Log24

Tuesday, October 3, 2006

Tuesday October 3, 2006

Filed under: General,Geometry — Tags: , , , — 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.

Saturday, July 29, 2006

Saturday July 29, 2006

Filed under: General,Geometry — Tags: , , , , — m759 @ 2:02 pm

Big Rock

Thanks to Ars Mathematicaa link to everything2.com:

“In mathematics, a big rock is a result which is vastly more powerful than is needed to solve the problem being considered. Often it has a difficult, technical proof whose methods are not related to those of the field in which it is applied. You say ‘I’m going to hit this problem with a big rock.’ Sard’s theorem is a good example of a big rock.”

Another example:

Properties of the Monster Group of R. L. Griess, Jr., may be investigated with the aid of the Miracle Octad Generator, or MOG, of R. T. Curtis.  See the MOG on the cover of a book by Griess about some of the 20 sporadic groups involved in the Monster:

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

The MOG, in turn, illustrates (via Abstract 79T-A37, Notices of the American Mathematical Society, February 1979) the fact that the group of automorphisms of the affine space of four dimensions over the two-element field is also the natural group of automorphisms of an arbitrary 4×4 array.

This affine group, of order 322,560, is also the natural group of automorphisms of a family of graphic designs similar to those on traditional American quilts.  (See the diamond theorem.)

This top-down approach to the diamond theorem may serve as an illustration of the “big rock” in mathematics.

For a somewhat simpler, bottom-up, approach to the theorem, see Theme and Variations.

For related literary material, see Mathematics and Narrative and The Diamond as Big as the Monster.

“The rock cannot be broken.
It is the truth.”

Wallace Stevens,
“Credences of Summer”

 

Sunday, May 7, 2006

Sunday May 7, 2006

Filed under: General,Geometry — Tags: , , — m759 @ 3:00 am

Bagombo Snuff Box
 
(in memory of
Burt Kerr Todd)


“Well, it may be the devil
or it may be the Lord
But you’re gonna have to
serve somebody.”

— “Bob Dylan”
(pseudonym of Robert Zimmerman),
quoted by “Bob Stewart”
on July 18, 2005

“Bob Stewart” may or may not be the same person as “crankbuster,” author of the “Rectangular Array Theorem” or “RAT.”  This “theorem” is intended as a parody of the “Miracle Octad Generator,” or “MOG,” of R. T. Curtis.  (See the Usenet group sci.math, “Steven Cullinane is a Crank,” July 2005, messages 51-60.)

“Crankbuster” has registered at Math Forum as a teacher in Sri Lanka (formerly Ceylon).   For a tall tale involving Ceylon, see the short story “Bagombo Snuff Box” in the book of the same title by Kurt Vonnegut, who has at times embodied– like Martin Gardner and “crankbuster“– “der Geist, der stets verneint.”

Here is my own version (given the alleged Ceylon background of “crankbuster”) of a Bagombo snuff box:

Related material:

Log24 entries of
April 16-30, 2005,

and the 5 Log24 entries
ending on Friday,
April 28, 2006.

Sunday, March 26, 2006

Sunday March 26, 2006

Filed under: General,Geometry — Tags: , — m759 @ 2:02 pm
'Nauts

(continued from
Life of the Party, March 24)

Exhibit A —

From (presumably) a Princeton student
(see Activity, March 24):

The image “http://www.log24.com/log/pix06/060324-Activity.jpg” cannot be displayed, because it contains errors.

Exhibit B —

From today's Sunday comics:

The image “http://www.log24.com/log/pix06/060326-Blondie2.gif” cannot be displayed, because it contains errors.

Exhibit C —

From a Smith student with the
same name as the Princeton student
(i.e., Dagwood's "Twisterooni" twin):

The image “http://www.log24.com/log/pix06/060326-Smith.jpg” cannot be displayed, because it contains errors.

Related illustrations
("Visual Stimuli") from
the Smith student's game —

The image “http://www.log24.com/log/pix06/060326-Psychonauts1.jpg” cannot be displayed, because it contains errors.

Literary Exercise:

Continuing the Smith student's
Psychonauts theme,
compare and contrast
two novels dealing with
similar topics:

A Wrinkle in Time,
by the Christian author
Madeleine L'Engle,
and
Psychoshop,
by the secular authors
Alfred Bester and
Roger Zelazny.

Presumably the Princeton student
would prefer the Christian fantasy,
the Smith student the secular.

Those who prefer reality to fantasy —
not as numerous as one might think —
may examine what both 4×4 arrays
illustrated above have in common:
their structure.

