry — Tags: , — m759 @ 5:01 AM

Popular novelist Dan Brown is to speak at Chautauqua Institution on August 1.

This suggests a review of some figures discussed here in a note on Brown from February 20, 2004

IMAGE- Like motions of a pattern's parts can induce motions of the whole. Escher-'Fishes and Scales,' Cullinane-'Invariance'

Related material: Notes from Nov. 5, 1981, and from Dec. 24, 1981.

For the lower figure in context, see the diamond theorem.

Saturday, May 28, 2011

Savage Detectives

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

IMAGE- Rubeus Hagrid and Jorn Barger


IMAGE- Cover of 'The Savage and Beautiful Country'

   Alan McGlashan

From Savage Logic

Sunday, March 15, 2009  5:24 PM

The Origin of Change

A note on the figure
from this morning's sermon:

Diamond Theory version of 'The Square Inch Space' with yin-yang symbol for comparison

"Two things of opposite natures seem to depend
On one another, as a man depends
On a woman, day on night, the imagined
On the real. This is the origin of change.
Winter and spring, cold copulars, embrace
And forth the particulars of rapture come."

Wallace Stevens,  
"Notes Toward a Supreme Fiction,"
Canto IV of "It Must Change"

Saturday, January 22, 2011

High School Squares*

Filed under: General,Geometry — Tags: — m759 @ 1:20 AM

The following is from the weblog of a high school mathematics teacher—

http://www.log24.com/log/pix11/110121-LatinSquares4x4.jpg

This is related to the structure of the figure on the cover of the 1976 monograph Diamond Theory

http://www.log24.com/log/pix11/110122-DiamondTheoryCover.jpg

Each small square pattern on the cover is a Latin square,
with elements that are geometric figures rather than letters or numerals.
All order-four Latin squares are represented.

For a deeper look at the structure of such squares, let the high-school
chart above be labeled with the letters A through X, and apply the
four-color decomposition theorem.  The result is 24 structural diagrams—

    Click to enlarge

IMAGE- The Order-4 (4x4) Latin Squares

Some of the squares are structurally congruent under the group of 8 symmetries of the square.

This can be seen in the following regrouping—

   Click to enlarge

IMAGE- The Order-4 (4x4) Latin Squares, with Congruent Squares Adjacent

      (Image corrected on Jan. 25, 2011– "seven" replaced "eight.")

* Retitled "The Order-4 (i.e., 4×4) Latin Squares" in the copy at finitegeometry.org/sc.

Tuesday, October 19, 2010

Savage Logic…

Filed under: General,Geometry — Tags: — m759 @ 2:22 AM

and the New York Lottery

IMAGE-- NY Lottery Oct. 18, 2010-- Midday 069, Evening 359

A search in this journal for yesterday's evening number in the New York Lottery, 359, leads to…

The Cerebral Savage: 
On the Work of Claude Lévi-Strauss

by Clifford Geertz

Shown below is 359, the final page of Chapter 13 in
The Interpretation of Cultures: Selected Essays by Clifford Geertz,
New York, 1973: Basic Books, pp. 345-359 —

http://www.log24.com/log/pix10B/101019-Geertz359.gif

This page number 359 also appears in this journal in an excerpt from Dan Brown's novel Angels & Demons

See this journal's entries for March 1-15, 2009, especially…

Sunday, March 15, 2009  5:24 PM

Philosophy and Poetry:

The Origin of Change

A note on the figure
from this morning's sermon:

Diamond Theory version of 'The Square Inch Space' with yin-yang symbol for comparison

"Two things of opposite natures seem to depend
On one another, as a man depends
On a woman, day on night, the imagined

On the real. This is the origin of change.
Winter and spring, cold copulars, embrace
And forth the particulars of rapture come."

