Tuesday, February 9, 2016


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

IMAGE- Redefining the cube's symmetry planes: 13 planes, not 9.

The hexagons above appear also in Gary W. Gibbons,
"The Kummer Configuration and the Geometry of Majorana Spinors," 
1993, in a cube model of the Kummer 166 configuration

From Gary W. Gibbons, 'The Kummer Configuration and the Geometry of Majorana Spinors,' 1993, a cube model of the Kummer 16_6 configuration

Related material — The Religion of Cubism (May 9, 2003).

Wednesday, January 19, 2011

Intermediate Cubism

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

The following is a new illustration for Cubist Geometries

IMAGE- A Galois cube: model of the 27-point affine 3-space

(For elementary cubism, see Pilate Goes to Kindergarten and The Eightfold Cube.
 For advanced, see Solomon's Cube and Geometry of the I Ching .)

Cézanne's Greetings.

Monday, September 28, 2020


Filed under: General — m759 @ 10:30 AM

Also on May 2, 2020 — A paper on Cubism as Religion is accepted:

Some may question the desirability of acceptance by MDPI.
Acceptance at the Pearly Gates is another matter.

Thursday, February 20, 2020

In Memory of Jack Youngerman…

Filed under: General — Tags: — m759 @ 11:54 PM

An abstract artist who reportedly died at 93 yesterday.

A search in this  journal for Shubnikov yields

"Raiders of the Lost Stone" (December 26, 2017).

Monday, August 12, 2019

Never-Ending Toy Story

Filed under: General — Tags: , — m759 @ 1:40 PM

" … this beautiful love story . . . ."

An image from the previous post:

The above line "From the producer of Transformers " suggests
a story from March 18, 2019 . . .

      Misreading the words of di Bonaventura
yields a phrase that might be applied to 
the Church of Rome . . .
     "A franchise based on release dates." 
See dies natalis  in this journal.

     For the Church of Synchronology, see
the above di Bonaventura date, March 18.

     Then there is the Church of Cubism . . .

     "Before time began, there was the Cube."
     — Optimus Prime, Transformers , 2007

Friday, June 2, 2017

Squares and Circles, 1936

Filed under: General — m759 @ 12:45 AM

"This essay and exhibition might well be dedicated
to those painters of squares and circles
(and the architects influenced by them)
who have suffered at the hands of
philistines with political power."

Alfred H. Barr, Jr. in Cubism and Abstract Art ,
The Museum of Modern Art, New York, 1936,
    page 18

Wednesday, January 18, 2017

An Associative Function …

Filed under: General — m759 @ 2:02 PM

Quoted here on December 16, 2006

'An associative function' in cubist collage and in Joyce's Ulysses, in a paper by Archie K. Loss

See also …

The date  of the "Seconds" review above, 16 Dec. 2006, was 
the reason for the requotation in the first paragraph above.

Saturday, November 29, 2014

Something Missing?

Filed under: General — m759 @ 7:59 AM

The title refers to this morning's previous post.

The above links from today's aldaily.com :  Cubism,  Bernstein,  Hell.

Thursday, June 20, 2013

ART WARS: Chesterton Thursday

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

The New York Times  philosophy column "The Stone"
last evening had an essay on art by a sarcastic anarchist,
one Crispin Sartwell

"… whole generations of art lovers have been
trained in modernist dogma, and arts institutions’
access to various forms of state or foundation
support depend on it completely. One goes to
the museum to gasp at stunning works of
incomparable, super-human genius by beings
who are infinitely more exalted and important
than the mere humans staring at their paintings.
That’s why ordinary people staring at a Picasso
(allegedly) experience a kind of transcendence
or re-articulation of their lives and world."

 Cubism Re-Articulated:

  Click image for some backstory.

(IMAGE: Walter Gropius and Froebel's Third Gift,
from a Google image search today)

Background: Cubism in this journal and
Pilate Goes to Kindergarten.

Related material: Chesterton + Thursday in this journal.

