Friday, February 19, 2010
Deep Play:
Mimzy vs. Mimsy
From a 2007 film, "The Last Mimzy," based on
the classic 1943 story by Lewis Padgett
"Mimsy Were the Borogoves"–
As the above mandala pictures show,
the film incorporates many New Age fashions.
The original story does not.
A more realistic version of the story
might replace the mandalas with
the following illustrations–
Click to enlarge.
For a commentary, see "NonEuclidean Blocks."
(Here "nonEuclidean" means simply
other than Euclidean. It does not imply any
violation of Euclid's parallel postulate.)
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Thursday, February 18, 2010
Truth, Geometry, Algebra
The following notes are related to A Simple Reflection Group of Order 168.
1. According to H.S.M. Coxeter and Richard J. Trudeau
“There is a pleasantly discursive treatment of Pontius Pilate’s unanswered question ‘What is truth?’.”
— Coxeter, 1987, introduction to Trudeau’s The NonEuclidean Revolution
1.1 Trudeau’s Diamond Theory of Truth
1.2 Trudeau’s Story Theory of Truth
2. According to Alexandre Borovik and Steven H. Cullinane
2.1 Coxeter Theory according to Borovik
2.1.1 The Geometry–
Mirror Systems in Coxeter Theory
2.1.2 The Algebra–
Coxeter Languages in Coxeter Theory
2.2 Diamond Theory according to Cullinane
2.2.1 The Geometry–
Examples: Eightfold Cube and Solomon’s Cube
2.2.2 The Algebra–
Examples: Cullinane and (rather indirectly related) Gerhard Grams
Summary of the story thus far:
Diamond theory and Coxeter theory are to some extent analogous– both deal with reflection groups and both have a visual (i.e., geometric) side and a verbal (i.e., algebraic) side. Coxeter theory is of course highly developed on both sides. Diamond theory is, on the geometric side, currently restricted to examples in at most three Euclidean (and six binary) dimensions. On the algebraic side, it is woefully underdeveloped. For material related to the algebraic side, search the Web for generators+relations+”characteristic two” (or “2“) and for generators+relations+”GF(2)”. (This last search is the source of the Grams reference in 2.2.2 above.)
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Tuesday, February 16, 2010
From today's NY Times—
Obituaries for mystery authors
Ralph McInerny and Dick Francis
From the date (Jan. 29) of McInerny's death–
"…although a work of art 'is formed around something missing,' this 'void is its vanishing point, not its essence.'"
– Harvard University Press on Persons and Things (Walpurgisnacht, 2008), by Barbara Johnson
From the date (Feb. 14) of Francis's death–
The EIghtfold Cube
The "something missing" in the above figure is an eighth cube, hidden behind the others pictured.
This eighth cube is not, as Johnson would have it, a void and "vanishing point," but is instead the "still point" of T.S. Eliot. (See the epigraph to the chapter on automorphism groups in Parallelisms of Complete Designs, by Peter J. Cameron. See also related material in this journal.) The automorphism group here is of course the order168 simple group of Felix Christian Klein.
For a connection to horses, see
a March 31, 2004, post
commemorating the birth of Descartes
and the death of Coxeter–
Putting Descartes Before Dehors
For a more Protestant meditation,
see The Cross of Descartes—
"I've been the front end of a horse
and the rear end. The front end is better."
— Old vaudeville joke
For further details, click on
the image below–
Notre Dame Philosophical Reviews
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Sunday, October 11, 2009
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Monday, September 14, 2009
Figure
The Sept. 8 entry on nonEuclidean* blocks ended with the phrase “Go figure.” This suggested a MAGMA calculation that demonstrates how Klein’s simple group of order 168 (cf. Jeremy Gray in The Eightfold Way) can be visualized as generated by reflections in a finite geometry.
* i.e., other than Euclidean. The phrase “nonEuclidean” is usually applied to only some of the geometries that are not Euclidean. The geometry illustrated by the blocks in question is not Euclidean, but is also, in the jargon used by most mathematicians, not “nonEuclidean.”
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Tuesday, September 8, 2009
Froebel's
Magic Box
Continued from
Dec. 7, 2008,
and from
yesterday.
NonEuclidean
Blocks
Passages from a classic story:
… he took from his pocket a gadget he had found in the box, and began to unfold it. The result resembled a tesseract, strung with beads….
Tesseract
"Your mind has been conditioned to Euclid," Holloway said. "So this– thing– bores us, and seems pointless. But a child knows nothing of Euclid. A different sort of geometry from ours wouldn't impress him as being illogical. He believes what he sees."
"Are you trying to tell me that this gadget's got a fourth dimensional extension?" Paradine demanded.
"Not visually, anyway," Holloway denied. "All I say is that our minds, conditioned to Euclid, can see nothing in this but an illogical tangle of wires. But a child– especially a baby– might see more. Not at first. It'd be a puzzle, of course. Only a child wouldn't be handicapped by too many preconceived ideas."
"Hardening of the thoughtarteries," Jane interjected.
Paradine was not convinced. "Then a baby could work calculus better than Einstein? No, I don't mean that. I can see your point, more or less clearly. Only–"
"Well, look. Let's suppose there are two kinds of geometry– we'll limit it, for the sake of the example. Our kind, Euclidean, and another, which we'll call x. X hasn't much relationship to Euclid. It's based on different theorems. Two and two needn't equal four in it; they could equal y, or they might not even equal. A baby's mind is not yet conditioned, except by certain questionable factors of heredity and environment. Start the infant on Euclid–"
"Poor kid," Jane said.
Holloway shot her a quick glance. "The basis of Euclid. Alphabet blocks. Math, geometry, algebra– they come much later. We're familiar with that development. On the other hand, start the baby with the basic principles of our x logic–"
"Blocks? What kind?"
Holloway looked at the abacus. "It wouldn't make much sense to us. But we've been conditioned to Euclid."
— "Mimsy Were the Borogoves," Lewis Padgett, 1943