-- Wallace Stevens,
  "Notes Toward a Supreme Fiction,"
   Canto IV of "It Must Change"

Sunday, March 15, 2009  11:00 AM

Ides of March Sermon:

Angels, Demons,
"Symbology"

"On Monday morning, 9 March, after visiting the Mayor of Rome and the Municipal Council on the Capitoline Hill, the Holy Father spoke to the Romans who gathered in the square outside the Senatorial Palace…

'… a verse by Ovid, the great Latin poet, springs to mind. In one of his elegies he encouraged the Romans of his time with these words:

"Perfer et obdura: multo graviora tulisti."

 "Hold out and persist:
  you have got through
  far more difficult situations."

 (Tristia, Liber  V, Elegia  XI, verse 7).'"
 

This journal
on 9 March:

Diamond Theory version of 'The Square Inch Space' with yin-yang symbol for comparison

Note the color-interchange
symmetry
of each symbol
under 180-degree rotation.

Related material:
The Illuminati Diamond:

IMAGE- Illuminati Diamond, pp. 359-360 in 'Angels & Demons,' Simon & Schuster Pocket Books 2005, 448 pages, ISBN 0743412397

The symmetry of the yin-yang symbol, of the diamond-theorem symbol, and of Brown's Illuminati Diamond is also apparent in yesterday's midday New York lottery number (see above).

"Savage logic works like a kaleidoscope…." — Clifford Geertz on Lévi-Strauss

Tuesday, June 15, 2010

Imago, Imago, Imago

Filed under: General,Geometry — Tags: , , , — m759 @ 11:07 AM

Recommended— an online book—

Flight from Eden: The Origins of Modern Literary Criticism and Theory,
by Steven Cassedy, U. of California Press, 1990.

See in particular

Valéry and the Discourse On His Method.

Pages 156-157—

Valéry saw the mind as essentially a relational system whose operation he attempted to describe in the language of group mathematics. “Every act of understanding is based on a group,” he says (C, 1:331). “My specialty—reducing everything to the study of a system closed on itself and finite” (C, 19: 645). The transformation model came into play, too. At each moment of mental life the mind is like a group, or relational system, but since mental life is continuous over time, one “group” undergoes a “transformation” and becomes a different group in the next moment. If the mind is constantly being transformed, how do we account for the continuity of the self? Simple; by invoking the notion of the invariant. And so we find passages like this one: “The S[elf] is invariant, origin, locus or field, it’s a functional property of consciousness” (C, 15:170 [2: 315]). Just as in transformational geometry, something remains fixed in all the projective transformations of the mind’s momentary systems, and that something is the Self (le Moi, or just M, as Valéry notates it so that it will look like an algebraic variable). Transformation theory is all over the place. “Mathematical science . . . reduced to algebra, that is, to the analysis of the transformations of a purely differential being made up of homogeneous elements, is the most faithful document of the properties of grouping, disjunction, and variation in the mind” (O, 1:36). “Psychology is a theory of transformations, we just need to isolate the invariants and the groups” (C, 1:915). “Man is a system that transforms itself” (C, 2:896).

Notes:

  Paul Valéry, Oeuvres (Paris: Pléiade, 1957-60)

C   Valéry, Cahiers, 29 vols. (Paris: Centre National de le Recherche Scientifique, 1957-61)

Compare Jung’s image in Aion  of the Self as a four-diamond figure:

http://www.log24.com/log/pix10A/100615-JungImago.gif

and Cullinane’s purely geometric four-diamond figure:

http://www.log24.com/log/pix10A/100615-FourD.gif

For a natural group of 322,560 transformations acting on the latter figure, see the diamond theorem.

What remains fixed (globally, not pointwise) under these transformations is the system  of points and hyperplanes from the diamond theorem. This system was depicted by artist Josefine Lyche in her installation “Theme and Variations” in Oslo in 2009.  Lyche titled this part of her installation “The Smallest Perfect Universe,” a phrase used earlier by Burkard Polster to describe the projective 3-space PG(3,2) that contains these points (at right below) and hyperplanes (at left below).