Thursday, April 11, 2013

Naked Art

Filed under: General,Geometry — m759 @ 9:48 PM

The New Yorker  on Cubism:

"The style wasn’t new, exactly— or even really a style,
in its purest instances— though it would spawn no end
of novelties in art and design. Rather, it stripped naked
certain characteristics of all pictures. Looking at a Cubist
work, you are forced to see how you see. This may be
gruelling, a gymnasium workout for eye and mind.
It pays off in sophistication."

Online "Culture Desk" weblog, posted today by Peter Schjeldahl

Non-style from 1911:

IMAGE- Britannica 11th edition on the symmetry axes and planes of the cube

See also Cube Symmetry Planes  in this  journal.

A comment at The New Yorker  related to Schjeldahl's phrase "stripped naked"—

"Conceptualism is the least seductive modern-art movement."

POSTED 4/11/2013, 3:54:37 PM BY CHRISKELLEY

(The "conceptualism" link was added to the quoted comment.)

Monday, November 5, 2012

Sitting Specially

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

Some webpages at finitegeometry.org discuss
group actions on Sylvester’s duads and synthemes.

Those pages are based on the square model of
PG(3,2) described in the 1980’s by Steven H. Cullinane.

A rival tetrahedral model of PG(3,2) was described
in the 1990’s by Burkard Polster.

Polster’s tetrahedral model appears, notably, in
a Mathematics Magazine  article from April 2009—

IMAGE- Figure from article by Alex Fink and Richard Guy on how the symmetric group of degree 5 'sits specially' in the symmetric group of degree 6

Click for a pdf of the article.

Related material:

The Religion of Cubism” (May 9, 2003) and “Art and Lies
(Nov. 16, 2008).

This  post was suggested by following the link in yesterday’s
Sunday School post  to High White Noon, and the link from
there to A Study in Art Education, which mentions the date of
Rudolf Arnheim‘s death, June 9, 2007. This journal
on that date


IMAGE- The ninefold square

— The Delphic Corporation

The Fink-Guy article was announced in a Mathematical
Association of America newsletter dated April 15, 2009.

Those who prefer narrative to mathematics may consult
a Log24 post from a few days earlier, “Where Entertainment is God”
(April 12, 2009), and, for some backstory, The Judas Seat
(February 16, 2007).

Wednesday, April 18, 2012

Adam in Eden

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

…. and John Golding, an authority on Cubism who "courted abstraction"—

"Adam in Eden was the father of Descartes." — Wallace Stevens

Fictional symbologist Robert Langdon and a cube

Symbologist Robert Langdon views a corner of Solomon's Cube

From a Log24 post, "Eightfold Cube Revisited,"
on the date of Golding's death—

Dynkin diagram D4 for triality

A related quotation—

"… quaternions provide a useful paradigm
  for studying the phenomenon of 'triality.'"

  — David A. Richter's webpage Zometool Triality

See also quaternions in another Log24 post
from the date of Golding's death— Easter Act.

Tuesday, February 1, 2011

The Search

Filed under: General — m759 @ 10:01 AM

An image suggested by last night's PBS hour "Chautauqua: An American Narrative"—


Click for larger versions of the image search and of the Hall of Philosophy.

Both the screenshot and the Chautauqua photo (by jbi46 at flickr.com) were taken on July 19th, 2010.

The screenshot appeared in the post "Pediments of Appearance" (which also included two much less complex images).

Some background —  A webpage on  Analytical Cubism and a related search in this journal.

From Wallace Stevens, who appears at top center in the image above—

An Ordinary Evening in New Haven, XXII

Professor Eucalyptus said, “The search
For reality is as momentous as
The search for god.” It is the philosopher’s search

For an interior made exterior
And the poet’s search for the same exterior made
Interior: breathless things broodingly abreath

With the Inhalations of original cold
And of original earliness. Yet the sense
Of cold and earliness is a daily sense,

Not the predicate of bright origin.
Creation is not renewed by images
Of lone wanderers. To re-create, to use

The cold and earliness and bright origin
Is to search. Likewise to say of the evening star,
The most ancient light in the most ancient sky,

That it is wholly an inner light, that it shines
From the sleepy bosom of the real, re-creates,
Searches a possible for its possibleness.