Padgett (pseudonym of a
husbandand
wife writing team) says that alphabet blocks are the intuitive "basis of Euclid."
Au contraire; they are the basis of
Gutenberg.
For the intuitive basis of one type of nonEuclidean* geometry– finite geometry over the twoelement Galois field– see the work of…
Friedrich Froebel
(17821852), who
invented kindergarten.
His "third gift" —
© 2005 The Institute for Figuring
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Friday, April 10, 2009
Pilate Goes
to Kindergarten
“There is a pleasantly discursive
treatment of Pontius Pilate’s
unanswered question
‘What is truth?’.”
— H. S. M. Coxeter, 1987,
introduction to Trudeau’s
remarks on the “Story Theory“
of truth as opposed to the
“Diamond Theory” of truth in
The NonEuclidean Revolution
Consider the following question in a paper cited by V. S. Varadarajan:
E. G. Beltrametti, “Can a finite geometry describe physical spacetime?” Universita degli studi di Perugia, Atti del convegno di geometria combinatoria e sue applicazioni, Perugia 1971, 57–62.
Simplifying:
“Can a finite geometry describe physical space?”
Simplifying further:
“Yes. Vide ‘The Eightfold Cube.'”
Comments Off on Friday April 10, 2009
Thursday, February 5, 2009
Through the
Looking Glass:
A Sort of Eternity
From the new president's inaugural address:
"… in the words of Scripture, the time has come to set aside childish things."
The words of Scripture:
9

For we know in part, and we prophesy in part.

10

But when that which is perfect is come, then that which is in part shall be done away.

11

When I was a child, I spake as a child, I understood as a child, I thought as a child: but when I became a man, I put away childish things.

12

For now we see through a glass, darkly, but then face to face: now I know in part; but then shall I know even as also I am known.
— First Corinthians 13

"through a glass"—
[di’ esoptrou].
By means of
a mirror [esoptron].
Childish things:
© 2005 The Institute for Figuring
Photo by Norman Brosterman
fom the Inventing Kindergarten
exhibit at The Institute for Figuring
(cofounded by Margaret Wertheim)
Notsochildish:
Three planes through
the center of a cube
that split it into
eight subcubes:
Through a glass, darkly:
A group of 8 transformations is
generated by affine reflections
in the above three planes.
Shown below is a pattern on
the faces of the 2x2x2 cube
that is symmetric under one of
these 8 transformations–
a 180degree rotation:
(Click on image
for further details.)
But then face to face:
A larger group of 1344,
rather than 8, transformations
of the 2x2x2 cube
is generated by a different
sort of affine reflections– not
in the infinite Euclidean 3space
over the field of real numbers,
but rather in the finite Galois
3space over the 2element field.
Galois age fifteen,
drawn by a classmate.
These transformations
in the Galois space with
finitely many points
produce a set of 168 patterns
like the one above.
For each such pattern,
at least one nontrivial
transformation in the group of 8
described above is a symmetry
in the Euclidean space with
infinitely many points.
For some generalizations,
see Galois Geometry.
Related material:
The central aim of Western religion–
"Each of us has something to offer the Creator...
the bridging of
masculine and feminine,
life and death.
It's redemption.... nothing else matters."
 Martha Cooley in The Archivist (1998)
The central aim of Western philosophy–
Dualities of Pythagoras
as reconstructed by Aristotle:
Limited Unlimited
Odd Even
Male Female
Light Dark
Straight Curved
... and so on ....
"Of these dualities, the first is the most important; all the others may be seen as different aspects of this fundamental dichotomy. To establish a rational and consistent relationship between the limited [man, etc.] and the unlimited [the cosmos, etc.] is… the central aim of all Western philosophy."
— Jamie James in The Music of the Spheres (1993)
"In the garden of Adding
live Even and Odd…
And the song of love's recision
is the music of the spheres."
— The Midrash Jazz Quartet in City of God, by E. L. Doctorow (2000)
A quotation today at art critic Carol Kino's website, slightly expanded:
"Art inherited from the old religion
the power of consecrating things
and endowing them with
a sort of eternity;
museums are our temples,
and the objects displayed in them
are beyond history."
— Octavio Paz,"Seeing and Using: Art and Craftsmanship," in Convergences: Essays on Art and Literature (New York: Harcourt Brace Jovanovich 1987), 52
From Brian O'Doherty's 1976 Artforum essays– not on museums, but rather on gallery space:
"Inside the White Cube"
"We have now reached
a point where we see
not the art but the space first….
An image comes to mind
of a white, ideal space
that, more than any single picture,
may be the archetypal image
of 20thcentury art."
"Space: what you
damn well have to see."
— James Joyce, Ulysses

