Eightfold Cube

Saturday, November 24, 2012

Will and Representation*

Filed under: General,Geometry — Tags: Eightfold Cube — m759 @ 2:56 pm

Robert A. Wilson, in an inaugural lecture in April 2008—

Representation theory

A group always arises in nature as the symmetry group of some object, and group
theory in large part consists of studying in detail the symmetry group of some
object, in order to throw light on the structure of the object itself (which in some
sense is the “real” object of study).

But if you look carefully at how groups are used in other areas such as physics
and chemistry, you will see that the real power of the method comes from turning
the whole procedure round: instead of starting from an object and abstracting
its group of symmetries, we start from a group and ask for all possible objects
that it can be the symmetry group of .

This is essentially what we call Representation theory . We think of it as taking a
group, and representing it concretely in terms of a symmetrical object.

Now imagine what you can do if you combine the two processes: we start with a
symmetrical object, and find its group of symmetries. We now look this group up
in a work of reference, such as our big red book (The ATLAS of Finite Groups),
and find out about all (well, perhaps not all) other objects that have the same
group as their group of symmetries.

We now have lots of objects all looking completely different, but all with the same
symmetry group. By translating from the first object to the group, and then to
the second object, we can use everything we know about the first object to tell
us things about the second, and vice versa.

As Poincaré said,

Mathematicians do not study objects, but relations between objects.
Thus they are free to replace some objects by others, so long as the
relations remain unchanged.

Par exemple—

Fano plane transformed to eightfold cube,
and partitions of the latter as points of the former:

* For the "Will" part, see the PyrE link at Talk Amongst Yourselves.

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Wednesday, November 14, 2012

Group Actions

Filed under: General,Geometry — Tags: Cube Quaternions, Cube School, Eightfold Cube, Time in Space — m759 @ 4:30 pm

The December 2012 Notices of the American
Mathematical Society has an ad on page 1564
(in a review of two books on vulgarized mathematics)
for three workshops next year on “Low-dimensional
Topology, Geometry, and Dynamics”—

(Only the top part of the ad is shown; for further details
see an ICERM page.)

(ICERM stands for Institute for Computational
and Experimental Research in Mathematics.)

The ICERM logo displays seven subcubes of
a 2x2x2 eight-cube array with one cube missing—

The logo, apparently a stylized image of the architecture
of the Providence building housing ICERM, is not unlike
a picture of Froebel’s Third Gift—

Froebel's third gift, the eightfold cube

Photo by Norman Brosterman from the Inventing Kindergarten
exhibit at The Institute for Figuring (co-founded by Margaret Wertheim)

The eighth cube, missing in the ICERM logo and detached in the
Froebel Cubes photo, may be regarded as representing the origin
(0,0,0) in a coordinatized version of the 2x2x2 array—
in other words the cube invariant under linear , as opposed to
more general affine , permutations of the cubes in the array.

These cubes are not without relevance to the workshops’ topics—
low-dimensional exotic geometric structures, group theory, and dynamics.

See The Eightfold Cube, A Simple Reflection Group of Order 168, and
The Quaternion Group Acting on an Eightfold Cube.

Those who insist on vulgarizing their mathematics may regard linear
and affine group actions on the eight cubes as the dance of
Snow White (representing (0,0,0)) and the Seven Dwarfs—

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Tuesday, June 26, 2012

Looking Deeply

Filed under: General,Geometry — Tags: Eightfold Cube, Looking Deeply, Triskele — m759 @ 3:48 pm

Last night's post on The Trinity of Max Black and the use of
the term "eightfold" by the Mathematical Sciences Research Institute
at Berkeley suggest a review of an image from Sept. 22, 2011—

The triskele detail above echoes a Buddhist symbol found,
for instance, on the Internet in an ad for meditation supplies—

Related remarks—

http://www.spencerart.ku.edu/about/dialogue/fdpt.shtml—

Mary Dusenbury (Radcliffe '64)—

"… I think a textile, like any work of art, holds a tremendous amount of information— technical, material, historical, social, philosophical— but beyond that, many works of art are very beautiful and they speak to us on many layers— our intellect, our heart, our emotions. I've been going to museums since I was a very small child, thinking about what I saw, and going back to discover new things, to see pieces that spoke very deeply to me, to look at them again, and to find more and more meaning relevant to me in different ways and at different times of my life. …

… I think I would suggest to people that first of all they just look. Linger by pieces they find intriguing and beautiful, and look deeply. Then, if something interests them, we have tried to put a little information around the galleries to give a bit of history, a bit of context, for each piece. But the most important is just to look very deeply."

http://en.wikipedia.org/wiki/Nikaya_Buddhism—

According to Robert Thurman, the term "Nikāya Buddhism" was coined by Professor Masatoshi Nagatomi of Harvard University, as a way to avoid the usage of the term Hinayana.^[12] "Nikaya Buddhism" is thus an attempt to find a more neutral way of referring to Buddhists who follow one of the early Buddhist schools, and their practice.

12. The Emptiness That is Compassion:
An Essay on Buddhist Ethics, Robert A. F. Thurman, 1980
[Religious Traditions , Vol. 4 No. 2, Oct.-Nov. 1981, pp. 11-34]

http://dsal.uchicago.edu/cgi-bin/philologic/getobject.pl?c.2:1:6.pali—

Nikāya [Sk. nikāya, ni+kāya]
collection ("body") assemblage, class, group

http://en.wiktionary.org/wiki/नि—

Sanskrit etymology for नि (ni)

1 From Proto-Indo-European *ni …

Prefix

नि (ni)

down
back
in, into

http://www.rigpawiki.org/index.php?title=Kaya—

Kaya (Skt. kāya ; སྐུ་, Tib. ku ; Wyl. sku ) —
the Sanskrit word kaya literally means ‘body’
but can also signify dimension, field or basis.

སྐུ། (Wyl. sku ) n. Pron.: ku

• structure, existentiality, founding stratum ▷HVG KBEU

• gestalt ▷HVG LD

Note that The Trinity of Max Black is a picture of a set—
i.e., of an "assemblage, class, group."

Note also the reference above to the word "gestalt."

"Was ist Raum, wie können wir ihn
erfassen und gestalten?"

— Walter Gropius

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Saturday, June 16, 2012

Chiral Problem

Filed under: General,Geometry — Tags: Cube Quaternions, Cube School, Eightfold Cube, Wheeler — m759 @ 1:06 am

In memory of William S. Knowles, chiral chemist, who died last Wednesday (June 13, 2012)—

Detail from the Harvard Divinity School 1910 bookplate in yesterday morning's post—

"ANDOVER–HARVARD THEOLOGICAL LIBRARY"

Detail from Knowles's obituary in this morning's New York Times—

William Standish Knowles was born in Taunton, Mass., on June 1, 1917. He graduated a year early from the Berkshire School, a boarding school in western Massachusetts, and was admitted to Harvard. But after being strongly advised that he was not socially mature enough for college, he did a second senior year of high school at another boarding school, Phillips Academy in Andover, N.H.