Image-- Josefine Lyche's combination of Polster's phrase with<br /> Cullinane's images in her gallery show, Oslo, 2009-- 'The Smallest<br /> Perfect Universe -- Points and Hyperplanes'

Although the system of points (at right above) and hyperplanes (at left above) exemplifies Valéry’s notion of invariant, it seems unlikely to be the sort of thing he had in mind as an image of the Self.

Sunday, March 29, 2009

Sunday March 29, 2009

Filed under: General,Geometry — Tags: — m759 @ 7:48 PM

Getting All
the Meaning In

Webpage heading for the
2009 meeting of the
American Comparative
Literature Association:

ACLA 2009 web page heading with map and alphabetic symbols

The mysterious symbols on
the above map suggest the
following reflections:


From A Cure of the Mind: The Poetics of Wallace Stevens, by Theodore Sampson, published by Black Rose Books Ltd., 2000–

Page x:

"… if what he calls 'the spirit's alchemicana' (CP [Collected Poems] 471) addresses itself to the irrational element in poetry, to what extent is such an element dominant in his theory and practice of poetry, and therefore in what way is Stevens' intricate verbal music dependent on his irrational use of language– a 'pure rhetoric of a language without words?' (CP 374)?"

Related material:

 

From "'When Novelists Become Cubists:' The Prose Ideograms of Guy Davenport," by Andre Furlani:

Laurence Zachar argues that Davenport's writing is situated "aux frontieres intergeneriques" where manifold modes are brought into concord: "L'etonnant chez Davenport est la facon don't ce materiau qui parait l'incarnation meme du chaos– hermetique, enigmatique, obscur, avec son tropplein de references– se revele en fait etre construit, ordonne, structure. Plus l'on s'y plonge, et plus l'on distingue de cohesion dans le texte." 'What astonishes in Davenport is the way in which material that seems the very incarnation of chaos– hermetic, enigmatic, obscure, with its proliferation of allusions– in fact reveals itself to be constructed, organized, structured. The more one immerses oneself in them the more one discerns the texts' cohesion.' (62).

Davenport also works along the intergeneric border between text and graphic, for he illustrates many of his texts. (1) "The prime use of words is for imagery: my writing is drawing," he states in an interview (Hoeppfner 123). Visual imagery is not subordinated to writing in Davenport, who draws on the assemblage practice of superimposing image and writing. "I trust the image; my business is to get it onto the page," he writes in the essay "Ernst Machs Max Ernst." "A page, which I think of as a picture, is essentially a texture of images. […] The text of a story is therefore a continuous graph, kin to the imagist poem, to a collage (Ernst, Willi Baumeister, El Lissitzky), a page of Pound, a Brakhage film" (Geography 374-75).

Note:

(1.) Davenport is an illustrator of books (such as Hugh Kenner's The Stoic Comedians and The Counterfeiters) and journals (such as The Kenyon Review, Parnassus, and Paideuma). His art is the subject of Erik Anderson Reece's monograph, A Balance of Quinces, which reveals the inseparable relationship between Davenport's literary and pictorial work.

References:

Davenport, Guy. The Geography of the Imagination. San Francisco: North Point Press, 1981. Rpt. New York: Pantheon, 1992.

Hoepffner, Bernard. "Pleasant Hill: An Interview with Guy Davenport." Conjunctions 24 (1995): 118-24.

Reece, Erik Anderson. A Balance of Quinces: The Paintings and Drawings of Guy Davenport. New York: New Directions, 1996.

Zachar, Laurence. "Guy Davenport: Une Mosaique du genres." Recherches Anglaises et Nord-Americaines 21 (1994): 51-63.

"… when novelists become Cubists; that is, when they see the possibilities of making a hieroglyph, a coherent symbol, an ideogram of the total work. A symbol comes into being when an artist sees that it is the only way to get all the meaning in."