Comments Off on Thursday February 5, 2009
Tuesday, January 6, 2009
Archetypes, Synchronicity,
and Dyson on Jung
The current (Feb. 2009) Notices of the American Mathematical Society has a written version of Freeman Dyson’s 2008 Einstein Lecture, which was to have been given in October but had to be canceled. Dyson paraphrases a mathematician on Carl Jung’s theory of archetypes:
“… we do not need to accept Jung’s theory as true in order to find it illuminating.”
The same is true of Jung’s remarks on synchronicity.
For example —
Yesterday’s entry, “A Wealth of Algebraic Structure,” lists two articles– each, as it happens, related to Jung’s fourdiamond figure from Aion as well as to my own Notes on Finite Geometry. The articles were placed online recently by Cambridge University Press on the following dates:
R. T. Curtis’s 1974 article defining his Miracle Octad Generator (MOG) was published online on Oct. 24, 2008.
Curtis’s 1987 article on geometry and algebraic structure in the MOG was published online on Dec. 19, 2008.
On these dates, the entries in this journal discussed…
Oct. 24:
Cube Space, 19842003
Material related to that entry:
Dec. 19:
Art and Religion: Inside the White Cube
That entry discusses a book by Mark C. Taylor:
The Picture in Question: Mark Tansey and the Ends of Representation (U. of Chicago Press, 1999).
In Chapter 3, “Sutures of Structures,” Taylor asks —
“What, then, is a frame, and what is frame work?”
One possible answer —
Hermann Weyl on the relativity problem in the context of the 4×4 “frame of reference” found in the above Cambridge University Press articles.
“Examples are the stainedglass
windows of knowledge.”
— Vladimir Nabokov
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Friday, December 19, 2008
Inside the
White Cube
Part I: The White Cube
Part II: Inside
Part III: Outside
Click to enlarge.
Mark Tansey, The Key (1984)
For remarks on religion
related to the above, see
Log24 on the Garden of Eden
and also Mark C. Taylor,
"What Derrida Really Meant"
(New York Times, Oct. 14, 2004).
For some background on Taylor,
see Wikipedia. Taylor, Chairman
of the Department of Religion at
Columbia University, has a
1973 doctorate in religion from
Harvard University. His opinion
of Derrida indicates that his
sympathies lie more with
the serpent than with the angel
in the Tansey picture above.
For some remarks by Taylor on
the art of Tansey relevant to the
structure of the white cube
(Part I above), see Taylor's
The Picture in Question:
Mark Tansey and the
Ends of Representation
(U. of Chicago Press, 1999):
From Chapter 3,
"Sutures* of Structures," p. 58:
"What, then, is a frame, and what is frame work?
This question is deceptive in its simplicity. A frame is, of course, 'a basic skeletal structure designed to give shape or support' (American Heritage Dictionary)…. when the frame is in question, it is difficult to determine what is inside and what is outside. Rather than being on one side or the other, the frame is neither inside nor outside. Where, then, Derrida queries, 'does the frame take place….'"
* P. 61:
"… the frame forms the suture of structure. A suture is 'a seamless [sic**] joint or line of articulation,' which, while joining two surfaces, leaves the trace of their separation."
** A dictionary says "a seamlike joint or line of articulation," with no mention of "trace," a term from Derrida's jargon.

