Thursday, February 27, 2014

Sacred Space, continued

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

"An image comes to mind of a white, ideal space
​that, more than any single picture, may be the
archetypal image of 20th-century art."

— Brian O'Doherty, "Inside the White Cube"

Cube  spaces exist also in mathematics.

Tuesday, February 25, 2014

The Argument

Filed under: General — Tags: — m759 @ 8:30 PM

The title of a review of Charles Taylor's book A Secular Age
was quoted here at noon last Saturday —

"The Place of the Sacred
in the Absence of God

My comment from last Saturday —

"The place of the sacred is not, perhaps,
Davos, but a more abstract location."

A sequel —

"Religious Experience and the Modern Self,"
by Ross Douthat in The New York Times 
today at 4:25 PM ET —

"The argument comes from the Canadian
philosopher Charles Taylor and his
doorstop-thick magnum opus A Secular Age …." 

Related material: 

Helprin Doors and Doorstop Thick.

Monday, December 24, 2012

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.

Monday, October 8, 2012

Issue 16

Filed under: General,Geometry — Tags: , , — m759 @ 8:14 PM

From triplecanopy, Issue 16 —

International Art English, by Alix Rule and David Levine (July 30, 2012)

… In what follows, we examine some of the curious lexical, grammatical, and stylistic features of what we call International Art English. We consider IAE’s origins, and speculate about the future of this language through which contemporary art is created, promoted, sold, and understood. Some will read our argument as an overelaborate joke. But there’s nothing funny about this language to its users. And the scale of its use testifies to the stakes involved. We are quite serious….*

Space  is an especially important word in IAE and can refer to a raft of entities not traditionally thought of as spatial (the space of humanity ) as well as ones that are in most circumstances quite obviously spatial (the space of the gallery ). An announcement for the 2010 exhibition “Jimmie Durham and His Metonymic Banquet,” at Proyecto de Arte Contemporáneo Murcia in Spain, had the artist “questioning the division between inside and outside in the Western sacred space”—the venue was a former church—“to highlight what is excluded in order to invest the sanctum with its spatial purity. Pieces of cement, wire, refrigerators, barrels, bits of glass and residues of ‘the sacred,’ speak of the space of the exhibition hall … transforming it into a kind of ‘temple of confusion.’”

Spatial and nonspatial space are interchangeable in IAE. The critic John Kelsey, for instance, writes that artist Rachel Harrison “causes an immediate confusion between the space of retail and the space of subjective construction.” The rules for space  in this regard also apply to field , as in “the field of the real”—which is where, according to art historian Carrie Lambert-Beatty, “the parafictional has one foot.” (Prefixes like para -, proto -, post -, and hyper – expand the lexicon exponentially and Germanly, which is to say without adding any new words.) It’s not just that IAE is rife with spacey terms like intersection , parallel , parallelism , void , enfold , involution , and platform …

* Footnote not in the original—
  See also Geometry and Death from the date of the above article.

Sunday, August 28, 2011

The Cosmic Part

Filed under: General,Geometry — Tags: — m759 @ 6:29 PM

Yesterday's midday post, borrowing a phrase from the theology of Marvel Comics,
offered Rubik's mechanical contrivance as a rather absurd "Cosmic Cube."

A simpler candidate for the "Cube" part of that phrase:


The Eightfold Cube

As noted elsewhere, a simple reflection group* of order 168 acts naturally on this structure.

"Because of their truly fundamental role in mathematics,
even the simplest diagrams concerning finite reflection groups
(or finite mirror systems, or root systems—
the languages are equivalent) have interpretations
of cosmological proportions."

Alexandre V. Borovik in "Coxeter Theory: The Cognitive Aspects"

Borovik has a such a diagram—


The planes in Borovik's figure are those separating the parts of the eightfold cube above.

In Coxeter theory, these are Euclidean hyperplanes. In the eightfold cube, they represent three of seven projective points that are permuted by the above group of order 168.

In light of Borovik's remarks, the eightfold cube might serve to illustrate the "Cosmic" part of the Marvel Comics phrase.

For some related theological remarks, see Cube Trinity in this journal.

Happy St. Augustine's Day.