— Guy Davenport, The Geography of the Imagination

See also last night's
commentary on the
 following symbols:

Diamond Theory version of 'The Square Inch Space' with yin-yang symbol for comparison

Sunday, March 15, 2009

Sunday March 15, 2009

Filed under: General,Geometry — Tags: — m759 @ 5:24 PM

The Origin of Change

A note on the figure
from this morning's sermon:

Diamond Theory version of 'The Square Inch Space' with yin-yang symbol for comparison

"Two things of opposite natures seem to depend
On one another, as a man depends
On a woman, day on night, the imagined

On the real. This is the origin of change.
Winter and spring, cold copulars, embrace
And forth the particulars of rapture come."

-- Wallace Stevens,
  "Notes Toward a Supreme Fiction,"
   Canto IV of "It Must Change"

Sunday March 15, 2009

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

Angels, Demons,
"Symbology"

"On Monday morning, 9 March, after visiting the Mayor of Rome and the Municipal Council on the Capitoline Hill, the Holy Father spoke to the Romans who gathered in the square outside the Senatorial Palace…

'… a verse by Ovid, the great Latin poet, springs to mind. In one of his elegies he encouraged the Romans of his time with these words:

"Perfer et obdura: multo graviora tulisti."

 "Hold out and persist:
  you have got through
  far more difficult situations."

 (Tristia, Liber  V, Elegia  XI, verse 7).'"

This journal
on 9 March:

Diamond Theory version of 'The Square Inch Space' with yin-yang symbol for comparison

Note the color-interchange
symmetry
of each symbol
under 180-degree rotation.

Related material:
The Illuminati Diamond:

IMAGE- Illuminati Diamond, pp. 359-360 in 'Angels & Demons,' Simon & Schuster Pocket Books 2005, 448 pages, ISBN 0743412397

Dan Brown's novel Angels & Demons introduced in the year 2000 the fictional academic discipline of "symbology" and a fictional Harvard professor of that discipline, Robert Langdon (named after ambigram* artist John Langdon).

Fictional Harvard professor of symbology Robert Langdon, as portrayed by Tom Hanks

Tom Hanks as Robert Langdon


A possible source for Brown's term "symbology" is a 1995 web page, "The Rotation of the Elements," by one "John Opsopaus." (Cf. Art History Club.)

"The four qualities are the key to understanding the rotation of the elements and many other applications of the symbology of the four elements." –John Opsopaus

* "…ambigrams were common in symbology…." —Angels & Demons
 

Monday, March 9, 2009

Monday March 9, 2009

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

Humorism

'The Manchurian Candidate' campaign button

"Always with a
little humor."
Dr. Yen Lo  

Diamond diagram of the four humors, the four qualities, the four elements, the four seasons, and four colors

From Temperament: A Brief Survey

For other interpretations
of the above shape, see
The Illuminati Diamond.

from Jung's Aion:

"From the circle and quaternity motif is derived the symbol of the geometrically formed crystal and the wonder-working stone. From here analogy formation leads on to the city, castle, church, house, room, and vessel. Another variant is the wheel. The former motif emphasizes the ego’s containment in the greater dimension of the self; the latter emphasizes the rotation which also appears as a ritual circumambulation. Psychologically, it denotes concentration on and preoccupation with a centre…." –Jung, Collected Works, Vol. 9, Part II, paragraph 352

As for rotation, see the ambigrams in Dan Brown's Angels & Demons (to appear as a film May 15) and the following figures:

Diamond Theory version of 'The Square Inch Space' with yin-yang symbol for comparison
 
Click on image
for a related puzzle.
For a solution, see
 The Diamond Theorem.

A related note on
"Angels & Demons"
director Ron Howard:

Director Ron Howard with illustration of the fictional discipline 'symbology'
 
Click image for details.

Monday, March 2, 2009

Monday March 2, 2009

Filed under: General,Geometry — Tags: — m759 @ 11:30 AM
Joyce's Nightmare
continues

Today in History – March 2

Today is Monday, March 2, the 61st day of 2009. There are 304 days left in the year.

