“PG(3,2)” « Search Results

Sunday, July 29, 2012

The Galois Tesseract

Filed under: General,Geometry — Tags: Defining Form, Galois's Space, Springer — m759 @ 11:00 pm

The three parts of the figure in today's earlier post "Defining Form"—

IMAGE- Hyperplanes (square and triangular) in PG(3,2), and coordinates for AG(4,2)

— share the same vector-space structure:

0	c	d	c + d
a	a + c	a + d	a + c + d
b	b + c	b + d	b + c + d
a + b	a + b + c	a + b + d	a + b + c + d

(This vector-space a b c d diagram is from Chapter 11 of
Sphere Packings, Lattices and Groups , by John Horton
Conway and N. J. A. Sloane, first published by Springer
in 1988.)

The fact that any 4×4 array embodies such a structure was implicit in
the diamond theorem (February 1979). Any 4×4 array, regarded as
a model of the finite geometry AG(4, 2), may be called a Galois tesseract.
(So called because of the Galois geometry involved, and because the
16 cells of a 4×4 array with opposite edges identified have the same
adjacency pattern as the 16 vertices of a tesseract (see, for instance,
Coxeter's 1950 "Self-Dual Configurations and Regular Graphs," figures
5 and 6).)

A 1982 discussion of a more abstract form of AG(4, 2):

Source:

The above 1982 remarks by Brouwer may or may not have influenced
the drawing of the above 1988 Conway-Sloane diagram.

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Defining Form

Filed under: General,Geometry — Tags: Defining Form, Figurate Geometry — m759 @ 2:00 pm

Background: Square-Triangle Theorem.

For a more literary approach, see "Defining Form" in this journal
and a bibliography from the University of Zaragoza.

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Monday, July 16, 2012

Mapping Problem continued

Filed under: General,Geometry — Tags: Figurate Geometry, Springer — m759 @ 2:56 am

Another approach to the square-to-triangle
mapping problem (see also previous post)—

IMAGE- Triangular analogs of the hyperplanes in the square model of PG(3,2)

For the square model referred to in the above picture, see (for instance)

Coordinates for the 16 points in the triangular arrays
of the corresponding affine space may be deduced
from the patterns in the projective-hyperplanes array above.

This should solve the inverse problem of mapping,
in a natural way, the triangular array of 16 points
to the square array of 16 points.

Update of 9:35 AM ET July 16, 2012:

Note that the square model's 15 hyperplanes S
and the triangular model's 15 hyperplanes T —

— share the following vector-space structure —

0	c	d	c + d
a	a + c	a + d	a + c + d
b	b + c	b + d	b + c + d
a + b	a + b + c	a + b + d	a + b + c + d

(This vector-space a b c d diagram is from
Chapter 11 of Sphere Packings, Lattices
and Groups , by John Horton Conway and
N. J. A. Sloane, first published by Springer
in 1988.)

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Thursday, July 12, 2012

Galois Space

Filed under: General,Geometry — Tags: Galois's Space, Geometry — m759 @ 6:01 pm

An example of lines in a Galois space * —

The 35 lines in the 3-dimensional Galois projective space PG(3,2)—

(Click to enlarge.)

There are 15 different individual linear diagrams in the figure above.
These are the points of the Galois space PG(3,2). Each 3-set of linear diagrams
represents the structure of one of the 35 4×4 arrays and also represents a line
of the projective space.

The symmetry of the linear diagrams accounts for the symmetry of the
840 possible images in the kaleidoscope puzzle.

* For further details on the phrase "Galois space," see
Beniamino Segre's "On Galois Geometries," Proceedings of the
International Congress of Mathematicians, 1958 [Edinburgh].
(Cambridge U. Press, 1960, 488-499.)

(Update of Jan. 5, 2013— This post has been added to finitegeometry.org.)

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Monday, September 12, 2011

Key Paper Now Online

Filed under: General — m759 @ 11:07 pm

The 3-Space PG(3,2) and Its Group

by George M. Conwell, Annals of Mathematics ,
Second Series, Vol. 11, No. 2 (Jan., 1910), pp. 60-76

Article Stable URL: http://www.jstor.org/stable/1967582

At least for now, this paper may be downloaded without
signing in or making a payment. Click the "View PDF" link.

Update of Sept. 13— From Library Journal on Sept. 7—

The JSTOR journal archive announced today that it is making nearly 500,000 public domain journal articles from more than 220 journals—or about six percent of JSTOR's total content—freely available for use by "anyone, without registration and regardless of institutional affiliation."