* I.e., one generated by reflections : group actions that fix a hyperplane pointwise. In the eightfold cube, viewed as a vector space of 3 dimensions over the 2-element Galois field, these hyperplanes are certain sets of four subcubes.

Friday, May 13, 2011

Apollo’s 13

Filed under: General,Geometry — Tags: — m759 @ 6:36 AM

Continued … See related previous posts.

IMAGE- The 13 symmetry axes of the cube

Those who prefer narrative to mathematics
may consult Wikipedia on The Cosmic Cube.

Tuesday, May 10, 2011

Groups Acting

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

The LA Times  on last weekend's film "Thor"—

"… the film… attempts to bridge director Kenneth Branagh's high-minded Shakespearean intentions with Marvel Entertainment's bottom-line-oriented need to crank out entertainment product."

Those averse to Nordic religion may contemplate a different approach to entertainment (such as Taymor's recent approach to Spider-Man).

A high-minded— if not Shakespearean— non-Nordic approach to groups acting—

"What was wrong? I had taken almost four semesters of algebra in college. I had read every page of Herstein, tried every exercise. Somehow, a message had been lost on me. Groups act . The elements of a group do not have to just sit there, abstract and implacable; they can do  things, they can 'produce changes.' In particular, groups arise naturally as the symmetries of a set with structure. And if a group is given abstractly, such as the fundamental group of a simplical complex or a presentation in terms of generators and relators, then it might be a good idea to find something for the group to act on, such as the universal covering space or a graph."

— Thomas W. Tucker, review of Lyndon's Groups and Geometry  in The American Mathematical Monthly , Vol. 94, No. 4 (April 1987), pp. 392-394

"Groups act "… For some examples, see

Related entertainment—

High-minded— Many Dimensions

Not so high-minded— The Cosmic Cube


One way of blending high and low—

The high-minded Charles Williams tells a story
in his novel Many Dimensions about a cosmically
significant cube inscribed with the Tetragrammaton—
the name, in Hebrew, of God.

The following figure can be interpreted as
the Hebrew letter Aleph inscribed in a 3×3 square—


The above illustration is from undated software by Ed Pegg Jr.

For mathematical background, see a 1985 note, "Visualizing GL(2,p)."

For entertainment purposes, that note can be generalized from square to cube
(as Pegg does with his "GL(3,3)" software button).

For the Nordic-averse, some background on the Hebrew connection—

Friday, June 15, 2007

Friday June 15, 2007

Filed under: General,Geometry — Tags: , — m759 @ 1:00 PM
A Study in
Art Education

Rudolf Arnheim, a student of Gestalt psychology (which, an obituary notes, emphasizes "the perception of forms as organized wholes") was the first Professor of the Psychology of Art at Harvard.  He died at 102 on Saturday, June 9, 2007.

The conclusion of yesterday's New York Times obituary of Arnheim:

"… in The New York Times Book Review in 1986, Celia McGee called Professor Arnheim 'the best kind of romantic,' adding, 'His wisdom, his patient explanations and lyrical enthusiasm are those of a teacher.'"

A related quotation:

"And you are teaching them a thing or two about yourself. They are learning that you are the living embodiment of two timeless characterizations of a teacher: 'I say what I mean, and I mean what I say' and 'We are going to keep doing this until we get it right.'"

Tools for Teaching

Here, yet again, is an illustration that has often appeared in Log24– notably, on the date of Arnheim's death:

The 3x3 square

Related quotations:

"We have had a gutful of fast art and fast food. What we need more of is slow art: art that holds time as a vase holds water: art that grows out of modes of perception and whose skill and doggedness make you think and feel; art that isn't merely sensational, that doesn't get its message across in 10 seconds, that isn't falsely iconic, that hooks onto something deep-running in our natures. In a word, art that is the very opposite of mass media. For no spiritually authentic art can beat mass media at their own game."

Robert Hughes, speech of June 2, 2004

"Whether the 3×3 square grid is fast art or slow art, truly or falsely iconic, perhaps depends upon the eye of the beholder."

Log24, June 5, 2004

If the beholder is Rudolf Arnheim, whom we may now suppose to be viewing the above figure in the afterlife, the 3×3 square is apparently slow art.  Consider the following review of his 1982 book The Power of the Center:

"Arnheim deals with the significance of two kinds of visual organization, the concentric arrangement (as exemplified in a bull's-eye target) and the grid (as exemplified in a Cartesian coordinate system)….