Today's Highlight in History:

On March 2, 1939, Roman Catholic Cardinal Eugenio Pacelli was elected Pope on his 63rd birthday; he took the name Pius XII.

IMAGE- Illuminati Diamond, pp. 359-360 in 'Angels & Demons,' Simon & Schuster Pocket Books 2005, 448 pages, ISBN 0743412397

 

Log24 on June 9, 2008

From Gravity's Rainbow (Penguin Classics, 1995), page 563:

"He brings out the mandala he found.
'What's it mean?'
[….]

Slothrop gives him the mandala. He hopes it will work like the mantra that Enzian told him once, mba-kayere (I am passed over), mba-kayere… a spell […]. A mezuzah. Safe passage through a bad night…."

 

In lieu of Slothrop's mandala, here is another…

Christ and the four elements, 1495
 

Christ and the Four Elements

This 1495 image is found in
The Janus Faces of Genius:
The Role of Alchemy
in Newton's Thought,
by B. J. T. Dobbs,
Cambridge University Press,
2002, p. 85


Related mandalas:Diamond arrangement of the four elements
and

Logo by Steven H. Cullinane for website on finite geometry

For further details,
click on any of the
three mandalas above.

 

Angels and Demons cross within a diamond (page 306), and Finite Geometry logo

Happy birthday to
Tom Wolfe, author of
The Painted Word.
 

Friday, May 25, 2007

Friday May 25, 2007

Filed under: General,Geometry — Tags: , , , — m759 @ 7:11 AM
Dance and the Soul

From Log24 on
this date last year:

"May there be an ennui
of the first idea?
What else,
prodigious scholar,
should there be?"

— Wallace Stevens,
"Notes Toward a
Supreme Fiction"

The Associated Press,
May 25, 2007–

Thought for Today:
"I hate quotations.
 Tell me what you know."
— Ralph Waldo Emerson

[Journals, on May 3, 1849]

The First Idea:

The Line, by S. H. Cullinane

Four Elements:
 

Four Elements (Diamond)

Square Dance:

Square Dance (Diamond Theorem)

This "telling of what
I know" will of course
mean little to those
who, like Emerson,
have refused to learn
through quotations.

For those less obdurate
than Emerson —Harold Bloom
on Wallace Stevens

and Paul Valery's
   "Dance and the Soul"–

"Stevens may be playful, yet seriously so, in describing desire, at winter's end, observing not only the emergence of the blue woman of early spring, but seeing also the myosotis, whose other name is 'forget-me-not.' Desire, hearing the calendar hymn, repudiates the negativity of the mind of winter, unable to bear what Valery's Eryximachus had called 'this cold, exact, reasonable, and moderate consideration of human life as it is.' The final form of this realization in Stevens comes in 1950, in The Course of a Particular, in the great monosyllabic line 'One feels the life of that which gives life as it is.' But even Stevens cannot bear that feeling for long. As Eryximachus goes on to say in Dance and the Soul:

A cold and perfect clarity is a poison impossible to combat. The real, in its pure state, stops the heart instantaneously….[…] To a handful of ashes is the past reduced, and the future to a tiny icicle. The soul appears to itself as an empty and measurable form. –Here, then, things as they are come together, limit one another, and are thus chained together in the most rigorous and mortal* fashion…. O Socrates, the universe cannot for one instant endure to be only what it is.

Valery's formula for reimagining the First Idea is, 'The idea introduces into what is, the leaven of what is not.' This 'murderous lucidity' can be cured only by what Valery's Socrates calls 'the intoxication due to act,' particularly Nietzschean or Dionysiac dance, for this will rescue us from the state of the Snow Man, 'the motionless and lucid observer.'" —Wallace Stevens: The Poems of Our Climate

* "la sorte… la plus mortelle":
    mortal in the sense
   "deadly, lethal"

Other quotations

(from March 28,
the birthday of
Reba McEntire):

Logical Songs

Reba McEntire, Saturday Evening Post, Mar/Apr 1995

Logical Song I
(Supertramp)

"When I was young, it seemed that
Life was so wonderful, a miracle,
Oh it was beautiful, magical
And all the birds in the trees,
Well they'd be singing so happily,
Joyfully, playfully watching me"

Logical Song II
(Sinatra)

"You make me feel so young,
You make me feel like
Spring has sprung
And every time I see you grin
I'm such a happy in-
dividual….