Dr. Knowles graduated from Harvard with a bachelor’s degree in chemistry in 1939….

"This is the relativity problem: to fix objectively a class of equivalent coordinatizations and to ascertain the group of transformations S mediating between them."

— Hermann Weyl, The Classical Groups, Princeton University Press, 1946, p. 16

From Pilate Goes to Kindergarten—

The six congruent quaternion actions illustrated above are based on the following coordinatization of the eightfold cube—

Problem: Is there a different coordinatization
that yields greater symmetry in the pictures of
quaternion group actions?

A paper written in a somewhat similar spirit—

"Chiral Tetrahedrons as Unitary Quaternions"—

ABSTRACT: Chiral tetrahedral molecules can be dealt [with] under the standard of quaternionic algebra. Specifically, non-commutativity of quaternions is a feature directly related to the chirality of molecules….

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Sunday, June 3, 2012

Child’s Play

Filed under: General,Geometry — Tags: Cube Descending, Cube School, Eightfold Cube, Geometric Theology — m759 @ 2:56 pm

(Continued)

“A set having three members is a single thing
wholly constituted by its members but distinct from them.
After this, the theological doctrine of the Trinity as
‘three in one’ should be child’s play.”

– Max Black, Caveats and Critiques: Philosophical Essays
in Language, Logic, and Art , Cornell U. Press, 1975

IMAGE- The Trinity of Max Black (a 3-set, with its eight subsets arranged in a Hasse diagram that is also a cube)

Related material—

The Trinity Cube

Comments (2)

Saturday, May 19, 2012

G8

Filed under: General,Geometry — Tags: Cube Quaternions, Cube School, Eightfold Cube — m759 @ 8:00 pm

"The group of 8" is a phrase from politics, not mathematics.
Of the five groups of order 8 (see today's noon post),

the one pictured* in the center, Z₂ × Z₂ × Z₂ , is of particular
interest. See The Eightfold Cube. For a connection of this
group of 8 to the last of the five pictured at noon, the
quaternion group, see Finite Geometry and Physical Space.

* The picture is of the group's cycle graph.

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Monday, May 7, 2012

More on Triality

Filed under: General,Geometry — Tags: Cube Quaternions, Cube School, Eightfold Cube — m759 @ 4:20 pm

John Baez wrote in 1996 ("Week 91") that

"I've never quite seen anyone come right out
and admit that triality arises from the
permutations of the unit vectors i, j, and k
in 3d Euclidean space."

Baez seems to come close to doing this with a
somewhat different i , j , and k — Hurwitz
quaternions— in his 2005 book review
quoted here yesterday.

See also the Log24 post of Jan. 4 on quaternions,
and the following figures. The actions on cubes
in the lower figure may be viewed as illustrating
(rather indirectly) the relationship of the quaternion
group's 24 automorphisms to the 24 rotational
symmetries of the cube.

IMAGE- Actions of the unit quaternions in finite geometry, on a ninefold square and on an eightfold cube

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Wednesday, April 18, 2012

Adam in Eden

Filed under: General,Geometry — Tags: Cullinane Cube, Eightfold Cube, on0418, Solomon's Cube — 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—

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.

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Monday, April 9, 2012

Eightfold Cube Revisited

Filed under: General,Geometry — Tags: Eightfold Cube — m759 @ 12:00 pm

A search today (Élie Cartan's birthday) for material related to triality*

yielded references to something that has been called a Bhargava cube .

Two pages from a 2006 paper by Bhargava—

Bhargava's reference [4] above for "the story of the cube" is to…

Higher Composition Laws I:
A New View on Gauss Composition,
and Quadratic Generalizations

Manjul Bhargava

The Annals of Mathematics
Second Series, Vol. 159, No. 1 (Jan., 2004), pp. 217-250
Published by: Annals of Mathematics
Article Stable URL: http://www.jstor.org/stable/3597249

A brief account in the context of embedding problems (click to enlarge)—

For more ways of slicing a cube,
see The Eightfold Cube —

* Note (1) some remarks by Tony Smith
related to the above Dynkin diagram
and (2) another colorful variation on the diagram.

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Thursday, March 1, 2012

Block That Metaphor:

Filed under: General,Geometry — Tags: Eightfold Cube, Gopnik, The Building-Block Metaphor — m759 @ 11:09 pm

The Cube Model and Peano Arithmetic

The eightfold cube model of the Fano plane may or may not have influenced a new paper (with the date Feb. 10, 2011, in its URL) on an attempted consistency proof of Peano arithmetic—

The Consistency of Arithmetic, by Storrs McCall

"Is Peano arithmetic (PA) consistent? This paper contains a proof that it is. …

Axiomatic proofs we may categorize as 'syntactic', meaning that they concern only symbols and the derivation of one string of symbols from another, according to set rules. 'Semantic' proofs, on the other hand, differ from syntactic proofs in being based not only on symbols but on a non-symbolic, non-linguistic component, a domain of objects. If the sole paradigm of 'proof ' in mathematics is 'axiomatic proof ', in which to prove a formula means to deduce it from axioms using specified rules of inference, then Gödel indeed appears to have had the last word on the question of PA-consistency. But in addition to axiomatic proofs there is another kind of proof. In this paper I give a proof of PA's consistency based on a formal semantics for PA. To my knowledge, no semantic consistency proof of Peano arithmetic has yet been constructed.

The difference between 'semantic' and 'syntactic' theories is described by van Fraassen in his book The Scientific Image :

"The syntactic picture of a theory identifies it with a body of theorems, stated in one particular language chosen for the expression of that theory. This should be contrasted with the alternative of presenting a theory in the first instance by identifying a class of structures as its models. In this second, semantic, approach the language used to express the theory is neither basic nor unique; the same class of structures could well be described in radically different ways, each with its own limitations. The models occupy centre stage." (1980, p. 44)

Van Fraassen gives the example on p. 42 of a consistency proof in formal geometry that is based on a non-linguistic model. Suppose we wish to prove the consistency of the following geometric axioms:

A1. For any two lines, there is at most one point that lies on both.
A2. For any two points, there is exactly one line that lies on both.
A3. On every line there lie at least two points.

The following diagram shows the axioms to be consistent:

Figure 1

The consistency proof is not a 'syntactic' one, in which the consistency of A1-A3 is derived as a theorem of a deductive system, but is based on a non-linguistic structure. It is a semantic as opposed to a syntactic proof. The proof constructed in this paper, like van Fraassen's, is based on a non-linguistic component, not a diagram in this case but a physical domain of three-dimensional cube-shaped blocks. ….