It is proposed that the two structures of grid and target are the symbolic vehicles par excellence for two metaphysical/psychological stances.  The concentric configuration is the visual/structural equivalent of an egocentric view of the world.  The self is the center, and all distances exist in relation to the focal spectator.  The concentric arrangement is a hermetic, impregnable pattern suited to conveying the idea of unity and other-worldly completeness.  By contrast, the grid structure has no clear center, and suggests an infinite, featureless extension…. Taking these two ideal types of structural scaffold and their symbolic potential (cosmic, egocentric vs. terrestrial, uncentered) as given, Arnheim reveals how their underlying presence organizes works of art."

— Review of Rudolf Arnheim's The Power of the Center: A Study of Composition in the Visual Arts (Univ. of Calif. Press, 1982). Review by David A. Pariser, Studies in Art Education, Vol. 24, No. 3 (1983), pp. 210-213

Arnheim himself says in this book (pp. viii-ix) that "With all its virtues, the framework of verticals and horizontals has one grave defect.  It has no center, and therefore it has no way of defining any particular location.  Taken by itself, it is an endless expanse in which no one place can be distinguished from the next.  This renders it incomplete for any mathematical, scientific, and artistic purpose.  For his geometrical analysis, Descartes had to impose a center, the point where a pair of coordinates [sic] crossed.  In doing so he borrowed from the other spatial system, the centric and cosmic one."

Students of art theory should, having read the above passages, discuss in what way the 3×3 square embodies both "ideal types of structural scaffold and their symbolic potential."

We may imagine such a discussion in an afterlife art class– in, perhaps, Purgatory rather than Heaven– that now includes Arnheim as well as Ernst Gombrich and Kirk Varnedoe.

Such a class would be one prerequisite for a more advanced course– Finite geometry of the square and cube.

Friday, July 2, 2004

Friday July 2, 2004

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

Is Nothing Sacred?


From a review in today’s

New York Times

of an L.A. art exhibit,

“Beyond Geometry”

By Michael Kimmelman
in Los Angeles

The roots of this work go back to Duchamp, the abiding spirit of “Beyond Geometry.” When he acquired his porcelain urinal in 1917 from a plumbing equipment manufacturer on lower Fifth Avenue in Manhattan, signed it R. Mutt and submitted the now infamous “Fountain” to the Society of Independent Artists exhibition, he set the stage for nearly every subsequent attempt to blur the difference between art and everyday life.

This was the great breakthrough of modernism or the end of culture as we know it, depending on your perspective. Either way, after Duchamp, as the artist Joseph Kosuth has put it, all art became conceptual.

Duchamp predicted that even a breath might end up being called a work of art, and he was right. Gilbert and George started calling their performances sculptures in the 70’s. Chris Burden, James Lee Byars and others said that their actions were sculptures. Smithson declared derelict factories and suburbs to be sculptures. Artists even made light, the ultimate intangible, into sculpture.

The show includes sculptures by Richard Serra and Barnett Newman. I recall Mr. Serra once talking about how Barnett Newman’s paintings invite you to walk past them, to experience them not in a single glance but over time, physically. He said the paintings, with their vertical stripes, or “zips,” are “about dividing and placing spaces next to one another, not about illusionism.”

“They’re great when you have to walk by them and immerse yourself in the divisions of their spaces,” he added. Meaning, they’re like sculptures.

Nomenclature is not the point. What matters is the ethos of countercultural disruption, looking at the world and art through the other end of the telescope, which is the heart of “Beyond Geometry” and the appeal of its best works to young artists.

Now is the time to put this period of postwar tumult into global perspective. The show here is a useful step in that direction.

Meanwhile, in Philadelphia,
other art events:

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(Click on logo for details.)

The reader may determine whether the Philadelphia nothing is the sort of nothing deemed, by some, sacred in my note of March 9, 2000.

I personally have a very low opinion of Kimmelman and his “ethos of countercultural disruption.”  The sort of light sculpture his words evoke is not that of the Pantheon (illustrated in an entry for St. Peter’s Day) but that of the current Philadelphia “Big Nothing” show, which in turn reminds me of that classic 1973 Hollywood art exhibit, The Exorcist:

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