You and I are
Just like a couple of tots
Running across the meadow
Picking up lots
Of forget-me-nots"

Tuesday, August 10, 2004

Tuesday August 10, 2004

Filed under: General,Geometry — Tags: — m759 @ 3:01 PM

Battle of Gods and Giants

In checking the quotations from Dante in the previous entry, I came across the intriguing site Gigantomachia:

"A gigantomachia or primordial battle between the gods has been retold in myth, cult, art and theory for thousands of years, from the Egyptians to Heidegger. This site will present the history of the theme. But it will do so in an attempt to raise the question of the contemporary relevance of it. Does the gigantomachia take place today? Where? When? In what relation to you and me?"

Perhaps atop the Empire State Building?

(See An Affair to Remember and  Empire State Building to Honor Fay Wray.)

Perhaps in relation to what the late poet Donald Justice called "the wood within"?

Perhaps in relation to T. S. Eliot's "The Waste Land" and the Feast of the Metamorphosis?

Or perhaps not.

Perhaps at Pergamon:

Perhaps at Pergamon Press:

Invariants 

"What modern painters are trying to do,
if they only knew it, is paint invariants."

— James J. Gibson in Leonardo
(Vol. 11, pp. 227-235.
Pergamon Press Ltd., 1978)

An example of invariant structure:

The three line diagrams above result from the three partitions, into pairs of 2-element sets, of the 4-element set from which the entries of the bottom colored figure are drawn.  Taken as a set, these three line diagrams describe the structure of the bottom colored figure.  After coordinatizing the figure in a suitable manner, we find that this set of three line diagrams is invariant under the group of 16 binary translations acting on the colored figure.

A more remarkable invariance — that of symmetry itself — is observed if we arbitrarily and repeatedly permute rows and/or columns and/or 2×2 quadrants of the colored figure above. Each resulting figure has some ordinary or color-interchange symmetry.

This sort of mathematics illustrates the invisible "form" or "idea" behind the visible two-color pattern.  Hence it exemplifies, in a way, the conflict described by Plato between those who say that "real existence belongs only to that which can be handled" and those who say that "true reality consists in certain intelligible and bodiless forms."

For further details, see a section on Plato in the Gigantomachia site.
 

Monday, April 5, 2004

Monday April 5, 2004

Filed under: General,Geometry — Tags: — m759 @ 4:03 AM

Ideas and Art

 
Motto of
Plato's Academy

 

From Minimalist Fantasies,
by Roger Kimball, May 2003:

All I want anyone to get out of my paintings, and all I ever get out of them, is the fact that you can see the whole idea without any confusion. … What you see is what you see.
—Frank Stella, 1966

Minimal Art remains too much a feat of ideation, and not enough anything else. Its idea remains an idea, something deduced instead of felt and discovered.
— Clement Greenberg, 1967

The artists even questioned whether art needed to be a tangible object. Minimalism … Conceptualism — suddenly art could be nothing more than an idea, a thought on a piece of paper….
— Michael Kimmelman, 2003

There was a period, a decade or two ago, when you could hardly open an art journal without encountering the quotation from Frank Stella I used as an epigraph. The bit about “what you see is what you see” was reproduced ad nauseam. It was thought by some to be very deep. In fact, Stella’s remarks—from a joint interview with him and Donald Judd—serve chiefly to underscore the artistic emptiness of the whole project of minimalism. No one can argue with the proposition that “what you see is what you see,” but there’s a lot to argue with in what he calls “the fact that you can see the whole idea without any confusion.” We do not, of course, see ideas. Stella’s assertion to the contrary might be an instance of verbal carelessness, but it is not merely verbal carelessness. At the center of minimalism, as Clement Greenberg noted, is the triumph of ideation over feeling and perception, over aesthetics.
— Roger Kimball, 2003