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Friday, December 12, 2008
On the Symmetric Group S_{8}
Wikipedia on Rubik's 2×2×2 "Pocket Cube"–
"Any permutation of the 8 corner cubies is possible (8! positions)."
Some pages related to this claim–
Simple Groups at Play
Analyzing Rubik's Cube with GAP
Online JavaScript Pocket Cube.
The claim is of course trivially true for the unconnected subcubes of Froebel's Third Gift:
© 2005 The Institute for Figuring
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Sunday, December 7, 2008
Space and
the Soul
On a book by Margaret Wertheim:
“She traces the history of space beginning with the cosmology of Dante. Her journey continues through the historical foundations of celestial space, relativistic space, hyperspace, and, finally, cyberspace.” –Joe J. Accardi, Northeastern Illinois Univ. Lib., Chicago, in Library Journal, 1999 (quoted at Amazon.com)
There are also other sorts of space.
© 2005 The Institute for Figuring
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Sunday, November 16, 2008
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 conartist, 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.
“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 threedimensional 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 
Comments Off on Sunday November 16, 2008
Friday, October 24, 2008
“
The Cube Space” is a name given to
the eightfold cube in a vulgarized mathematics text,
Discrete Mathematics: Elementary and Beyond, by
Laszlo Lovasz et al., published by Springer in 2003. The identification in a natural way of the eight points of the linear 3space over the 2element field GF(2) with the eight vertices of a cube is an elementary and rather obvious construction, doubtless found in a number of discussions of discrete mathematics. But the lessobvious generation of
the affine group AGL(3,2) of order 1344 by permutations of parallel edges in such a cube may (or may not) have originated with me. For descriptions of this process I wrote in 1984, see
Diamonds and Whirls and Binary Coordinate Systems. For a vulgarized description of this process by Lovasz, without any acknowledgement of his sources, see an excerpt from his book.
Comments Off on Friday October 24, 2008
Wednesday, October 22, 2008
Euclid vs. Galois
On May 4, 2005, I wrote a note about how to visualize the 7point Fano plane within a cube.
Last month, John Baez showed slides that touched on the same topic. This note is to clear up possible confusion between our two approaches.
From Baez’s Rankin Lectures at the University of Glasgow:
Note that Baez’s statement (
pdf) “Lines in the Fano plane correspond to planes through the origin [the vertex labeled ‘1’] in this cube” is, if taken (wrongly) as a statement about a cube in Euclidean 3space, false.
The statement is, however, true of the eightfold cube, whose eight subcubes correspond to points of the linear 3space over the twoelement field, if “planes through the origin” is interpreted as planes within that linear 3space, as in Galois geometry, rather than within the Euclidean cube that Baez’s slides seem to picture.
This Galoisgeometry interpretation is, as an article of his from 2001 shows, actually what Baez was driving at. His remarks, however, both in 2001 and 2008, on the planecube relationship are both somewhat trivial– since “planes through the origin” is a standard definition of lines in projective geometry– and also unrelated– apart from the possibility of confusion– to my own efforts in this area. For further details, see The Eightfold Cube.
Comments Off on Wednesday October 22, 2008
Friday, September 26, 2008
Christmas Knotfor T.S. Eliot’s birthday
(Continued from Sept. 22–
“A Rose for Ecclesiastes.”)
From Kibler’s
“Variations on a Theme of
Heisenberg, Pauli, and Weyl,”
July 17, 2008:
“It is to be emphasized
that the 15 operators…
are underlaid by the geometry
of the generalized quadrangle
of order 2…. In this geometry,
the five sets… correspond to
a spread of this quadrangle,
i.e., to a set of 5 pairwise
skew lines….”
— Maurice R. Kibler,
July 17, 2008
For ways to visualize
this quadrangle,
see Inscapes.
Related material
A remark of Heisenberg quoted here on Christmas 2005:
“… die Schönheit… [ist] die richtige Übereinstimmung der Teile miteinander und mit dem Ganzen.”
“Beauty is the proper conformity of the parts to one another and to the whole.”