The material, entitled Early Journal Content, will be rolled out in batches starting today over the course of one week. It includes content published in the United States before 1923 and international content published before 1870, which ensures that all the content is firmly in the public domain. JSTOR, in an announcement, said that the move was "a first step in a larger effort to provide more access options" to JSTOR content for independent scholars and others unaffiliated with universities.

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Saturday, August 20, 2011

Castle Rock

Filed under: General,Geometry — Tags: 2x4 — m759 @ 6:29 pm

Happy birthday to Amy Adams
(actress from Castle Rock, Colorado)

"The metaphor for metamorphosis…" —Endgame

Related material:

"The idea that reality consists of multiple 'levels,' each mirroring all others in some fashion, is a diagnostic feature of premodern cosmologies in general…."

— Scholarly paper on "Correlative Cosmologies"

"How many layers are there to human thought? Sometimes in art, just as in people’s conversations, we’re aware of only one at a time. On other occasions, though, we realize just how many layers can be in simultaneous action, and we’re given a sense of both revelation and mystery. When a choreographer responds to music— when one artist reacts in detail to another— the sensation of multilayering can affect us as an insight not just into dance but into the regions of the mind.

The triple bill by the Mark Morris Dance Group at the Rose Theater, presented on Thursday night as part of the Mostly Mozart Festival, moves from simple to complex, and from plain entertainment to an astonishingly beautiful and intricate demonstration of genius….

'Socrates' (2010), which closed the program, is a calm and objective work that has no special dance excitement and whips up no vehement audience reaction. Its beauty, however, is extraordinary. It’s possible to trace in it terms of arithmetic, geometry, dualism, epistemology and ontology, and it acts as a demonstration of art and as a reflection of life, philosophy and death."

— Alastair Macaulay in today's New York Times

SOCRATES: Let us turn off the road a little….

— Libretto for Mark Morris's 'Socrates'

See also Amy Adams's new film "On the Road"
in a story from Aug. 5, 2010 as well as a different story,
Eightgate, from that same date:

A 2x4 array of squares

The above reference to "metamorphosis" may be seen,
if one likes, as a reference to the group of all projectivities
and correlations in the finite projective space PG(3,2)—
a group isomorphic to the 40,320 transformations of S₈
acting on the above eight-part figure.

See also The Moore Correspondence from last year
on today's date, August 20.

For some background, see a book by Peter J. Cameron,
who has figured in several recent Log24 posts—

"At the still point, there the dance is."
— Four Quartets

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Friday, March 18, 2011

Defining Configurations*

Filed under: General,Geometry — Tags: Six-Set, Springer — m759 @ 7:00 pm

The On-Line Encyclopedia of Integer Sequences has an article titled "Number of combinatorial configurations of type (n_3)," by N.J.A. Sloane and D. Glynn.

From that article:

DEFINITION: A combinatorial configuration of type (n_3) consists of an (abstract) set of n points together with a set of n triples of points, called lines, such that each point belongs to 3 lines and each line contains 3 points.
EXAMPLE: The unique (8_3) configuration consists of the triples 125, 148, 167, 236, 278, 347, 358, 456.

The following corrects the word "unique" in the example.

* This post corrects an earlier post, also numbered 14660 and dated 7 PM March 18, 2011, that was in error.
The correction was made at about 11:50 AM on March 20, 2011.

_____________________________________________________________

Update of March 21

The problem here is of course with the definition. Sloane and Glynn failed to include in their definition a condition that is common in other definitions of configurations, even abstract or purely "combinatorial" configurations. See, for instance, Configurations of Points and Lines , by Branko Grunbaum (American Mathematical Society, 2009), p. 17—

In the most general sense we shall consider combinatorial (or abstract) configurations; we shall use the term set-configurations as well. In this setting "points" are interpreted as any symbols (usually letters or integers), and "lines" are families of such symbols; "incidence" means that a "point" is an element of a "line". It follows that combinatorial configurations are special kinds of general incidence structures. Occasionally, in order to simplify and clarify the language, for "points" we shall use the term marks, and for "lines" we shall use blocks. The main property of geometric configurations that is preserved in the generalization to set-configurations (and that characterizes such configurations) is that two marks are incident with at most one block, and two blocks with at most one mark.

Whether or not omitting this "at most one" condition from the definition is aesthetically the best choice, it dramatically changes the number of configurations in the resulting theory, as the above (8_3) examples show.