Both Princeton and Smith might benefit
from an application of Plato's dictum:

The image “http://www.log24.com/theory/images/motto2.jpg” cannot be displayed, because it contains errors.

Thursday, January 26, 2006

Thursday January 26, 2006

Filed under: General,Geometry — m759 @ 9:00 am
In honor of Paul Newman’s age today, 81:

On Beauty

Elaine Scarry, On Beauty (pdf), page 21:

“Something beautiful fills the mind yet invites the search for something beyond itself, something larger or something of the same scale with which it needs to be brought into relation. Beauty, according to its critics, causes us to gape and suspend all thought. This complaint is manifestly true: Odysseus does stand marveling before the palm; Odysseus is similarly incapacitated in front of Nausicaa; and Odysseus will soon, in Book 7, stand ‘gazing,’ in much the same way, at the season-immune orchards of King Alcinous, the pears, apples, and figs that bud on one branch while ripening on another, so that never during the cycling year do they cease to be in flower and in fruit. But simultaneously what is beautiful prompts the mind to move chronologically back in the search for precedents and parallels, to move forward into new acts of creation, to move conceptually over, to bring things into relation, and does all this with a kind of urgency as though one’s life depended on it.”

The image “http://www.log24.com/theory/images/grid3x3.gif” cannot be displayed, because it contains errors.

The above symbol of Apollo suggests, in accordance with Scarry’s remarks, larger structures.   Two obvious structures are the affine 4-space over GF(3), with 81 points, and the affine plane over GF(32), also with 81 points.  Less obvious are some related projective structures.  Joseph Malkevitch has discussed the standard method of constructing GF(32) and the affine plane over that field, with 81 points, then constructing the related Desarguesian projective plane of order 9, with 92 + 9 + 1 = 91 points and 91 lines.  There are other, non-Desarguesian, projective planes of order 9.  See Visualizing GL(2,p), which discusses a spreadset construction of the non-Desarguesian translation plane of order 9.  This plane may be viewed as illustrating deeper properties of the 3×3 array shown above. To view the plane in a wider context, see The Non-Desarguesian Translation Plane of Order 9 and a paper on Affine and Projective Planes (pdf). (Click to enlarge the excerpt beow).

The image “http://www.log24.com/theory/images/060126-planes2.jpg” cannot be displayed, because it contains errors.

See also Miniquaternion Geometry: The Four Projective Planes of Order 9 (pdf), by Katie Gorder (Dec. 5, 2003), and a book she cites:

Miniquaternion geometry: An introduction to the study of projective planes, by T. G. Room and P. B. Kirkpatrick. Cambridge Tracts in Mathematics and Mathematical Physics, No. 60. Cambridge University Press, London, 1971. viii+176 pp.

For “miniquaternions” of a different sort, see my entry on Visible Mathematics for Hamilton’s birthday last year:

The image “http://www.log24.com/theory/images/Quaternions2.jpg” cannot be displayed, because it contains errors.

 

Sunday, January 15, 2006

Sunday January 15, 2006

Filed under: General,Geometry — Tags: , — m759 @ 7:59 am

Inscape

My entry for New Year's Day links to a paper by Robert T. Curtis*
from The Arabian Journal for Science and Engineering
(King Fahd University, Dhahran, Saudi Arabia),
Volume 27, Number 1A, January 2002.

From that paper:

"Combinatorially, an outer automorphism [of S6] can exist because the number of unordered pairs of 6 letters is equal to the number of ways in which 6 letters can be partitioned into three pairs. Which is to say that the two conjugacy classes of odd permutations of order 2 in S6 contain the same number of elements, namely 15. Sylvester… refers to the unordered pairs as duads and the partitions as synthemes. Certain collections of five synthemes… he refers to as synthematic totals or simply totals; each total is stabilized within S6 by a subgroup acting triply transitively on the 6 letters as PGL2(5) acts on the projective line. If we draw a bipartite graph on (15+15) vertices by joining each syntheme to the three duads it contains, we obtain the famous 8-cage (a graph of valence 3 with minimal cycles of length 8)…."

Here is a way of picturing the 8-cage and a related configuration of points and lines:

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

Diamond Theory shows that this structure
can also be modeled by an "inscape"
made up of subsets of a
4×4 square array:

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

The illustration below shows how the
points and lines of the inscape may
be identified with those of the
Cremona-Richmond configuration.

The image “http://www.log24.com/theory/images/Inscape2.gif” cannot be displayed, because it contains errors.