Comments Off on Friday September 26, 2008
Tuesday, August 19, 2008
Three Times
"Credences of Summer," VII,
by Wallace Stevens, from
Transport to Summer (1947)
"Three times the concentred
self takes hold, three times
The thrice concentred self,
having possessed
The object, grips it
in savage scrutiny,
Once to make captive,
once to subjugate
Or yield to subjugation,
once to proclaim
The meaning of the capture,
this hard prize,
Fully made, fully apparent,
fully found." 
Stevens does not say what object he is discussing.
One possibility —
Bertram Kostant, Professor Emeritus of Mathematics at MIT, on an object discussed in a recent New Yorker:
"A word about E(8). In my opinion, and shared by others, E(8) is the most magnificent 'object' in all of mathematics. It is like a diamond with thousands of facets. Each facet offering a different view of its unbelievable intricate internal structure."
Another possibility —
A more modest object —
the 4×4 square.
Update of Aug. 2021 —
Symmetries and Facets
Kostant's poetic comparison might be applied also to this object.
The natural rearrangements (symmetries) of the 4×4 array might also be described poetically as "thousands of facets, each facet offering a different view of… internal structure."
More precisely, there are 322,560 natural rearrangements– which a poet might call facets*— of the array, each offering a different view of the array's internal structure– encoded as a unique ordered pair of symmetric graphic designs. The symmetry of the array's internal structure is reflected in the symmetry of the graphic designs. For examples, see the Diamond 16 Puzzle.
For an instance of Stevens's "three times" process, see the three parts of the 2004 web page Ideas and Art.
* For the metaphor of rearrangements as facets, note that each symmetry (rearrangement) of a Platonic solid corresponds to a rotated facet: the number of symmetries equals the number of facets times the number of rotations (edges) of each facet–
The metaphor of rearrangements as facets breaks down, however, when we try to use it to compute, as above with the Platonic solids, the
number of natural rearrangements, or symmetries, of the 4×4 array. Actually, the true analogy is between the 16 unit squares of the 4×4 array, regarded as the 16 points of a finite 4space (which has finitely many symmetries), and the infinitely many points of Euclidean 4space (which has infinitely many symmetries).
If Greek geometers had started with a finite space (as in The Eightfold Cube), the history of mathematics might have dramatically illustrated Halmos's saying (Aug. 16) that
"The problem is– the genius is– given an infinite question, to think of the right finite question to ask. Once you thought of the finite answer, then you would know the right answer to the infinite question."
The Greeks, of course, answered the infinite questions first– at least for Euclidean space. Halmos was concerned with more general modern infinite spaces (such as Hilbert space) where the intuition to be gained from finite questions is still of value.
Comments Off on Tuesday August 19, 2008
Friday, August 8, 2008
Click on image for details.
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Friday, July 25, 2008
56 Triangles
"This wonderful picture was drawn by Greg Egan with the help of ideas from Mike Stay and Gerard Westendorp. It's probably the best way for a nonmathematician to appreciate the symmetry of Klein's quartic. It's a 3holed torus, but drawn in a way that emphasizes the tetrahedral symmetry lurking in this surface! You can see there are 56 triangles: 2 for each of the tetrahedron's 4 corners, and 8 for each of its 6 edges."
Exercise:
Click on image for further details.
Note that if eight points are arranged
in a cube (like the centers of the
eight subcubes in the figure above),
there are 56 triangles formed by
the 8 points taken 3 at a time.
Baez's discussion says that the Klein quartic's 56 triangles can be partitioned into 7 eighttriangle Egan "cubes" that correspond to the 7 points of the Fano plane in such a way that automorphisms of the Klein quartic correspond to automorphisms of the Fano plane. Show that the 56 triangles within the
eightfold cube can also be partitioned into 7 eighttriangle sets that correspond to the 7 points of the Fano plane in such a way that (affine) transformations of the
eightfold cube induce (projective) automorphisms of the Fano plane.
Comments Off on Friday July 25, 2008
Monday, July 21, 2008
Knight Moves:
The Relativity Theory
of Kindergarten Blocks
(Continued from
January 16, 2008)
Something:
From Friedrich Froebel,
who invented kindergarten:
Click on image for details.
An Unusually
Complicated Theory:
From Christmas 2005:
Click on image for details.
For the eightfold cube
as it relates to Klein’s
simple group, see
“A Reflection Group
of Order 168.”
For an even more
complicated theory of
Klein’s simple group, see
Click on image for details.
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Friday, July 4, 2008
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Friday, June 27, 2008
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Thursday, May 22, 2008
The Undertaking:
An Exercise in
Conceptual Art
Hexagram 54:
THE JUDGMENT
Undertakings bring misfortune.
Nothing that would further.
“Brian O’Doherty, an Irishborn artist,
before the [Tuesday, May 20] wake
of his alter ego* ‘Patrick Ireland’
on the grounds of the
Irish Museum of Modern Art.”
— New York Times, May 22, 2008
THE IMAGE
Thus the superior man
understands the transitory
in the light of
the eternity of the end.
Another version of
the image:
See 2/22/08
and 4/19/08.
Related material:
Michael Kimmelman in today’s New York Times—
“An essay from the ’70s by Mr. O’Doherty, ‘Inside the White Cube,’ became famous in art circles for describing how modern art interacted with the gallery spaces in which it was shown.”
Brian O’Doherty, “Inside the White Cube,” 1976 Artforum essays on the gallery space and 20thcentury art:
“The history of modernism is intimately framed by that space. Or rather the history of modern art can be correlated with changes in that space and in the way we see it. We have now reached a point where we see not the art but the space first…. An image comes to mind of a white, ideal space that, more than any single picture, may be the archetypal image of 20thcentury art.”
An archetypal image
THE SPACE:
A nonarchetypal image
THE ART:
Natasha Wescoat, 2004
See also
Epiphany 2008:
“Nothing that would further.”
— Hexagram 54
Lear’s fool:
…. Now thou art an 0 without a figure. I am better than thou art, now. I am a fool; thou art nothing….