Update of March 22 (itself updated on March 25)

For further background on configurations, see Dolgachev—

Note that the two examples Dolgachev mentions here, with 16 points and 9 points, are not unrelated to the geometry of 4×4 and 3×3 square arrays. For the Kummer and related 16-point configurations, see section 10.3, "The Three Biplanes of Order 4," in Burkard Polster's A Geometrical Picture Book (Springer, 1998). See also the 4×4 array described by Gordon Royle in an undated web page and in 1980 by Assmus and Sardi. For the Hesse configuration, see (for instance) the passage from Coxeter quoted in Quaternions in an Affine Galois Plane.

Update of March 27

See the above link to the (16,6) 4×4 array and the (16,6) exercises using this array in R.D. Carmichael's classic Introduction to the Theory of Groups of Finite Order (1937), pp. 42-43. For a connection of this sort of 4×4 geometry to the geometry of the diamond theorem, read "The 2-subsets of a 6-set are the points of a PG(3,2)" (a note from 1986) in light of R.W.H.T. Hudson's 1905 classic Kummer's Quartic Surface , pages 8-9, 16-17, 44-45, 76-77, 78-79, and 80.

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Tuesday, June 15, 2010

Imago, Imago, Imago

Filed under: General,Geometry — Tags: Aion Art, Change Logos, Eden, Self, Strange Change, Valéry — m759 @ 11:07 am

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).

Notes:

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.

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Sunday, May 23, 2010

For Your Consideration —

Filed under: General — Tags: Death Story — m759 @ 10:10 am

Cannes Festival Readies for Awards Night

Uncertified Copy

The pictures in the detail are copies of
figures created by S. H. Cullinane in 1986.
They illustrate his model of hyperplanes
and points in the finite projective space
known as PG(3,2) that underlies
Cullinane's diamond theorem.

The title of the pictures in the detail
is that of a film by Burkard Polster
that portrays a rival model of PG(3,2).

The artist credits neither Cullinane nor Polster.

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Friday, May 21, 2010

The Oslo Version

Filed under: General,Geometry — Tags: Death Story — m759 @ 9:29 am

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.

Wednesday, April 28, 2010

Eightfold Geometry

Filed under: General,Geometry — Tags: Folklore, four-set, Geometry, Natural Diagram, Octad, Octad Generator — m759 @ 11:07 am

Image-- Analysis of structure of the 35 partitions of an 8-set into two 4-sets

Image-- Miracle Octad Generator of R.T. Curtis

Related web pages:

Miracle Octad Generator,
Generating the Octad Generator,
Geometry of the 4×4 Square

Related folklore:

"It is commonly known that there is a bijection between the 35 unordered triples of a 7-set [i.e., the 35 partitions of an 8-set into two 4-sets] and the 35 lines of PG(3,2) such that lines intersect if and only if the corresponding triples have exactly one element in common." –"Generalized Polygons and Semipartial Geometries," by F. De Clerck, J. A. Thas, and H. Van Maldeghem, April 1996 minicourse, example 5 on page 6

The Miracle Octad Generator may be regarded as illustrating the folklore.

Update of August 20, 2010–

For facts rather than folklore about the above bijection, see The Moore Correspondence.

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Wednesday, May 20, 2009

Wednesday May 20, 2009

Filed under: General,Geometry — Tags: Geometry, Octad, Octad Generator — m759 @ 4:00 pm

From Quilt Blocks to the
Mathieu Group M₂₄

_{Diamonds
(a traditional

quilt block):

_Octads:}

Click on illustrations for details.

The connection:

The four-diamond figure is related to the finite geometry PG(3,2). (See "Symmetry Invariance in a Diamond Ring," AMS Notices, February 1979, A193-194.) PG(3,2) is in turn related to the 759 octads of the Steiner system S(5,8,24). (See "Generating the Octad Generator," expository note, 1985.)

The relationship of S(5,8,24) to the finite geometry PG(3,2) has also been discussed in–

"A Geometric Construction of the Steiner System S(4,7,23)," by Alphonse Baartmans, Walter Wallis, and Joseph Yucas, Discrete Mathematics 102 (1992) 177-186.

Abstract: "The Steiner system S(4,7,23) is constructed from the geometry of PG(3,2)."

"A Geometric Construction of the Steiner System S(5,8,24)," by R. Mandrell and J. Yucas, Journal of Statistical Planning and Inference 56 (1996), 223-228.

Abstract: "The Steiner system S(5,8,24) is constructed from the geometry of PG(3,2)."