 

 

From How Not Much Is a Whole World,
by Michael Kimmelman, April 2, 2004

Decades on, it's curious how much Minimalism, the last great high modern movement, still troubles people who just can't see why … a plain white canvas with a line painted across it


"William Clark,"
by Patricia Johanson, 1967

should be considered art. That line might as well be in the sand: on this side is art, it implies. Go ahead. Cross it.

….

The tug of an art that unapologetically sees itself as on a par with science and religion is not to be underestimated, either. Philosophical ambition and formal modesty still constitute Minimalism's bottom line.

If what results can sometimes be more fodder for the brain than exciting to look at, it can also have a serene and exalted eloquence….

That line in the sand doesn't separate good art from bad, or art from nonart, but a wide world from an even wider one.

 

I maintain that of course
we can see ideas.

Example: the idea of
invariant structure.

"What modern painters
are trying to do,
if they only knew it,
is paint invariants."

— James J. Gibson, Leonardo,
    Vol. 11, pp. 227-235.
    Pergamon Press Ltd., 1978

For a discussion
of how this works, see
Block Designs,
4×4 Geometry, and
Diamond Theory.

Incidentally, structures like the one shown above are invariant under an important subgroup of the affine group AGL(4,2)…  That is to say, they are not lost in translation.  (See previous entry.)
 

Tuesday, December 3, 2002

Tuesday December 3, 2002

Filed under: General,Geometry — Tags: — m759 @ 1:45 PM

Symmetry, Invariance, and Objectivity

The book Invariances: The Structure of the Objective World, by Harvard philosopher Robert Nozick, was reviewed in the New York Review of Books issue dated June 27, 2002.

On page 76 of this book, published by Harvard University Press in 2001, Nozick writes:

"An objective fact is invariant under various transformations. It is this invariance that constitutes something as an objective truth…."

Compare this with Hermann Weyl's definition in his classic Symmetry (Princeton University Press, 1952, page 132):

"Objectivity means invariance with respect to the group of automorphisms."

It has finally been pointed out in the Review, by a professor at Göttingen, that Nozick's book should have included Weyl's definition.

I pointed this out on June 10, 2002.

For a survey of material on this topic, see this Google search on "nozick invariances weyl" (without the quotes).

Nozick's omitting Weyl's definition amounts to blatant plagiarism of an idea.

Of course, including Weyl's definition would have required Nozick to discuss seriously the concept of groups of automorphisms. Such a discussion would not have been compatible with the current level of philosophical discussion at Harvard, which apparently seldom rises above the level of cocktail-party chatter.

A similarly low level of discourse is found in the essay "Geometrical Creatures," by Jim Holt, also in the issue of the New York Review of Books dated December 19, 2002. Holt at least writes well, and includes (if only in parentheses) a remark that is highly relevant to the Nozick-vs.-Weyl discussion of invariance elsewhere in the Review:

"All the geometries ever imagined turn out to be variations on a single theme: how certain properties of a space remain unchanged when its points get rearranged."  (p. 69)

This is perhaps suitable for intelligent but ignorant adolescents; even they, however, should be given some historical background. Holt is talking here about the Erlangen program of Felix Christian Klein, and should say so. For a more sophisticated and nuanced discussion, see this web page on Klein's Erlangen Program, apparently by Jean-Pierre Marquis, Département de Philosophie, Université de Montréal. For more by Marquis, see my later entry for today, "From the Erlangen Program to Category Theory."

Saturday, July 20, 2002

Saturday July 20, 2002

Filed under: General,Geometry — Tags: , — m759 @ 10:13 PM
 

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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