Wednesday, January 19, 2011

A Shade Out of Synch

Filed under: General — m759 @ 5:24 PM

Review of a 1968 novel by Wilfrid Sheed, who died today—



Sheed on his boyhood in My Life as a Fan: A Memoir

"So it was back to… tinkering with my batting stance and praying that some one of my aged neighbors would miraculously rear back and give birth, like Sarah in the Bible, to a boy who would even more miraculously emerge at about my own age and not turn out to be a butterfly collector or other form of creep." (Simon & Schuster, 1993; in 2001 edition, page 84)

See also Shadow (September 23rd) —

"I was the shadow of the waxwing slain" — John Shade in Pale Fire , a novel by butterfly collector Vladimir Nabokov

—as well as Intermediate Cubism and Ironic Butterfly.


"This is called the transformation of things."

Sunday, November 16, 2008

Sunday November 16, 2008

Filed under: General,Geometry — m759 @ 8:00 PM
Art and Lies

Observations suggested by an article on author Lewis Hyde– “What is Art For?“–  in today’s New York Times Magazine:

Margaret Atwood (pdf) on Lewis Hyde’s
Trickster Makes This World: Mischief, Myth, and Art

“Trickster,” says Hyde, “feels no anxiety when he deceives…. He… can tell his lies with creative abandon, charm, playfulness, and by that affirm the pleasures of fabulation.” (71) As Hyde says, “…  almost everything that can be said about psychopaths can also be said about tricksters,” (158), although the reverse is not the case. “Trickster is among other things the gatekeeper who opens the door into the next world; those who mistake him for a psychopath never even know such a door exists.” (159)

What is “the next world”? It might be the Underworld….

The pleasures of fabulation, the charming and playful lie– this line of thought leads Hyde to the last link in his subtitle, the connection of the trickster to art. Hyde reminds us that the wall between the artist and that American favourite son, the con-artist, can be a thin one indeed; that craft and crafty rub shoulders; and that the words artifice, artifact, articulation and art all come from the same ancient root, a word meaning to join, to fit, and to make. (254) If it’s a seamless whole you want, pray to Apollo, who sets the limits within which such a work can exist. Tricksters, however, stand where the door swings open on its hinges and the horizon expands: they operate where things are joined together, and thus can also come apart.

For more about
“where things are
joined together,” see
 Eight is a Gate and
The Eightfold Cube.
Related material:

The Trickster
and the Paranormal

Martin Gardner on
   a disappearing cube —

“What happened to that… cube?”

Apollinax laughed until his eyes teared. “I’ll give you a hint, my dear. Perhaps it slid off into a higher dimension.”

“Are you pulling my leg?”

“I wish I were,” he sighed. “The fourth dimension, as you know, is an extension along a fourth coordinate perpendicular to the three coordinates of three-dimensional space. Now consider a cube. It has four main diagonals, each running from one corner through the cube’s center to the opposite corner. Because of the cube’s symmetry, each diagonal is clearly at right angles to the other three. So why shouldn’t a cube, if it feels like it, slide along a fourth coordinate?”

— “Mr. Apollinax Visits New York,” by Martin Gardner, Scientific American, May 1961, reprinted in The Night is Large

For such a cube, see

Cube with its four internal diagonals


this illustration in

The Religion of Cubism
(and the four entries
preceding it —
 Log24, May 9, 2003).

Beware of Gardner’s
“clearly” and other lies.

Tuesday, July 8, 2008

Tuesday July 8, 2008

Filed under: General — Tags: , — m759 @ 3:33 AM

Yesterday's entry discussed T.E. Hulme— a co-founder, with Ezra Pound, of the Imagist school of poetry. Recent entries on randomness, using the New York Lottery as a source of examples, together with Hulme's approach to poetry discussed yesterday, suggest the following meditation– what Charles Cameron might call a "bead game."

Part I:

Ezra Pound on Imagism (from Gaudier-Brzeska,* 1916):

Three years ago in Paris I got out of a "metro" train at La Concorde, and saw suddenly a beautiful face, and then another and another, and then a beautiful child’s face, and then another beautiful woman, and I tried all that day to find words for what this had meant to me, and I could not find any words that seemed to me worthy, or as lovely as that sudden emotion. [….]