* "A fresh approach to the exceptional automorphism and covers of the symmetric groups"

Wednesday, December 14, 2005

Wednesday December 14, 2005

Filed under: General — Tags: — m759 @ 1:00 am
From Here
to Eternity

For Loomis Dean

The image “http://www.log24.com/log/pix05B/051214-MorenoCover.jpg” cannot be displayed, because it contains errors.

See also
For Rita Moreno
on Her Birthday

(Dec. 11, 2005)

Los Angeles Times
Tuesday, Dec. 13, 2005

OBITUARIES

The image “http://www.log24.com/log/pix05B/051214-LoomisDean.jpg” cannot be displayed, because it contains errors.

LOOMIS DEAN
After many years at Life magazine,
he continued to find steady work
as a freelancer and as a still
photographer on film sets.
(Dean Family)

Loomis Dean, 88;
Life Magazine Photographer
Known for Pictures of
Celebrities and Royalty

By Jon Thurber, Times Staff Writer

Loomis Dean, a Life magazine photographer who made memorable pictures of the royalty of both Europe and Hollywood, has died. He was 88.

Dean died Wednesday [December 7, 2005] at Sonoma Valley Hospital in Sonoma, Calif., of complications from a stroke, according to his son, Christopher.

In a photographic career spanning six decades, Dean's leading images included shirtless Hollywood mogul Darryl F. Zanuck trying a one-handed chin-up on a trapeze bar, the ocean liner Andrea Doria listing in the Atlantic and writer Ernest Hemingway in Spain the year before he committed suicide. One of his most memorable photographs for Life was of cosmopolitan British playwright and composer Noel Coward in the unlikely setting of the Nevada desert.

Dean shot 52 covers for Life, either as a freelance photographer or during his two stretches as a staffer with the magazine, 1947-61 and 1966-69. After leaving the magazine, Dean found steady freelance work in magazines and as a still photographer on film sets, including several of the early James Bond movies starring Sean Connery.

Born in Monticello, Fla., Dean was the son of a grocer and a schoolteacher.

When the Dean family's business failed during the Depression, they moved to Sarasota, Fla., where Dean's father worked as a curator and guide at the John and Mable Ringling Museum of Art.

Dean studied engineering at the University of Florida but became fascinated with photography after watching a friend develop film in a darkroom. He went off to what is now the Rochester Institute of Technology, which was known for its photography school.

After earning his degree, Dean went to work for the Ringling circus as a junior press agent and, according to his son, cultivated a side job photographing Ringling's vast array of performers and workers.

He worked briefly as one of Parade magazine's first photographers but left after receiving an Army Air Forces commission during World War II. During the war, he worked in aerial reconnaissance in the Pacific and was along on a number of air raids over Japan.

His first assignment for Life in 1946 took him back to the circus: His photograph of clown Lou Jacobs with a giraffe looking over his shoulder made the magazine's cover and earned Dean a staff job.

In the era before television, Life magazine photographers had some of the most glamorous work in journalism. Life assigned him to cover Hollywood. In 1954, the magazine published one of his most memorable photos, the shot of Coward dressed for a night on the town in New York but standing alone in the stark Nevada desert.

Dean had the idea of asking Coward, who was then doing a summer engagement at the Desert Inn in Las Vegas, to pose in the desert to illustrate his song "Mad Dogs and Englishmen Go Out in the Midday Sun."

As Dean recalled in an interview with John Loengard for the book "Life Photographers: What They Saw," Coward wasn't about to partake of the midday sun. "Oh, dear boy, I don't get up until 4 o'clock in the afternoon," Dean recalled him saying.

But Dean pressed on anyway. As he related to Loengard, he rented a Cadillac limousine and filled the back seat with a tub loaded with liquor, tonic and ice cubes — and Coward.

The temperature that day reached 119 as Coward relaxed in his underwear during the drive to a spot about 15 miles from Las Vegas. According to Dean, Coward's dresser helped him into his tuxedo, resulting in the image of the elegant Coward with a cigarette holder in his mouth against his shadow on the dry lake bed.

"Splendid! Splendid! What an idea! If we only had a piano," Coward said of the shoot before hopping back in the car and stripping down to his underwear for the ride back to Las Vegas.

In 1956, Life assigned Dean to Paris. While sailing to Europe on the Ile de France, he was awakened with the news that the Andrea Doria had collided with another liner, the Stockholm.

The accident occurred close enough to Dean's liner that survivors were being brought aboard.