“…. in the last mystery of all the single figure of what is called the World goes joyously dancing in a state beyond moon and sun, and the number of the Trumps is done. Save only for that which has no number and is called the Fool, because mankind finds it folly till it is known. It is sovereign or it is nothing, and if it is nothing then man was born dead.”
— The Greater Trumps,
by Charles Williams, Ch. 14
Comments Off on Thursday May 22, 2008
Saturday, May 10, 2008
MoMA Goes to
Kindergarten
"… the startling thesis of Mr. Brosterman's new book, 'Inventing Kindergarten' (Harry N. Abrams, $39.95): that everything the giants of modern art and architecture knew about abstraction they learned in kindergarten, thanks to building blocks and other educational toys designed by Friedrich Froebel, a German educator, who coined the term 'kindergarten' in the 1830's."
— "Was Modernism Born
in Toddler Toolboxes?"
by Trip Gabriel, New York Times,
April 10, 1997
RELATED MATERIAL
Figure 1 —
Concept from 1819:
(Footnotes 1 and 2)
Figure 2 —
The Third Gift, 1837:
Froebel's Third Gift
Froebel, the inventor of
kindergarten, worked as
an assistant to the
crystallographer Weiss
mentioned in Fig. 1.
(Footnote 3)
Figure 3 —
The Third Gift, 1906:
Figure 4 —
Solomon's Cube,
1981 and 1983:
Figure 5 —
Design Cube, 2006:
For some mathematical background, see
Footnotes:
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Tuesday, April 8, 2008
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Wednesday, January 16, 2008
Knight Moves:
Geometry of the
Eightfold Cube
Click on the image for a larger version
and an expansion of some remarks
quoted here on Christmas 2005.
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Sunday, January 6, 2008
Comments Off on Sunday January 6, 2008
Thursday, July 5, 2007
In Defense of
Plato’s Realism
(vs. sophists’ nominalism–
see recent entries.)
Plato cited geometry,
notably in the Meno,
in defense of his realism.
Consideration of the
Meno’s diamond figure
leads to the following:
Click on image for details.
As noted in an entry,
Plato, Pegasus, and
the Evening Star,
linked to
at the end of today’s
previous entry,
the “universals”
of Platonic realism
are exemplified by
the hexagrams of
the I Ching,
which in turn are
based on the seven
trigrams above and
on the eighth trigram,
of all yin lines,
not shown above:
K’un
The Receptive
_____________________________________________
Update of Nov. 30, 2013:
From a littleknown website in Kuala Lumpur:
(Click to enlarge.)
The remarks on Platonic realism are from Wikipedia.
The eightfold cube is apparently from this post.
Comments Off on Thursday July 5, 2007
Monday, June 25, 2007
Object Lesson
“… the best definition
I have for Satan
is that it is a real
spirit of unreality.”
M. Scott Peck,
People of the Lie
“Far in the woods they sang their unreal songs, Secure. It was difficult to sing in face Of the object. The singers had to avert themselves Or else avert the object.”
— Wallace Stevens, “Credences of Summer”

Today is June 25,
anniversary of the
birth in 1908 of
Willard Van Orman Quine.
Quine died on
Christmas Day, 2000.
Today, Quine’s birthday, is,
as has been noted by
Quine’s son, the point of the
calendar opposite Christmas–
i.e., “AntiChristmas.”
If the AntiChrist is,
as M. Scott Peck claims,
a spirit of unreality, it seems
fitting today to invoke
Quine, a student of reality,
and to borrow the title of
Quine’s Word and Object…
Word:
An excerpt from
“Credences of Summer”
by Wallace Stevens:
“Three times the concentred self takes hold, three times The thrice concentred self, having possessed
The object, grips it in savage scrutiny, Once to make captive, once to subjugate Or yield to subjugation, once to proclaim The meaning of the capture, this hard prize, Fully made, fully apparent, fully found.”
— “Credences of Summer,” VII, by Wallace Stevens, from Transport to Summer (1947) 
Object:
From Friedrich Froebel,
who invented kindergarten:
From Christmas 2005:
Click on the images
for further details.
For a larger and
more sophisticaled
relative of this object,
see yesterday’s entry
At Midsummer Noon.
The object is real,
not as a particular
physical object, but
in the way that a
mathematical object
is real — as a
pure Platonic form.
“It’s all in Plato….”
— C. S. Lewis
Comments Off on Monday June 25, 2007
Saturday, April 7, 2007
Comments Off on Saturday April 7, 2007
Saturday, December 23, 2006
Black Mark
Bernard Holland in The New York Times on Monday, May 20, 1996:
“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….”
Lottery on Friday, Dec. 22, 2006:

Analysis of the structure
of a 2x2x2 cube
via trinities of
projective points
in a Fano plane.
Comments Off on Saturday December 23, 2006
Friday, November 24, 2006
Galois’s Window:
Geometry
from Point
to Hyperspace
by Steven H. Cullinane
Euclid is “the most famous
geometer ever known
and for good reason:
for millennia it has been
his window
that people first look through
when they view geometry.”
— Euclid’s Window:
The Story of Geometry
from Parallel Lines
to Hyperspace,
by Leonard Mlodinow
“…the source of
all great mathematics
is the special case,
the concrete example.
It is frequent in mathematics
that every instance of a
concept of seemingly
great generality is
in essence the same as
a small and concrete
special case.”
— Paul Halmos in
I Want To Be a Mathematician
Euclid’s geometry deals with affine
spaces of 1, 2, and 3 dimensions
definable over the field
of real numbers.
Each of these spaces
has infinitely many points.
Some simpler spaces are those
defined over a finite field–
i.e., a “Galois” field–
for instance, the field
which has only two
elements, 0 and 1, with
addition and multiplication
as follows:
We may picture the smallest
affine spaces over this simplest
field by using square or cubic
cells as “points”:
From these five finite spaces,
we may, in accordance with
Halmos’s advice,
select as “a small and
concrete special case”
the 4point affine plane,
which we may call
Galois’s Window.
The interior lines of the picture
are by no means irrelevant to
the space’s structure, as may be
seen by examining the cases of
the above Galois affine 3space
and Galois affine hyperplane
in greater detail.
For more on these cases, see
The Eightfold Cube,
Finite Relativity,
The Smallest Projective Space,
LatinSquare Geometry, and
Geometry of the 4×4 Square.
(These documents assume that
the reader is familar with the
distinction between affine and
projective geometry.)
These 8 and 16point spaces
may be used to
illustrate the action of Klein’s
simple group of order 168
and the action of
a subgroup of 322,560 elements
within the large Mathieu group.
The view from Galois’s window
also includes aspects of
quantum information theory.
For links to some papers
in this area, see
Elements of Finite Geometry.
Comments Off on Friday November 24, 2006
Sunday, October 8, 2006
Today’s Birthday:
Matt Damon
“The man who lives in contact with what he believes to be a living Church is a man always expecting to meet Plato and Shakespeare tomorrow at breakfast.”
— G. K. Chesterton
See also works by the late Arthur Loeb of Harvard’s Department of Visual and Environmental Studies.
“I don’t want to be a product of my environment. I want my environment to be a product of me.” — Frank Costello in The Departed
For more on the Harvard environment,
see today’s online Crimson:
The Harvard Crimson, Online Edition 
Sunday, Oct. 8, 2006 
POMP AND CIRCUSSTANCE
CRIMSON/ MEGHAN T. PURDY
Friday, Oct. 6:
The Ringling Bros. Barnum & Bailey Circus has come to town, and yesterday the animals were disembarked near MIT and paraded to their temporary home at the Banknorth Garden.

OPINION
At Last, a Guiding Philosophy The General Education report is a strong cornerstone, though further scrutiny is required.
By THE CRIMSON STAFF After four long years, the Curricular Review has finally found its heart.
The Trouble With the Germans The College is a little undereducated these days.
By SAHIL K. MAHTANI Harvard College– in the best formulation I’ve heard– promulgates a Japanesestyle education, where the professoriate pretend to teach, the students pretend to learn, and everyone is happy.