… The semantics presented in this paper I call 'block semantics', for reasons that will become clear…. Block semantics is based on domains consisting of cube-shaped objects of the same size, e.g. children's wooden building blocks. These can be arranged either in a linear array or in a rectangular array, i.e. either in a row with no space between the blocks, or in a rectangle composed of rows and columns. A linear array can consist of a single block, and the order of individual blocks in a linear or rectangular array is irrelevant. Given three blocks A, B and C, the linear arrays ABC and BCA are indistinguishable. Two linear arrays can be joined together or concatenated into a single linear array, and a rectangle can be re-arranged or transformed into a linear array by successive concatenation of its rows. The result is called the 'linear transformation' of the rectangle. An essential characteristic of block semantics is that every domain of every block model is finite. In this respect it differs from Tarski’s semantics for first-order logic, which permits infinite domains. But although every block model is finite, there is no upper limit to the number of such models, nor to the size of their domains.

It should be emphasized that block models are physical models, the elements of which can be physically manipulated. Their manipulation differs in obvious and fundamental ways from the manipulation of symbols in formal axiomatic systems and in mathematics. For example the transformations described above, in which two linear arrays are joined together to form one array, or a rectangle of blocks is re-assembled into a linear array, are physical transformations not symbolic transformations. …"

— Storrs McCall, Department of Philosophy, McGill University

Tuesday, January 3, 2012

Theorum

Filed under: General,Geometry — Tags: Cube Quaternions, Cube School, Eightfold Cube — m759 @ 7:48 am

In memory of artist Ronald Searle—

Searle reportedly died at 91 on December 30th.

From Log24 on that date—

Click the above image for some context.

Update of 9:29 PM EST Jan. 3, 2012—

Theorum

From RationalWiki

Theorum (rhymes with decorum, apparently) is a neologism proposed by Richard Dawkins in The Greatest Show on Earth to distinguish the scientific meaning of theory from the colloquial meaning. In most of the opening introduction to the show, he substitutes "theorum" for "theory" when referring to the major scientific theories such as evolution.

Problems with "theory"

Dawkins notes two general meanings for theory; the scientific one and the general sense that means a wild conjecture made up by someone as an explanation. The point of Dawkins inventing a new word is to get around the fact that the lay audience may not thoroughly understand what scientists mean when they say "theory of evolution". As many people see the phrase "I have a theory" as practically synonymous with "I have a wild guess I pulled out of my backside", there is often confusion about how thoroughly understood certain scientific ideas are. Hence the well known creationist argument that evolution is "just a theory" – and the often cited response of "but gravity is also just a theory".

To convey the special sense of thoroughness implied by the word theory in science, Dawkins borrowed the mathematical word "theorem". This is used to describe a well understood mathematical concept, for instance Pythagoras' Theorem regarding right angled triangles. However, Dawkins also wanted to avoid the absolute meaning of proof associated with that word, as used and understood by mathematicians. So he came up with something that looks like a spelling error. This would remove any person's emotional attachment or preconceptions of what the word "theory" means if it cropped up in the text of The Greatest Show on Earth , and so people would (in "theory ") have no other choice but to associate it with only the definition Dawkins gives.

This phrase has completely failed to catch on, that is, if Dawkins intended it to catch on rather than just be a device for use in The Greatest Show on Earth . When googled, Google will automatically correct the spelling to theorem instead, depriving this very page its rightful spot at the top of the results.

Friday, December 30, 2011

Quaternions on a Cube

Filed under: General,Geometry — Tags: Cube Quaternions, Cube School, Eightfold Cube, Finite Geometry, Quantum Suffering — m759 @ 5:48 am

The following picture provides a new visual approach to
the order-8 quaternion group's automorphisms.

Click the above image for some context.

Here the cube is called "eightfold" because the eight vertices,
like the eight subcubes of a 2×2×2 cube,* are thought of as
independently movable. See The Eightfold Cube.

Wednesday, September 21, 2011

Symmetric Generation

Filed under: General,Geometry — Tags: Aitchison, Crucible Raiders, Eightfold Cube, Generation, Geometry, Symmetric Generation — m759 @ 2:00 pm

Suggested by yesterday's Relativity Problem Revisited and by Cassirer on Objectivity—

From Symmetric Generation of Groups , by R.T. Curtis (Cambridge U. Press, 2007)—

"… we are saying much more than that G ≅ M₂₄ is generated by
some set of seven involutions, which would be a very weak
requirement. We are asserting that M₂₄ is generated by a set
of seven involutions which possesses all the symmetries of L₃(2)
acting on the points of the 7-point projective plane…."
— Symmetric Generation , p. 41

"It turns out that this approach is particularly revealing and that
many simple groups, both sporadic and classical, have surprisingly
simple definitions of this type."
— Symmetric Generation , p. 42

Sunday, September 18, 2011

Anatomy of a Cube

Filed under: General,Geometry — Tags: Cube School, Eightfold Cube — m759 @ 12:00 pm

R.D. Carmichael’s seminal 1931 paper on tactical configurations suggests
a search for later material relating such configurations to block designs.
Such a search yields the following—

“… it seems that the relationship between
BIB [balanced incomplete block ] designs
and tactical configurations, and in particular,
the Steiner system, has been overlooked.”
— D. A. Sprott, U. of Toronto, 1955

The figure by Cullinane included above shows a way to visualize Sprott’s remarks.

For the group actions described by Cullinane, see “The Eightfold Cube” and
“A Simple Reflection Group of Order 168.”

Update of 7:42 PM Sept. 18, 2011—

From a Summer 2011 course on discrete structures at a Berlin website—

A different illustration of the eightfold cube as the Steiner system S(3, 4, 8)—

Note that only the static structure is described by Felsner, not the
168 group actions discussed (as above) by Cullinane. For remarks on
such group actions in the literature, see “Cube Space, 1984-2003.”

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Sunday, August 28, 2011

The Cosmic Part

Filed under: General,Geometry — Tags: Eightfold Cube, Geometric Theology, Rubik Adoration, Sacred Space — 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.

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Monday, July 11, 2011

Accentuate the Positive

Filed under: General,Geometry — Tags: Eightfold Cube, Geometric Theology, The Positive — m759 @ 2:02 pm

An image that may be viewed as
a cube with a “+“ on each face—

The eightfold cube

Underlying structure

For the Pope and others on St. Benedict’s Day
who prefer narrative to mathematics—

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Saturday, June 25, 2011

The Fano Entity

Filed under: General,Geometry — Tags: Eightfold Cube — m759 @ 2:02 am

The New York Times at 9 PM ET June 23, 2011—

ROBERT FANO: I’m trying to think briefly how to put it.