The "one image poem" is a form of super-position, that is to say, it is one idea set on top of another. I found it useful in getting out of the impasse in which I had been left by my metro emotion. I wrote a thirty-line poem, and destroyed it because it was what we call work "of second intensity." Six months later I made a poem half that length; a year later I made the following hokku-like sentence: —

"The apparition of these
    faces in the crowd:
 Petals, on a
    wet, black bough."


I dare say it is meaningless unless one has drifted into a certain vein of thought. In a poem of this sort one is trying to record the precise instant when a thing outward and objective transforms itself, or darts into a thing inward and subjective.

Part II:

Eleanor Goodman on translation (in a July 7, 2008, weblog entry, "Pound and Process: An Introduction"):

"… all translations exist on an axis. Indeed, they exist in a manifold of many axes intersecting. One axis is that of foreignness and familiarity. One axis is that of structural mimicry, another of melodic mimicry. And one axis is that of semantic fidelity."

Goodman's use of the word "manifold" here is of course poetic, not mathematical.

Part III:

New York Lottery, mid-day on July 7, 2008: 771.

Part IV:

A Google search on manifold 771 reveals that 771 is, according to Google's scanners, an alternate form (a "translation," via structural mimicry) of a script version of the letter M. (See Part V below.)

Part V:

Long version of a 
one-image poem —

"Random apparition:
  manifold translated."

This poem summarizes the
relationship (See Part IV above) of
the (apparently) random number 771
to the rather non-random concept of
a linear manifold:

Paul R. Halmos, Finite Dimensional Vector Spaces, Princeton, 1948-- Definition of linear manifold (denoted by script M)

[Such lines and planes have not
been, in mathematical language,

— Paul R. Halmos,
Finite Dimensional Vector Spaces,
Princeton University Press, 1948

Short version of the   
above one-image poem

Script M

* Gaudier-Brzeska created the artifact shown on the cover of Solid Objects, a work of literary theory by Douglas Mao. For more on that artifact and on the New York Lottery, see Sermon for St. Peter's Day. "It is not in the premise that reality/ Is a solid…." –Wallace Stevens

"I was like, Oh My God." —Poet Billy Collins at Chautauqua Institution, morning of July 7, 2008

Friday, May 9, 2008

Friday May 9, 2008

Filed under: General — Tags: — m759 @ 10:31 AM
Cubist Language Game

"Philosophers ponder the idea
 of identity: what it is to give
 something a name on Monday
 and have it respond to 
  that name on Friday…."

Bernard Holland 


From Log24 on
August 19, 2003
and on
Ash Wednesday, 2004:
a reviewer on
An Instance of the Fingerpost::

"Perhaps we are meant to
 see the story as a cubist
 retelling of the crucifixion."

Related material
for today's anniversay
of the birth of philosopher
Jose Ortega y Gasset:

Cubism as Multispeech
Halloween Meditations
(illustrated below)

Cover of 'The Gameplayers of Zan,' by M.A. Foster

"Modern art…
will always have
the masses against it."
Ortega y Gasset, 1925    

Monday, February 25, 2008

Monday February 25, 2008

Filed under: General,Geometry — Tags: , , , — m759 @ 4:00 PM
A System of SymbolsA book from
Yale University Press
discussed in Log24
four years ago today:

Inside Modernism: Relativity Theory, Cubism, Narrative

Click on image for details.

The book is titled
Inside Modernism:
Relativity Theory,
Cubism, Narrative

For a narrative about relativity
and cubes, see Knight Moves.

Related material:

Geek chic in
this week’s New Yorker

“… it takes a system of symbols
to make numbers precise–
to ‘crystallize’ them….”

— and a mnemonic for three
days in October 2006
following a memorial to
the Amish schoolchildren
slain that month:

Seven is Heaven,
Eight is a Gate,
Nine is a Vine.

Saturday, April 28, 2007

Saturday April 28, 2007

Filed under: General,Geometry — m759 @ 11:07 PM

PA Lottery April 28, 2007: Midday 510, Evening 223

See last year’s   
entries for 5/10

My Space

4x4x4 cube

and for 2/23

Cubist Epiphany

4x4x4 cube

“This is a crazy world and
the only way to enjoy it
is to treat it as a joke.”