Comments Off on Sunday October 8, 2006
Friday, May 26, 2006
A Living Church
continued from March 27
"The man who lives in contact with what he believes to be a living Church is a man always expecting to meet Plato and Shakespeare tomorrow at breakfast."
— G. K. Chesterton
Comments Off on Friday May 26, 2006
Sunday, December 25, 2005
Eight is a Gate
(continued)
Compare and contrast:
Click on pictures for details.
"… die Schönheit… [ist] die
richtige Übereinstimmung
der Teile miteinander
und mit dem Ganzen."
"Beauty is the proper conformity
of the parts to one another
and to the whole."
— Werner Heisenberg,
"Die Bedeutung des Schönen
in der exakten Naturwissenschaft,"
address delivered to the
Bavarian Academy of Fine Arts,
Munich, 9 Oct. 1970, reprinted in
Heisenberg's Across the Frontiers,
translated by Peter Heath,
Harper & Row, 1974
Comments Off on Sunday December 25, 2005
Tuesday, August 2, 2005
Today's birthday:
Peter O'Toole
"What is it, Major Lawrence,
that attracts you personally
to the desert?"
"It's clean."
Visible Mathematics,
continued —
From May 18:
Lindbergh's Eden
"The Garden of Eden is behind us
and there is no road
back to innocence;
we can only go forward."
— Anne Morrow Lindbergh,
Earth Shine, p. xii
"Beauty is the proper conformity
of the parts to one another
and to the whole."
— Werner Heisenberg,
"Die Bedeutung des Schönen
in der exakten Naturwissenschaft,"
address delivered to the
Bavarian Academy of Fine Arts,
Munich, 9 Oct. 1970, reprinted in
Heisenberg's Across the Frontiers,
translated by Peter Heath,
Harper & Row, 1974
Related material:
The Eightfold Cube
(in Arabic, ka'b)
and
Comments Off on Tuesday August 2, 2005
Tuesday, June 7, 2005
“A SINGLE VERSE by Rimbaud,”
writes Dominique de Villepin,
the new French Prime Minister,
“shines like a powder trail
on a day’s horizon.
It sets it ablaze all at once,
explodes all limits,
draws the eyes
to other heavens.”
— Ben Macintyre,
The London Times, June 4:
When Rimbaud Meets Rambo
“Room 101 was the place where
your worst fears were realised
in George Orwell’s classic
Nineteen EightyFour.
[101 was also]
Professor Nash’s office number
in the movie ‘A Beautiful Mind.'”
— Prime Curios
Classics Illustrated —
Comments Off on Tuesday June 7, 2005
Wednesday, May 18, 2005
“Beauty is the proper conformity
of the parts to one another
and to the whole.”
— Werner Heisenberg,
“
Die Bedeutung des Schönen in der exakten Naturwissenschaft,”
address delivered to the
Bavarian Academy of Fine Arts,
Munich, 9 Oct. 1970, reprinted in
Heisenberg’s
Across the Frontiers,
translated by Peter Heath,
Harper & Row, 1974
Related material:
The Eightfold Cube
Comments Off on Wednesday May 18, 2005
Wednesday, May 4, 2005
The Fano Plane
Revisualized:
or, The Eightfold Cube
Here is the usual model of the seven points and seven lines (including the circle) of the smallest finite projective plane (the
Fano plane):
Every permutation of the plane's points that preserves collinearity is a symmetry of the plane. The group of symmetries of the Fano plane is of order 168 and is isomorphic to the group PSL(2,7) = PSL(3,2) = GL(3,2). (See Cameron on linear groups (pdf).)
The above model indicates with great clarity six symmetries of the plane– those it shares with the equilateral triangle. It does not, however, indicate where the other 162 symmetries come from.
Shown below is a new model of this same projective plane, using partitions of cubes to represent points:
The cubes' partitioning planes are added in binary (
1+1=0) fashion. Three partitioned cubes are collinear if and only if their partitioning planes' binary sum equals zero.
The second model is useful because it lets us generate naturally all 168 symmetries of the Fano plane by splitting a cube into a set of four parallel 1x1x2 slices in the three ways possible, then arbitrarily permuting the slices in each of the three sets of four. See examples below.
For a proof that such permutations generate the 168 symmetries, see
Binary Coordinate Systems.
(Note that this procedure, if regarded as acting on the set of eight individual subcubes of each cube in the diagram, actually generates a group of 168*8 = 1,344 permutations. But the group's action on the diagram's seven partitions of the subcubes yields only 168 distinct results. This illustrates the difference between affine and projective spaces over the binary field GF(2). In a related 2x2x2 cubic model of the affine 3space over GF(2) whose "points" are individual subcubes, the group of eight translations is generated by interchanges of parallel 2x2x1 cubeslices. This is clearly a subgroup of the group generated by permuting 1x1x2 cubeslices. Such translations in the affine 3space have no effect on the projective plane, since they leave each of the plane model's seven partitions– the "points" of the plane– invariant.)
To view the cubes model in a wider context, see Galois Geometry, Block Designs, and FiniteGeometry Models.
Comments Off on Wednesday May 4, 2005
Saturday, July 20, 2002
ABSTRACT: Finite projective geometry explains the surprising symmetry properties of some simple graphic designs– found, for instance, in quilts. Links are provided for applications to sporadic simple groups (via the "Miracle Octad Generator" of R. T. Curtis), to the connection between orthogonal Latin squares and projective spreads, and to symmetry of Walsh functions.

We regard the fourdiamond figure D above as a 4×4 array of twocolor diagonallydivided 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 Gimage of D (as at right, below) has some ordinary or colorinterchange 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 4space over GF(2) and that the 35 structures of the 840 = 35 x 24 Gimages of D are isomorphic to the 35 lines in the 3dimensional projective space over GF(2).
This can be seen by viewing the 35 structures as threesets of line diagrams, based on the three partitions of the fourset of square twocolor tiles into two twosets, and indicating the locations of these twosets 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 threeset of line diagrams sums to zero– i.e., each diagram in a threeset is the binary sum of the other two diagrams in the set. Thus, the 35 threesets of line diagrams correspond to the 35 threepoint lines of the finite projective 3space 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 3space PG(3,2).


The symmetry of the line diagrams accounts for the symmetry of the twocolor patterns. (A proof shows that a 2nx2n twocolor triangular halfsquares 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 Latinsquare 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 quartercentury 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 Gimages 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 79TA37, "Symmetry invariance in a diamond ring," by Steven H. Cullinane, Notices of the American Mathematical Society, February 1979, pages A193, 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
LatinSquare 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)

This work is licensed under a
Creative Commons AttributionNonCommercialNoDerivs 2.5 License.
Page created Jan. 6, 2006, by Steven H. Cullinane diamondtheorem.com



Initial Xanga entry. Updated Nov. 18, 2006.
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