For the connection of S(5,8,24) with the Mathieu group M₂₄, see the references in The Miracle Octad Generator.

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Tuesday, October 3, 2006

Tuesday October 3, 2006

Filed under: General,Geometry — Tags: Fields, Octad, Octad Generator, Sundown Ending — m759 @ 9:26 am

Serious

"I don't think the 'diamond theorem' is anything serious, so I started with blitzing that."

— Charles Matthews at Wikipedia, Oct. 2, 2006

"The 'seriousness' of a mathematical theorem lies, not in its practical consequences, which are usually negligible, but in the significance of the mathematical ideas which it connects. We may say, roughly, that a mathematical idea is 'significant' if it can be connected, in a natural and illuminating way, with a large complex of other mathematical ideas."

— G. H. Hardy, A Mathematician's Apology

Matthews yesterday deleted references to the diamond theorem and related material in the following Wikipedia articles:

Affine group‎
Reflection group‎
Symmetry in mathematics‎
Incidence structure‎
Invariant (mathematics)‎
Symmetry‎
Finite geometry‎
Group action‎
History of geometry‎

This would appear to be a fairly large complex of mathematical ideas.

See also the following "large complex" cited, following the above words of Hardy, in Diamond Theory:

Affine geometry, affine planes, affine spaces, automorphisms, binary codes, block designs, classical groups, codes, coding theory, collineations, combinatorial, combinatorics, conjugacy classes, the Conwell correspondence, correlations, design theory, duads, duality, error correcting codes, exceptional groups, finite fields, finite geometry, finite groups, finite rings, Galois fields, generalized quadrangles, generators, geometry, GF(2), GF(4), the (24,12) Golay code, group actions, group theory, Hadamard matrices, hypercube, hyperplanes, hyperspace, incidence structures, invariance, Karnaugh maps, Kirkman's schoolgirl problem, Latin squares, Leech lattice, linear groups, linear spaces, linear transformations, Mathieu groups, matrix theory, Meno, Miracle Octad Generator, MOG, multiply transitive groups, octads, the octahedral group, orthogonal arrays, outer automorphisms, parallelisms, partial geometries, permutation groups, PG(3,2), polarities, Polya-Burnside theorem, projective geometry, projective planes, projective spaces, projectivities, Reed-Muller codes, the relativity problem, Singer cycle, skew lines, sporadic simple groups, Steiner systems, symmetric, symmetry, symplectic, synthemes, synthematic, tesseract, transvections, Walsh functions, Witt designs.

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Monday, September 4, 2006

Monday September 4, 2006

Filed under: General,Geometry — Tags: Six-Set, The Nutshell — m759 @ 7:20 pm

In a Nutshell:

The Seed

"The symmetric group S₆ 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…."

— Noam Elkies, February 2006

This "seed" may be pictured as

The outer automorphism of a six-set in action

group actions on a linear complex

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.

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Friday, April 28, 2006

Friday April 28, 2006

Filed under: General,Geometry — Tags: Geometry, Octad, Octad Generator — m759 @ 12:00 pm

Exercise

Review the concepts of integritas, consonantia, and claritas in Aquinas:

"For in respect to beauty three things are essential: first of all, integrity or completeness, since beings deprived of wholeness are on this score ugly; and [secondly] a certain required design, or patterned structure; and finally a certain splendor, inasmuch as things are called beautiful which have a certain 'blaze of being' about them…."

— Summa Theologiae Sancti Thomae Aquinatis, I, q. 39, a. 8, as translated by William T. Noon, S.J., in Joyce and Aquinas, Yale University Press, 1957

Review the following three publications cited in a note of April 28, 1985 (21 years ago today):

(1) Cameron, P. J.,
Parallelisms of Complete Designs,
Cambridge University Press, 1976.

(2) Conwell, G. M.,
The 3-space PG(3,2) and its group,
Ann. of Math. 11 (1910) 60-76.

(3) Curtis, R. T.,
   A new combinatorial approach to M₂₄,
     Math. Proc. Camb. Phil. Soc.
   79 (1976) 25-42.

Discuss how the sextet parallelism in (1) illustrates integritas, how the Conwell correspondence in (2) illustrates consonantia, and how the Miracle Octad Generator in (3) illustrates claritas.

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Sunday, April 25, 2004

Sunday April 25, 2004

Filed under: General,Geometry — m759 @ 3:31 pm

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).

The Smallest Perfect Universe

“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 PG(3,2) — the smallest three-dimensional projective space. It plays an important role in many core mathematical disciplines such as combinatorics, group theory, and geometry.”