GINO FANO: "On the Fundamental Postulates"—

"E la prova di questo si ha precisamente nel fatto che si è potuto costruire (o, dirò meglio immaginare) un ente per cui sono verificati tutti i postulati precedenti…."

"The proof of this is precisely the fact that you could build (or, to say it better, imagine) an entity by which are verified all previous assumptions…."

Also from the Times article quoted above…

"… like working on a cathedral. We laid our bricks and knew that others might later replace them with better bricks. We believed in the cause even if we didn’t completely understand the implications.”

— Tom Van Vleck

Some art that is related, if only by a shared metaphor, to Van Vleck's cathedral—

The art is also related to the mathematics of Gino Fano.

For an explanation of this relationship (implicit in the above note from 1984),
see "The Fano plane revisualized—or: the eIghtfold cube."

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Thursday, May 26, 2011

Prime Cubes

Filed under: General,Geometry — Tags: Eightfold Cube — m759 @ 11:09 am

The title refers not to numbers of the form p³, p prime, but to geometric cubes with p ³ subcubes.

Such cubes are natural models for the finite vector spaces acted upon by general linear groups viewed as permutation groups of degree (not order ) p³.

For the case p =2, see The Eightfold Cube.

For the case p =3, see the "External links" section of the Nov. 30, 2009, version of Wikipedia article "General Linear Group." (That is the version just prior to the Dec. 14, 2009, revision by anonymous user "Greenfernglade.")

For symmetries of group actions for larger primes, see the related 1985 remark* on two -dimensional linear groups—

"Actions of GL(2,p ) on a p ×p coordinate-array
have the same sorts of symmetries,
where p is any odd prime."

* Group Actions, 1984-2009

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Wednesday, January 19, 2011

Intermediate Cubism

Filed under: General,Geometry — Tags: Cézanne's Greetings, Eightfold Cube — 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.

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Friday, December 17, 2010

Fare Thee Well

Filed under: General,Geometry — Tags: Eightfold Cube, Jack in the Box — m759 @ 4:01 am

Excerpt from a post of 8 AM May 26, 2006 —

A Living Church
continued from March 27, 2006

"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 to-morrow at breakfast."

– G. K. Chesterton

Platonic Solid

The image “http://www.log24.com/log/pix06A/060526-JackInTheBox.jpg” cannot be displayed, because it contains errors.

Shakespearean Fool
© 2004 Natasha Wescoat

A related scene from the opening of Blake Edwards's "S.O.B." —

Click for Julie Andrews in the full video.

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Sunday, October 3, 2010

Search for the Basic Picture

Filed under: General,Geometry — Tags: Eightfold Cube, Sacerdotal Jargon — m759 @ 5:01 pm

(Click to enlarge.)

The above is the result of a (fruitless) image search today for a current version of Giovanni Sambin's "Basic Picture: A Structure for Topology."

That search was suggested by the title of today's New York Times op-ed essay "Found in Translation" and an occurrence of that phrase in this journal on January 5, 2007.

Further information on one of the images above—

A search in this journal on the publication date of Giaquinto's Visual Thinking in Mathematics yields the following—

Thursday July 5, 2007

m759 @ 7:11 PM

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:

For the Meno 's diamond figure in Giaquinto, see a review—

— Review by Jeremy Avigad (preprint)

Finite geometry supplies a rather different context for Plato's "basic picture."

In that context, the Klein four-group often cited by art theorist Rosalind Krauss appears as a group of translations in the mathematical sense. (See Kernel of Eternity and Sacerdotal Jargon at Harvard.)

The Times op-ed essay today notes that linguistic translation "… is not merely a job assigned to a translator expert in a foreign language, but a long, complex and even profound series of transformations that involve the writer and reader as well."

The list of four-group transformations in the mathematical sense is neither long nor complex, but is apparently profound enough to enjoy the close attention of thinkers like Krauss.

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Monday, June 21, 2010

Cube Spaces

Filed under: General,Geometry — Tags: Appropriation, Change in China, Cube School, Eightfold Cube, Groups and Spaces, Interplay — m759 @ 11:30 am

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

Example 1— The 2×2×2 Cube—

also known as the eightfold cube—

2x2x2 cube

Group actions on the eightfold cube, 1984—

Version by Laszlo Lovasz et al., 2003—

Lovasz et al. go on to describe the same group actions
as in the 1984 note, without attribution.

Example 2— The 3×3×3 Cube

A note from 1985 describing group actions on a 3×3 plane array—

Undated software by Ed Pegg Jr. displays
group actions on a 3×3×3 cube that extend the
3×3 group actions from 1985 described above—

Pegg gives no reference to the 1985 work on group actions.

Example 3— The 4×4×4 Cube

A note from 27 years ago today—

As far as I know, this version of the
group-actions theorem has not yet been ripped off.

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Sunday, April 4, 2010

URBI ET ORBI

Filed under: General,Geometry — Tags: Eightfold Cube — m759 @ 11:11 am

URBI
(Toronto)–

Click on image for some background.

ORBI
(Globe and Mail)–

See also Baaad Blake and
Fearful Symmetry.

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Tuesday, March 30, 2010

Eightfold Symmetries

Filed under: General,Geometry — Tags: Eightfold Cube, Geometric Theology, Graphic Resonance, Triskele — m759 @ 9:48 pm

Harvard Crimson headline today–
“Deconstructing Design“

Reconstructing Design

The phrase “eightfold way” in today’s
previous entry has a certain
graphic resonance…

For instance, an illustration from the
Wikipedia article “Noble Eightfold Path” —

Adapted detail–

Adapted Dharma Wheel detail

Sunday, March 21, 2010

Galois Field of Dreams

Filed under: General,Geometry — Tags: Eightfold Cube, Springer — m759 @ 10:01 am

It is well known that the seven (2² + 2 +1) points of the projective plane of order 2 correspond to 2-point subspaces (lines) of the linear 3-space over the two-element field Galois field GF(2), and may be therefore be visualized as 2-cube subsets of the 2×2×2 cube.

Similarly, recent posts* have noted that the thirteen (3² + 3 + 1) points of the projective plane of order 3 may be seen as 3-cube subsets in the 3×3×3 cube.

The twenty-one (4² + 4 +1) points of the (unique) projective plane of order 4 may also be visualized as subsets of a cube– in this case, the 4×4×4 cube. This visualization is somewhat more complicated than the 3×3×3 case, since the 4×4×4 cube has no central subcube, and each projective-plane point corresponds to four, not three, subcubes.

These three cubes, with 8, 27, and 64 subcubes, thus serve as geometric models in a straightforward way– first as models of finite linear spaces, hence as models for small Galois geometries derived from the linear spaces. (The cubes with 8 and 64 subcubes also serve in a less straightforward, and new, way as finite-geometry models– see The Eightfold Cube, Block Designs, and Solomon's Cube.)