— Robert A. Heinlein,
The Number of the Beast

Saturday, December 16, 2006

Saturday December 16, 2006

Filed under: General,Geometry — m759 @ 10:31 AM
Cubism1 as Multispeech2
The image “http://www.log24.com/log/pix06B/061216-Cubism.gif” cannot be displayed, because it contains errors.

— From Pedagogy, Praxis, Ulysses

A quotation omitted from the above excerpt:

In Ulysses, there is "… the same quality of simultaneity as in cubist collage. Thus, for example, Bloom surveys the tombstones at Paddy Dignam's funeral and, in the midst of platitudinous and humorous thoughts, remembers Molly 'wanting to do it at the window'…."

Related material from quotations at the poetry journal eratio:

"The guiding law of the great variations in painting is one of disturbing simplicity.  First things are painted; then, sensations; finally, ideas.  This means that in the beginning the artist's attention was fixed on external reality; then, on the subjective; finally, on the intrasubjective.  These three stages are three points on a straight line."

— Jose Ortega y Gasset ("On Point of View in the Arts," an essay on the development of cubism)

Related material on
tombstones and windows:

Geometry's Tombstones,
Galois's Window, and
Architecture of Eternity.

The image “http://www.log24.com/theory/images/GaloisWindow.gif” cannot be displayed, because it contains errors.

See also the following part
of the eratio quotations:

The image “http://www.log24.com/log/pix06B/061216-Dilemma.jpg” cannot be displayed, because it contains errors.

Quotations arranged by
Gregory Vincent St. Thomasino

1 Or hypercubism: See 10/31/06.

2 Or "Wake" speech: See 10/31/05.

Monday, December 11, 2006

Monday December 11, 2006

Filed under: General,Geometry — m759 @ 7:20 AM
Geometry and Death

J. G. Ballard on “the architecture of death“:

“… a huge system of German fortifications that included the Siegfried line, submarine pens and huge flak towers that threatened the surrounding land like lines of Teutonic knights. Almost all had survived the war and seemed to be waiting for the next one, left behind by a race of warrior scientists obsessed with geometry and death.”

The Guardian, March 20, 2006

Edward Hirsch on Lorca:

“For him, writing is a struggle both with geometry and death.”

— “The Duende,” American Poetry Review, July/August 1999

“Rosenblum writes with
absolute intellectual honesty,
and the effect is sheer liberation….
The disposition of the material is
a model of logic and clarity.”

Harper’s Magazine review
quoted on back cover of
Cubism and Twentieth-Century Art,
by Robert Rosenblum
(Abrams paperback, 2001)

“Are Children the Ultimate Literary Critics?”
 — Top of the News 29 (Nov. 1972): 32-36.
“Sets forth his own aims in writing for children
 and laments ‘slice of life’ and chaos in
children’s literature. Maintains that children
like good plots, logic, and clarity,
and that they have a concern for
‘so-called eternal questions.'”

An Annotated Listing of Criticism
by Linnea Hendrickson

“She returned the smile, then looked
across the room to her youngest brother,
Charles Wallace, and to their father,
who were deep in concentration, bent
over the model they were building
of a tesseract: the square squared,
and squared again: a construction
of the dimension of time.”

A Swiftly Tilting Planet,
by Madeleine L’Engle

The image “http://www.log24.com/log/pix06B/061211-Swiftly2.gif” cannot be displayed, because it contains errors.

For “the dimension of time,”
see A Fold in Time,
Time Fold, and
Diamond Theory in 1937

A Swiftly Tilting Planet is a fantasy for children set partly in Vespugia, a fictional country bordered by Chile and Argentina.

For a more adult audience —

In memory of General Augusto Pinochet, who died yesterday in Santiago, Chile, a quotation from Federico Garcia Lorca‘s lecture on “the Duende” (Buenos Aires, Argentina, 1933):

“… Philip of Austria… longing to discover the Muse and the Angel in theology, found himself imprisoned by the Duende of cold ardors in that masterwork of the Escorial, where geometry abuts with a dream and the Duende wears the mask of the Muse for the eternal chastisement of the great king.”

Perhaps. Or perhaps Philip, “the lonely
hermit of the Escorial,” is less lonely now.