— Burkard Polster, May 2001

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Monday, August 4, 2003

Monday August 4, 2003

Filed under: General — Tags: Milioti's Razr — m759 @ 1:00 pm

Venn's Trinity

Today is the birthday of logician John Venn.

From the St. Andrews History of Mathematics site:

"Venn considered three discs R, S, and T as typical subsets of a set U. The intersections of these discs and their complements divide U into 8 non-overlapping regions, the unions of which give 256 different Boolean combinations of the original sets R, S, T."

Last night's entry, "A Queer Religion," gave a Catholic view of the Trinity. Here are some less interesting but more fruitful thoughts inspired by Venn's diagram of the Trinity (or, indeed, of any three entities):

"To really know a subject you've got to learn a bit of its history…."
— John Baez, August 4, 2002

"We both know what memories can bring;
They bring diamonds and rust."
— Joan Baez, April 1975

For the "diamonds" brought by memories of the 2⁸combinations described above, consider how the symmetric group S₈ is related to the symmetries of the finite projective space PG(3,2). (See Diamond Theory.)

For the "rust," consider the following:

"Lay not up for yourselves treasures upon earth, where moth and rust doth corrupt…."
— Matthew 6:19

The letters R, U, S, T in the Venn diagram above are perhaps relevant here, symbolizing, if you will, the earthly confusion of language, as opposed to the heavenly clarity of mathematics.

As for MOTH, see the article Hometown Zeroes (which brings us yet again to the Viper Room, scene of River Phoenix's death) and the very skillfully designed website MOTHEMATICS.

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Wednesday, April 2, 2003

Wednesday April 2, 2003

Filed under: General,Geometry — Tags: 1982 J9, Logos and Branding, Reinhardt, Sein Feld — m759 @ 2:30 pm

Symmetries…. May 15, 1998

The following journal note, from the day after Sinatra died, was written before I heard of his death. Note particularly the quote from Rilke. Other material was suggested, in part, by Alasdair Gray's Glasgow novel 1982 Janine. The "Sein Feld" heading is a reference to the Seinfeld final episode, which aired May 14, 1998. The first column contains a reference to angels — apparently Hell's Angels — and the second column provides a somewhat more serious look at this theological topic.

Sein Feld

1984 Janine

"But Angels love their own
And they're reaching out
    for you
Janine… Oh Janine
— Kim Wilde lyric,
    Teases & Dares album,
    1984, apparently about
    a British biker girl

"Logos means above all relation."
— Simone Weil,
Gateway to God,
Glasgow, 1982

"Gesang ist Dasein….
Ein Hauch um nichts.
Ein Wehn im Gott.
Ein Wind."
— Not Heidegger but Rilke:
Sonnets to Orpheus, I, 3

Geometry and Theology

PA lottery May 14, 1998:
256

S₈ The group of all projectivities and correlations of PG(3,2).

The above isomorphism implies the geometry of the Mathieu group M₂₄.

"The Leech lattice is a blown-up version of
S(5,8,24)."
— W. Feit

"We have strong evidence that the creator of the universe loves symmetry."
— Freeman Dyson

"Mackey presents eight axioms from which he deduces the [quantum] theory."
— M. Schechter

"Theology is about words; science is about things."
— Freeman Dyson, New York Review of Books, 5/28/98

What is "256" about?

Tape purchased 12/23/97:

Django
Reinhardt
Gypsy Jazz

"In the middle of 1982 Janine there are pages in which Jock McLeish is fighting with drugs and alcohol, attempting to either die or come through and get free of his fantasies. In his delirium, he hears the voice of God, which enters in small print, pushing against the larger type of his ravings. Something God says is repeated on the first and last pages of Unlikely Stories, Mostly, complete with illustration and the words 'Scotland 1984' beside it. God's statement is 'Work as if you were in the early days of a better nation.' It is the inherent optimism in that statement that perhaps best captures the strength of Aladair Gray's fiction, its straightforwardness and exuberance."
— Toby Olson, "Eros in Glasgow," in Book World, The Washington Post, December 16, 1984

For another look at angels, see "Winging It," by Christopher R. Miller, The New York Times Book Review Bookend page for Sunday, May 24, 1998. May 24 is the feast day of Sara (also known by the Hindu name Kali), patron saint of Gypsies.

For another, later (July 16, 1998) reply to Dyson, from a source better known than myself, see Why Religion Matters, by Huston Smith, Harper Collins, 2001, page 66.

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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