A group of collineations** of the 21-point plane is one of two nonisomorphic simple groups of order 20,160. The other is the linear group acting on the linear 4-space over the two-element Galois field GF(2). The 1899 paper establishing the nonisomorphism notes that "the expression Galois Field is perhaps not yet in general use."

Coordinates of the 4×4×4 cube's subcubes can, of course, be regarded as elements of the Galois field GF(64).

The preceding remarks were purely mathematical. The "dreams" of this post's title are not. See…

See also Geometry of the I Ching and a search in this journal for "Galois + Ching."

* February 27 and March 13

** G₂₀₁₆₀ in Mitchell 1910, LF(3,2²) in Edge 1965

— Mitchell, Ulysses Grant, "Geometry and Collineation Groups
of the Finite Projective Plane PG(2,2²),"
Princeton Ph.D. dissertation (1910)

— Edge, W. L., "Some Implications of the Geometry of
the 21-Point Plane," Math. Zeitschr. 87, 348-362 (1965)

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Monday, March 1, 2010

Visual Group Theory

Filed under: General,Geometry — Tags: Eightfold Cube — m759 @ 12:00 pm

The current article on group theory at Wikipedia has a Rubik's Cube as its logo–

Wikipedia article 'Group theory' with Rubik Cube and quote from Nathan Carter-- 'What is symmetry?'

The article quotes Nathan C. Carter on the question "What is symmetry?"

This naturally suggests the question "Who is Nathan C. Carter?"

A search for the answer yields the following set of images…

Click image for some historical background.

Carter turns out to be a mathematics professor at Bentley University. His logo– an eightfold-cube labeling (in the guise of a Cayley graph)– is in much better taste than Wikipedia's.

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Saturday, February 27, 2010

Cubist Geometries

Filed under: General,Geometry — Tags: Eightfold Cube, Interplay, Line at Infinity — m759 @ 2:01 pm

"The cube has…13 axes of symmetry:
6 C₂ (axes joining midpoints of opposite edges),
4 C₃ (space diagonals), and
3C₄ (axes joining opposite face centroids)."
–Wolfram MathWorld article on the cube

These 13 symmetry axes can be used to illustrate the interplay between Euclidean and Galois geometry in a cubic model of the 13-point Galois plane.

The geometer's 3×3×3 cube–
27 separate subcubes unconnected
by any Rubik-like mechanism–

The 13 symmetry axes of the (Euclidean) cube–
exactly one axis for each pair of opposite
subcubes in the (Galois) 3×3×3 cube–

A closely related structure–
the finite projective plane
with 13 points and 13 lines–

A later version of the 13-point plane
by Ed Pegg Jr.–

A group action on the 3×3×3 cube
as illustrated by a Wolfram program
by Ed Pegg Jr. (undated, but closely
related to a March 26, 1985 note
by Steven H. Cullinane)–

The above images tell a story of sorts.
The moral of the story–

Galois projective geometries can be viewed
in the context of the larger affine geometries
from which they are derived.

The standard definition of points in a Galois projective plane is that they are lines through the (arbitrarily chosen) origin in a corresponding affine 3-space converted to a vector 3-space.

If we choose the origin as the center cube in coordinatizing the 3×3×3 cube (See Weyl's relativity problem ), then the cube's 13 axes of symmetry can, if the other 26 cubes have properly (Weyl's "objectively") chosen coordinates, illustrate nicely the 13 projective points derived from the 27 affine points in the cube model.

The 13 lines of the resulting Galois projective plane may be derived from Euclidean planes through the cube's center point that are perpendicular to the cube's 13 Euclidean symmetry axes.

The above standard definition of points in a Galois projective plane may of course also be used in a simpler structure– the eightfold cube.

(The eightfold cube also allows a less standard way to picture projective points that is related to the symmetries of "diamond" patterns formed by group actions on graphic designs.)

See also Ed Pegg Jr. on finite geometry on May 30, 2006
at the Mathematical Association of America.

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Sunday, February 21, 2010

Reflections

Filed under: General,Geometry — Tags: Eightfold Cube, Fields, Springer — m759 @ 12:06 pm

From the Wikipedia article "Reflection Group" that I created on Aug. 10, 2005— as revised on Nov. 25, 2009—

Historically, (Coxeter 1934) proved that every reflection group [Euclidean, by the current Wikipedia definition] is a Coxeter group (i.e., has a presentation where all relations are of the form r_i² or $(r i r j) k$ ), and indeed this paper introduced the notion of a Coxeter group, while (Coxeter 1935) proved that every finite Coxeter group had a representation as a reflection group [again, Euclidean], and classified finite Coxeter groups.

Finite fields

This section requires expansion.

When working over finite fields, one defines a "reflection" as a map that fixes a hyperplane (otherwise for example there would be no reflections in characteristic 2, as $-1=1$ so reflections are the identity). Geometrically, this amounts to including shears in a hyperplane. Reflection groups over finite fields of characteristic not 2 were classified in (Zalesskiĭ & Serežkin 1981).

Related material:

"A Simple Reflection Group of Order 168," by Steven H. Cullinane, and

"Determination of the Finite Primitive Reflection Groups over an Arbitrary Field of Characteristic Not 2,"

by Ascher Wagner, U. of Birmingham, received 27 July 1977

Journal	Geometriae Dedicata
Publisher	Springer Netherlands
Issue	Volume 9, Number 2 / June, 1980

Ascher Wagner's 1977 dismissal of reflection groups over fields of characteristic 2

[A primitive permuation group preserves
no nontrivial partition of the set it acts upon.]

Clearly the eightfold cube is a counterexample.

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Friday, February 19, 2010

Mimzy vs. Mimsy

Filed under: General,Geometry — Tags: Cube School, Eightfold Cube, Strange Awards, Strange Change — m759 @ 11:00 am

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 "Non-Euclidean Blocks."

(Here "non-Euclidean" means simply
other than Euclidean. It does not imply any
violation of Euclid's parallel postulate.)

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Thursday, February 18, 2010

Theories: An Outline

Filed under: General,Geometry — Tags: Cullinane Cube, Eightfold Cube, Fitch, Pattern Theory, Solomon's Cube — m759 @ 10:31 am

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 Non-Euclidean 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

Mysteries of Faith

Filed under: General,Geometry — Tags: Eightfold Cube, Springer — m759 @ 9:00 am

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–

2x2x2 cube

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 order-168 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

Binary coordinates for a 4x2 array Chess knight formed by a Singer 7-cycle

For a more Protestant meditation,
see The Cross of Descartes—

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

Sunday October 11, 2009

Filed under: General,Geometry — Tags: Eightfold Cube — m759 @ 7:00 pm

Concepts of Space

Today I revised the illustrations
in Finite Geometry of the
Square and Cube
for consistency in labeling
the eightfold cube.