Wednesday, February 25, 2004

Wednesday February 25, 2004

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

Modernism as a Religion

In light of the controversy over Mel Gibson's bloody passion play that opens today, some more restrained theological remarks seem in order.  Fortunately, Yale University Press has provided a framework — uniting physics, art, and literature in what amounts to a new religion — for making such remarks.  Here is some background.

From a review by Adam White Scoville of Iain Pears's novel titled An Instance of the Fingerpost:

"Perhaps we are meant to see the story as a cubist retelling of the crucifixion, as Pilate, Barabbas, Caiaphas, and Mary Magdalene might have told it. If so, it is sublimely done so that the realization gradually and unexpectedly dawns upon the reader. The title, taken from Sir Francis Bacon, suggests that at certain times, 'understanding stands suspended' and in that moment of clarity (somewhat like Wordsworth's 'spots of time,' I think), the answer will become apparent as if a fingerpost were pointing at the way."

Recommended related material —

By others:

Inside Modernism:  Relativity Theory, Cubism, Narrative, Thomas Vargish and Delo E. Mook, Yale University Press, 1999

Signifying Nothing: The Fourth Dimension in Modernist Art and Literature

Corpus Hypercubus,
by Dali.  Not cubist,
perhaps "hypercubist."

By myself: 

Finite Relativity

The Crucifixion of John O'Hara

Block Designs

The Da Vinci Code and Symbology at Harvard

The Crimson Passion

Material that is related, though not recommended —

The Aesthetics of the Machine

Connecting Physics and the Arts

Monday, May 12, 2003

Monday May 12, 2003

Filed under: General — m759 @ 6:10 PM

The Tony Nominations

Dannie Abse quoting Robert Penn Warren:

“The name of the story will be Time,
But you must not pronounce its name.
Tell me a story of deep delight.”

 Dannie Abse

Abse deserves a Tony Smith award¹ for his play Pythagoras.

Frank Rich on Bush’s Top Gun speech:

“Only hours before President Bush’s prime-time speech came news of what Variety headlined on Page 1 as ‘Regime Change’ in Hollywood — the departure of the [West Wing] creator, the writer Aaron Sorkin.”

 George W. Bush

President Bush deserves a Tony Smith award² for his performance aboard the aircraft carrier Abraham Lincoln.

 Madeleine L’Engle on the religion of Cubism:

“There is such a thing as a tesseract.”

 Madeleine L’Engle

L’Engle, former librarian at the Cathedral of St. John the Divine in New York City, deserves a Tony Smith award³ for insisting on the existence of the tesseract, or 4-dimensional cube, as an object of conceptual art.

L’Engle is perhaps the best defender of the religious, or “story,” theory of truth, as opposed to the “diamond” theory of truth. (See my earlier May 12 entry, “Death and Truth,” which deals with the bishop of L’Engle’s cathedral.)

¹ See Tony Smith on mathematics.

² See Tony Smith on foreign policy.

³ See Tony Smith on conceptual art.

Friday, May 9, 2003

Friday May 9, 2003

Filed under: General,Geometry — Tags: — m759 @ 7:20 PM

The Religion of Cubism

In the dome of the Capitol at Washington, DC, a painting depicts The Apotheosis of Washington .  Personally, I prefer the following pair of pictures, which might be titled Apotheosis of the Cube.





A New York Times article says Tony Smith's instructions for fabricating Die  were as follows:

"a six-foot cube of quarter-inch hot-rolled steel with diagonal internal bracing."

The transparent cube in the upper picture above shows the internal diagonals.  The fact that there are four of these may be used to demonstrate the isomorphism of the group of rotations of the cube with the group of permutations on an arbitrary set of four elements.  For deeper results, see Diamond Theory.

For an explanation of why our current president might feel that the cube deserves an apotheosis, see the previous entry, "The Rhetoric of Power."

See, too, Nabokov's Transparent Things :

"Its ultimate vision was the incandescence of a book or a box grown completely transparent and hollow.  This is, I believe, it: not the crude anguish of physical death but the incomparable pangs of the mysterious mental maneuver needed to pass from one state of being to another.  Easy, you know, does it, son."

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