See "Smallest Perfect" in this journal.

## Thursday, April 13, 2017

## Thursday, September 15, 2016

### The Smallest Perfect Number/Universe

The smallest perfect number,* six, meets

"the smallest perfect universe,"** PG(3,2).

* For the definition of "perfect number," see any introductory

number-theory text that deals with the history of the subject.

** The phrase "smallest perfect universe" as a name for PG(3,2),

the projective 3-space over the 2-element Galois field GF(2),

was coined by math writer Burkard Polster. Cullinane's square

model of PG(3,2) differs from the earlier tetrahedral model

discussed by Polster.

## Tuesday, August 9, 2016

## Monday, May 30, 2016

### Perfect Universe

(A sequel to the previous post, Perfect Number)

Since antiquity, six has been known as

"the smallest perfect number." The word "perfect"

here means that a number is the sum of its

proper divisors — in the case of six: 1, 2, and 3.

The properties of a six-element set (a "6-set")

divided into three 2-sets and divided into two 3-sets

are those of what Burkard Polster, using the same

adjective in a different sense, has called

"the smallest perfect universe" — PG(3,2), the projective

3-dimensional space over the 2-element Galois field.

A Google search for the phrase "smallest perfect universe"

suggests a turnaround in *meaning *, if not in finance,

that might please Yahoo CEO Marissa Mayer on her birthday —

The *semantic* turnaround here in the *meaning* of "perfect"

is accompanied by a *model* turnaround in the *picture *of PG(3,2) as

Polster's *tetrahedral* model is replaced by Cullinane's *square* model.

Further background from the previous post —

See also Kirkman's Schoolgirl Problem.

## Friday, December 21, 2012

### Analogies*

The Moore correspondence may be regarded

as an analogy between the 35 partitions of an

8-set into two 4-sets and the 35 lines in the

finite projective space PG(3,2).

Closely related to the Moore correspondence

is a correspondence (or analogy) between the

15 2-subsets of a 6-set and the 15 points of PG(3,2).

An analogy *between* the two above analogies

is supplied by the exceptional outer automorphism of S_{6}.

See…

The 2-subsets of a 6-set are the points of a PG(3,2),

Picturing outer automorphisms of S_{6}, and

A linear complex related to M_{24}.

(Background: Inscapes, Inscapes III: PG(2,4) from PG(3,2),

and Picturing the smallest projective 3-space.)

* For some context, see Analogies and

"Smallest Perfect Universe" in this journal.

## Thursday, February 9, 2012

### ART WARS continued

**On the Complexity of Combat—**

The above article (see original pdf), clearly of more

theoretical than practical interest, uses the concept

of "symmetropy" developed by some Japanese

researchers.

For some background from finite geometry, see

Symmetry of Walsh Functions. For related posts

in this journal, see Smallest Perfect Universe.

**Update of 7:00 PM EST Feb. 9, 2012—**

**Background on Walsh-function symmetry in 1982—**

(Click image to enlarge. See also original pdf.)

Note the somewhat confusing resemblance to

a four-color decomposition theorem

used in the proof of the diamond theorem.

## Tuesday, June 15, 2010

### Imago, Imago, Imago

Recommended— an online book—

*Flight from Eden: The Origins of Modern Literary Criticism and Theory*,

by Steven Cassedy, U. of California Press, 1990.

See in particular

#### Valéry and the Discourse On His Method.

Pages 156-157—

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

** O ** Paul Valéry,

*Oeuvres*(Paris: Pléiade, 1957-60)

** C ** Valéry,

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

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

and Cullinane’s purely geometric four-diamond figure:

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

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

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

## Friday, May 21, 2010

### The Oslo Version

**From an art exhibition in Oslo last year–**

The artist's description above is not in correct left-to-right order.

Actually the hyperplanes above are at left, the points at right.

Compare to "Picturing the Smallest Projective 3-Space,"

a note of mine from April 26, 1986—

Click for the original full version.

Compare also to Burkard Polster's original use of

the phrase "the smallest perfect universe."

Polster's tetrahedral model of points and hyperplanes

is quite different from my own square version above.

See also Cullinane on Polster.

Here are links to the gallery press release

and the artist's own photos.

## Wednesday, February 28, 2007

### Wednesday February 28, 2007

of Geometry

*Elements*of Geometry

The title of Euclid’s *Elements* is, in Greek, *Stoicheia*.

From* Lectures on the Science of Language*,

by Max Muller, fellow of All Souls College, Oxford.

New York: Charles Scribner’s Sons, 1890, pp. 88-90 –

*Stoicheia*

“The question is, why were the elements, or the component primary parts of things, called *stoicheia* by the Greeks? It is a word which has had a long history, and has passed from Greece to almost every part of the civilized world, and deserves, therefore, some attention at the hand of the etymological genealogist.

*Stoichos*, from which *stoicheion*, means a row or file, like *stix* and *stiches* in Homer. The suffix *eios* is the same as the Latin *eius*, and expresses what belongs to or has the quality of something. Therefore, as *stoichos* means a row, *stoicheion* would be what belongs to or constitutes a row….

Hence *stoichos* presupposes a root *stich*, and this root would account in Greek for the following derivations:–

*stix*, gen.*stichos*, a row, a line of soldiers*stichos*, a row, a line;*distich*, a couplet*steicho*,*estichon*, to march in order, step by step; to mount*stoichos*, a row, a file;*stoichein*, to march in a line

In German, the same root yields *steigen*, to step, to mount, and in Sanskrit we find *stigh*, to mount….

*Stoicheia* are the degrees or steps from one end to the other, the constituent parts of a whole, forming a complete series, whether as hours, or letters, or numbers, or parts of speech, or physical elements, provided always that such elements are held together by a systematic order.”

**Example:**

**The Miracle Octad Generator of R. T. Curtis**

For the geometry of these *stoicheia*, see

The Smallest Perfect Universe and

Finite Geometry of the Square and Cube.

## Monday, September 4, 2006

### Monday September 4, 2006

**In a Nutshell:**

**The Seed**

"The symmetric group S_{6} of permutations of 6 objects is the only symmetric group with an outer automorphism….

This outer automorphism can be regarded as **the seed** from which grow about half of the sporadic simple groups…."

This "seed" may be pictured as

within what Burkard Polster has called "the smallest perfect universe"– PG(3,2), the projective 3-space over the 2-element field.

Related material: yesterday's entry for Sylvester's birthday.

## Sunday, April 25, 2004

### Sunday April 25, 2004

**Small World**

Added a note to 4×4 Geometry:

The 4×4 square model lets us visualize the projective space PG(3,2) as well as the affine space AG(4,2). For tetrahedral and circular models of PG(3,2), see the work of Burkard Polster. The following is from an advertisement of a talk by Polster on PG(3,2).

“After a short introduction to finite geometries, I’ll take you on a… guided tour of the smallest perfect universe — a complex universe of breathtaking abstract beauty, consisting of only 15 points, 35 lines and 15 planes — a space whose overall design incorporates and improves many of the standard features of the three-dimensional Euclidean space we live in…. Among mathematicians our perfect universe is known as — Burkard Polster, May 2001 |