Related material:

Inside the White Cube:
The Ideology of
the Gallery Space

Dagger Definitions

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Monday, September 14, 2009

Monday September 14, 2009

Filed under: General,Geometry — Tags: Eightfold Cube — m759 @ 3:09 pm

Figure

Generating permutations for the Klein simple group of order 168 acting on the eightfold cube

The Sept. 8 entry on non-Euclidean* 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 “non-Euclidean” 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 “non-Euclidean.”

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Tuesday, September 8, 2009

Tuesday September 8, 2009

Filed under: General,Geometry — Tags: Cube School, Eightfold Cube, Super-8 — m759 @ 12:25 pm

Froebel's
Magic Box

Box containing Froebel's Third Gift-- The Eightfold Cube

Continued from Dec. 7, 2008,
and from yesterday.

Non-Euclidean
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 thought-arteries," 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 husband-and-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 non-Euclidean* geometry– finite geometry over the two-element Galois field– see the work of…

Friedrich Froebel
(1782-1852), who
invented kindergarten.

His "third gift" —

Froebel's Third Gift-- The Eightfold Cube

Photo by Norman Brosterman
fom the Inventing Kindergarten
exhibit at The Institute for Figuring

Go figure.

* i.e., other than Euclidean

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Friday, April 10, 2009

Friday April 10, 2009

Filed under: General,Geometry — Tags: Eightfold Cube — m759 @ 8:00 am

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 Non-Euclidean Revolution

Consider the following question in a paper cited by V. S. Varadarajan:

E. G. Beltrametti, “Can a finite geometry describe physical space-time?” 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.'”

Froebel's 'Third Gift' to kindergarteners: the 2x2x2 cube, in 'Paradise of Childhood'

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Thursday, February 5, 2009

Thursday February 5, 2009

Filed under: General,Geometry — Tags: Cube School, Design Cube, Eightfold Cube, Geometric Theology, Namespace, White Cube — m759 @ 1:00 pm

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:

Photo by Norman Brosterman
fom the Inventing Kindergarten
exhibit at The Institute for Figuring
(co-founded by Margaret Wertheim)

Not-so-childish:

Three planes through
the center of a cube
that split it into
eight subcubes:
Cube subdivided into 8 subcubes by planes through the center
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 180-degree 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 3-space
over the field of real numbers,
but rather in the finite Galois
3-space over the 2-element field.

Galois age fifteen, drawn by a classmate.

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 20th-century art.”

“Space: what you
damn well have to see.”

— James Joyce, Ulysses

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Tuesday, January 6, 2009

Tuesday January 6, 2009

Filed under: General,Geometry — Tags: Eightfold Cube, Octad, Octad Generator — m759 @ 12:00 am

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 four-diamond 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, 1984-2003

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 stained-glass
windows of knowledge.”
— Vladimir Nabokov

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Friday, December 19, 2008

Friday December 19, 2008

Filed under: General,Geometry — Tags: Alyssa Milano Born, Eden, Eightfold Cube, Serpent, Valéry — m759 @ 1:06 pm

Inside the
White Cube

Part I: The White Cube

The Eightfold Cube

Part II: Inside

The Paradise of Childhood'-- Froebel's Third Gift

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

Friday December 12, 2008

Filed under: General,Geometry — Tags: Eightfold Cube, Reinhardt — m759 @ 3:09 pm

On the Symmetric Group S₈

Wikipedia on Rubik's 2×2×2 "Pocket Cube"–

http://www.log24.com/log/pix08A/081212-PocketCube.jpg

"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:

Photo by Norman Brosterman
fom the Inventing Kindergarten
exhibit at The Institute for Figuring
(co-founded by Margaret Wertheim)

and

"Ad Reinhardt and Tony Smith:
A Dialogue,"
an exhibition opening today
at Pace Wildenstein.

For a different sort
of dialogue, click on the
artists' names above.

For a different
approach to S₈,
see Symmetries.

"With humor, my dear Zilkov.
Always with a little humor."

-- The Manchurian Candidate

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Sunday, December 7, 2008

Sunday December 7, 2008

Filed under: General,Geometry — Tags: Eightfold Cube — m759 @ 11:00 am

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.

Photo by Norman Brosterman
fom the Inventing Kindergarten
exhibit at The Institute for Figuring
(co-founded by Margaret Wertheim)

This photo may serve as an
introduction to a different
sort of space.

See The Eightfold Cube.

For the religious meaning
of this small space, see

Richard Wilhelm on
the eight I Ching trigrams.

For a related larger space,
see the entry and links of
St. Augustine’s Day, 2006.

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Sunday, November 16, 2008

Sunday November 16, 2008

Filed under: General,Geometry — Tags: Eightfold Cube, Logos and Branding — 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
and
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

ashevillecreative.com

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.

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Friday, October 24, 2008

Friday October 24, 2008

Filed under: General,Geometry — Tags: Cube School, Eightfold Cube, Springer — m759 @ 8:08 am

“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 3-space over the 2-element 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 less-obvious 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.

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Wednesday, October 22, 2008

Wednesday October 22, 2008

Filed under: General,Geometry — Tags: Eightfold Cube — m759 @ 9:26 am

Euclid vs. Galois

On May 4, 2005, I wrote a note about how to visualize the 7-point 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:

(Click to enlarge)

John Baez, drawing of seven vertices of a cube corresponding to Fano-plane points

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 3-space, false.

The statement is, however, true of the eightfold cube, whose eight subcubes correspond to points of the linear 3-space over the two-element field, if “planes through the origin” is interpreted as planes within that linear 3-space, as in Galois geometry, rather than within the Euclidean cube that Baez’s slides seem to picture.

This Galois-geometry 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 plane-cube 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.

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Friday, September 26, 2008

Friday September 26, 2008

Filed under: General,Geometry — Tags: Eightfold Cube — m759 @ 3:17 pm

Christmas Knot
for 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.”

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Tuesday, August 19, 2008

Tuesday August 19, 2008

Filed under: General,Geometry — Tags: Aug19, Eightfold Cube, Facets — m759 @ 8:30 am

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. 20-21 —

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 4-space (which has finitely many symmetries), and the infinitely many points of Euclidean 4-space (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.

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Friday, August 8, 2008

Friday August 8, 2008

Filed under: General,Geometry — Tags: Eightfold Cube — m759 @ 8:08 am

Weyl on symmetry, the eightfold cube, the Fano plane, and trigrams of the I Ching

Click on image for details.

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Friday, July 25, 2008

Friday July 25, 2008

Filed under: General,Geometry — Tags: Eightfold Cube, Geometry of Even Subsets, Triangles-Spreads-Mathieu — m759 @ 6:01 pm

56 Triangles

Greg Egan's drawing of the 56 triangles on the Klein quartic 3-hole torus

John Baez on
Klein's quartic:

"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 3-holed 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 eight-triangle 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 eight-triangle 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.

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Monday, July 21, 2008

Monday July 21, 2008

Filed under: General,Geometry — Tags: Change Arises, Cube Moves, Cube School, Eightfold Cube, Knight Move, Strange Change — m759 @ 12:00 pm

Knight Moves:

The Relativity Theory
of Kindergarten Blocks

(Continued from
January 16, 2008)

"Hmm, next paper… maybe
'An Unusually Complicated
Theory of Something.'"

— Garrett Lisi at
Physics Forums, July 16

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

Friday July 4, 2008

Filed under: General,Geometry — Tags: Cartoon Graveyards, Croft, Eightfold Cube — m759 @ 8:00 am

REDEMPTION

"I need a photo-opportunity,
I want a shot at redemption.
Don't want to end up a cartoon
In a cartoon graveyard."
— Paul Simon

From Log24 on June 27, 2008,
the day that comic-book artist
Michael Turner died at 37 —

Van Gogh (by Ed Arno) in
The Paradise of Childhood
(by Edward Wiebé):

For Turner's photo-opportunity,
click on Lara.

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Friday, June 27, 2008

Friday June 27, 2008

Filed under: General,Geometry — Tags: Eightfold Cube — m759 @ 8:07 am

Deadpan

Obituary in today’s New York Times
of New Yorker cartoonist Ed Arno:
“Mr. Arno… dealt in whimsy
and deadpan surrealism.”

In his memory:
a cartoon by Arno combined
with material shown here,
under the heading
“From the Cartoon Graveyard,”
on May 27, the date of
Arno’s death —

'Dear Theo' cartoon of van Gogh by Ed Arno, adapted to illustrate the eightfold cube

Related material:

Yesterday’s entry. The key part of
that entry is of course the phrase
“the antics of a drunkard.”

Ray Milland in
“The Lost Weekend”
(see June 25, 10:31 AM)–

“I’m van Gogh
painting pure sunlight.”

It is not advisable,
in all cases,
to proceed thus far.

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Thursday, May 22, 2008

Thursday May 22, 2008

Filed under: General,Geometry — Tags: Eightfold Cube, Jack in the Box — m759 @ 9:00 am

The Undertaking:
An Exercise in
Conceptual Art

Hexagram 54:
THE JUDGMENT

Undertakings bring misfortune.
Nothing that would further.

The image “http://www.log24.com/log/pix08/080522-Irelandslide1.jpg” cannot be displayed, because it contains errors.

“Brian O’Doherty, an Irish-born 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:

Images of time and eternity in memory of Michelangelo

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 20th-century 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 20th-century art.”

An archetypal image

THE SPACE:

A non-archetypal image

THE ART:

Natasha Wescoat, 2004

See also Epiphany 2008:

How the eightfold cube works

“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

* For a different, Jungian, alter ego, see Irish Fourplay (Jan. 31, 2003) and “Outside the Box,” a New York Times review of O’Doherty’s art (featuring a St. Bridget’s Cross) by Bridget L. Goodbody dated April 25, 2007. See also Log24 on that date.

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Saturday, May 10, 2008

Saturday May 10, 2008

Filed under: General,Geometry — Tags: Antonelli, Cube School, Cullinane Cube, Design Cube, Eightfold Cube, Geometry, Solomon's Cube, Solomon's Mines, The Building-Block Metaphor — m759 @ 8:00 am

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:

Cubic crystal system
(Footnotes 1 and 2)

Figure 2 —
The Third Gift, 1837:

Froebel's third gift

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:

Seven partitions of the eightfold cube in 'Paradise of Childhood,' 1906

Figure 4 —
Solomon's Cube,
1981 and 1983:

Solomon's Cube - A 1981 design by Steven H. Cullinane

Figure 5 —
Design Cube, 2006:

Design Cube 4x4x4 by Steven H. Cullinane

The above screenshot shows a
moveable JavaScript display
of a space of six dimensions
(over the two-element field).

(To see how the display works,
try the Kaleidoscope Puzzle first.)

For some mathematical background, see

Footnotes:

1. Image said to be after Holden and Morrison, Crystals and Crystal Growing, 1982
2. Curtis Schuh, "The Library: Biobibliography of Mineralogy," article on Mohs
3. Bart Kahr, "Crystal Engineering in Kindergarten" (pdf), Crystal Growth & Design, Vol. 4 No. 1, 2004, 3-9

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Tuesday, April 8, 2008

Tuesday April 8, 2008

Filed under: General,Geometry — Tags: Eightfold Cube — m759 @ 8:00 am

Eight is a Gate

Part I:

December 2002

Part II:

Epiphany 2008

How the eightfold cube works
This figure is related to
the mathematics of
reflection groups.

Part III:

“The capacity of music to operate simultaneously along horizontal and vertical axes, to proceed simultaneously in opposite directions (as in inverse canons), may well constitute the nearest that men and women can come to absolute freedom. Music does ‘keep time’ for itself and for us.”

— George Steiner in Grammars of Creation

Inverse Canon —

From Werner Icking Music Archive:

Bach, Fourteen Canons
on the First Eight Notes
of the Goldberg Ground,
No. 11 —

Click to enlarge.

Play midi of Canon 11.

At a different site —
an mp3 of the 14 canons.

Part IV:

That Crown of Thorns,
by Timothy A. Smith

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Wednesday, January 16, 2008

Wednesday January 16, 2008

Filed under: General,Geometry — Tags: Chess Problem, Cube Moves, Cube School, Eightfold Cube, Knight Move — m759 @ 12:25 pm

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

Sunday January 6, 2008

Filed under: General,Geometry — Tags: Eightfold Cube — m759 @ 1:00 am

The following illustration of
how the eightfold cube works
was redone.

How the eightfold cube works

For further details, see
Finite Geometry of
the Square and Cube
and The Eightfold Cube.

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Thursday, July 5, 2007

Thursday July 5, 2007

Filed under: General,Geometry — Tags: Eightfold Cube — m759 @ 7:11 pm

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:

Trigram of K'un, the Receptive

K’un
The Receptive

_____________________________________________

Update of Nov. 30, 2013:

From a little-known website in Kuala Lumpur:
(Click to enlarge.)

The remarks on Platonic realism are from Wikipedia.
The eightfold cube is apparently from this post.

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Monday, June 25, 2007

Monday June 25, 2007

Filed under: General,Geometry — Tags: Eightfold Cube — m759 @ 3:00 pm

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 Anti-Christ 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

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Saturday, April 7, 2007

Saturday April 7, 2007

Filed under: General,Geometry — Tags: Eightfold Cube, Reinventing — m759 @ 12:25 pm

Today's birthdays:
Francis Ford Coppola
and Russell Crowe

Gift of the Third Kind

Background:
Art Wars and
Russell Crowe as
Santa's Helper.

From Friedrich Froebel,
who invented kindergarten:

From Christmas 2005:

Related material from
Pittsburgh:

… and from Grand Rapids:

Click on pictures for details.

Related material
for Holy Saturday:

Harrowing,
"Hey, Big Spender,"
and
Santa Versus the Volcano.

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Saturday, December 23, 2006

Saturday December 23, 2006

Filed under: General,Geometry — Tags: Black Mark, Eightfold Cube, Reinhardt — m759 @ 9:00 am

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….”

Log24 on Monday,
Dec. 18, 2006:

“I did a column in
Scientific American
on minimal art, and
I reproduced one of
Ed Rinehart’s [sic]
black paintings.”

— Martin Gardner (pdf)

“… the entire profession
has received a very public
and very bad black mark.”

— Joan S. Birman (pdf)

Lottery on Friday,
Dec. 22, 2006:

The image “http://www.log24.com/log/pix06B/061222-PAlottery.jpg” cannot be displayed, because it contains errors.

5/04, 2005:

Analysis of the structure
of a 2x2x2 cube

via trinities of
projective points
in a Fano plane.

7/15, 2005:

“Art history was very personal
through the eyes of Ad Reinhardt.”

— Robert Morris,
Smithsonian Archives
of American Art

Also on 7/15, 2005,
a quotation on Usenet:

“A set having three members is a
single thing wholly constituted by
its members but distinct from them.
After this, the theological doctrine
of the Trinity as ‘three in one’
should be child’s play.”

— Max Black,
Caveats and Critiques:
Philosophical Essays in
Language, Logic, and Art

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Friday, November 24, 2006

Friday November 24, 2006

Filed under: General,Geometry — Tags: Eightfold Cube — m759 @ 1:06 pm

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:

+	0	1
0	0	1
1	1	0

*	0	1
0	0	0
1	0	1

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 4-point affine plane,
which we may call

Galois's Window

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 3-space
and Galois affine hyperplane
in greater detail.

For more on these cases, see

The Eightfold Cube,
Finite Relativity,
The Smallest Projective Space,
Latin-Square 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 16-point 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.

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Sunday, October 8, 2006

Sunday October 8, 2006

Filed under: General,Geometry — Tags: Eightfold Cube, Jack in the Box — m759 @ 12:00 am

Today’s Birthday:
Matt Damon

Enlarge this image

“Cubistic”

— New York Times review
of Scorsese’s The Departed

Related material:

Log24, May 26, 2006 —

“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 to-morrow at breakfast.”

— G. K. Chesterton

Platonic solid	Natasha Wescoat, 2004 Shakespearean Fool
Not to mention Euclid and Picasso (Log24, Oct. 6, 2006) —

(Click on pictures for details. Euclid is represented by Alexander Bogomolny, Picasso by Robert Foote.)

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
CIRCUS-STANCE

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 under-educated these days.

By SAHIL K. MAHTANI
Harvard College– in the best formulation I’ve heard– promulgates a Japanese-style education, where the professoriate pretend to teach, the students pretend to learn, and everyone is happy.

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Friday, May 26, 2006

Friday May 26, 2006

Filed under: General,Geometry — Tags: Eightfold Cube, Jack in the Box — m759 @ 8:00 am

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 to-morrow at breakfast."

— G. K. Chesterton

Platonic
solid

The image “http://www.log24.com/log/pix06A/060526-JackInTheBox.jpg” cannot be displayed, because it contains errors.

Shakespearean
Fool

Related material:

Yesterday's entries
and their link to
The Line

as well as

Galois Geometry

and the remarks
of Oxford professor
Marcus du Sautoy,
who claims that
"the right side of the brain
is responsible for mathematics."

Let us hope that Professor du Sautoy
is more reliable on zeta functions,
his real field of expertise,
than on neurology.

The picture below may help
to clear up his confusion
between left and right.

His confusion about
pseudoscience may not
be so easily remedied.

The image “http://www.log24.com/log/pix06A/060526-BrainLR1.jpg” cannot be displayed, because it contains errors.
flickr.com/photos/jaycross/3975200/

(Any resemblance to the film
"Hannibal" is purely coincidental.)

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Sunday, December 25, 2005

Sunday December 25, 2005

Filed under: General,Geometry — Tags: Eightfold Cube — m759 @ 8:00 pm

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

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Tuesday, August 2, 2005

Tuesday August 2, 2005

Filed under: General,Geometry — Tags: Eden, Eightfold Cube — m759 @ 7:00 am

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

On Beauty

"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

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Tuesday, June 7, 2005

Tuesday June 7, 2005

Filed under: General,Geometry — Tags: Eightfold Cube — m759 @ 1:01 pm

The Sequel to Rhetoric 101:

101 101

“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 Eighty-Four.

[101 was also]
Professor Nash’s office number
in the movie ‘A Beautiful Mind.'”

— Prime Curios

Classics Illustrated —

The image “http://www.log24.com/log/pix05A/050607-Nightmare.jpg” cannot be displayed, because it contains errors.

Click on picture for details.

(For some mathematics that is actually
from 1984, see Block Designs
and the 2005 followup
The Eightfold Cube.)

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Wednesday, May 18, 2005

Wednesday May 18, 2005

Filed under: General,Geometry — Tags: Eightfold Cube — m759 @ 11:07 pm

On Beauty

“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

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Wednesday, May 4, 2005

Wednesday May 4, 2005

Filed under: General,Geometry — Tags: Alperin, Eightfold Cube, May 6 2005 — m759 @ 1:00 pm

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):

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

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.

Fano plane group - generating permutations

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 3-space over GF(2) whose "points" are individual subcubes, the group of eight translations is generated by interchanges of parallel 2x2x1 cube-slices. This is clearly a subgroup of the group generated by permuting 1x1x2 cube-slices. Such translations in the affine 3-space 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 Finite-Geometry Models.

For another application of the points-as-partitions technique, see Latin-Square Geometry: Orthogonal Latin Squares as Skew Lines.

For more on the plane's symmetry group in another guise, see John Baez on Klein's Quartic Curve and the online book The Eightfold Way. For more on the mathematics of cubic models, see Solomon's Cube.

For a large downloadable folder with many other related web pages, see Notes on Finite Geometry.

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Saturday, July 20, 2002

Saturday July 20, 2002

Filed under: General,Geometry — Tags: Change Logos, Eightfold Cube, four-set, Map Systems, Octad, Octad Generator, Strange Change — 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