His photographs of the shaken voyagers and the sinking Andrea Doria were some of the first on the accident published in a U.S. magazine.

During his years in Europe, Dean photographed communist riots and fashion shows in Paris, royal weddings throughout Europe and noted authors including James Jones and William S. Burroughs.

He spent three weeks with Hemingway in Spain in 1960 for an assignment on bullfighting. In 1989, Dean published "Hemingway's Spain," about his experiences with the great writer.

In 1965, Dean won first prize in a Vatican photography contest for a picture of Pope Paul VI. The prize included an audience with the pope and $750. According to his son, it was Dean's favorite honor.

In addition to his son, he is survived by a daughter, Deborah, and two grandsons.

Instead of flowers, donations may be made to the American Child Photographer's Charity Guild (www.acpcg.com) or the Make-A-Wish Foundation.

Related material:
The Big Time

(Log 24, July 29, 2003):

A Story That Works

 
  • "There is the dark, eternally silent, unknown universe;
  • there are the friend-enemy minds shouting and whispering their tales and always seeking the three miracles —

    • that minds should really touch, or
    • that the silent universe should speak, tell minds a story, or (perhaps the same thing)
    • that there should be a story that works, that is all hard facts, all reality, with no illusions and no fantasy;
  • and lastly, there is lonely, story-telling, wonder-questing, mortal me."

    Fritz Leiber in "The Button Molder"

 

Sunday, November 20, 2005

Sunday November 20, 2005

Filed under: General,Geometry — Tags: , — m759 @ 4:04 pm

An Exercise
of Power

Johnny Cash:
“And behold,
a white horse.”

The image “http://www.log24.com/log/pix05B/051120-SpringerLogo9.gif” cannot be displayed, because it contains errors.
Adapted from
illustration below:

The image “http://www.log24.com/log/pix05B/051120-NonEuclideanRev.jpg” cannot be displayed, because it contains errors.

“There is a pleasantly discursive treatment of Pontius Pilate’s unanswered question ‘What is truth?'”

H. S. M. Coxeter, 1987, introduction to Richard J. Trudeau’s remarks on the “Story Theory” of truth as opposed to  the “Diamond Theory” of truth in The Non-Euclidean Revolution

“A new epistemology is emerging to replace the Diamond Theory of truth. I will call it the ‘Story Theory’ of truth: There are no diamonds. People make up stories about what they experience. Stories that catch on are called ‘true.’ The Story Theory of truth is itself a story that is catching on. It is being told and retold, with increasing frequency, by thinkers of many stripes*….”

Richard J. Trudeau in
The Non-Euclidean Revolution

“‘Deniers’ of truth… insist that each of us is trapped in his own point of view; we make up stories about the world and, in an exercise of power, try to impose them on others.”

— Jim Holt in The New Yorker.

(Click on the box below.)

The image “http://www.log24.com/log/pix05B/050819-Critic4.jpg” cannot be displayed, because it contains errors.

Exercise of Power:

Show that a white horse–

A Singer 7-Cycle

a figure not unlike the
symbol of the mathematics
publisher Springer–
is traced, within a naturally
arranged rectangular array of
polynomials, by the powers of x
modulo a polynomial
irreducible over a Galois field.

This horse, or chess knight–
“Springer,” in German–
plays a role in “Diamond Theory”
(a phrase used in finite geometry
in 1976, some years before its use
by Trudeau in the above book).

Related material

On this date:

 In 1490, The White Knight
 (Tirant lo Blanc The image “http://www.log24.com/images/asterisk8.gif” cannot be displayed, because it contains errors. )–
a major influence on Cervantes–
was published, and in 1910

The image “http://www.log24.com/log/pix05B/051120-Caballo1.jpg” cannot be displayed, because it contains errors.

the Mexican Revolution began.

Illustration:
Zapata by Diego Rivera,
Museum of Modern Art,
New York

The image “http://www.log24.com/images/asterisk8.gif” cannot be displayed, because it contains errors. Description from Amazon.com

“First published in the Catalan language in Valencia in 1490…. Reviewing the first modern Spanish translation in 1969 (Franco had ruthlessly suppressed the Catalan language and literature), Mario Vargas Llosa hailed the epic’s author as ‘the first of that lineage of God-supplanters– Fielding, Balzac, Dickens, Flaubert, Tolstoy, Joyce, Faulkner– who try to create in their novels an all-encompassing reality.'”

Monday, August 22, 2005

Monday August 22, 2005

Filed under: General — Tags: , , — m759 @ 4:07 pm
The Hole

Part I: Mathematics and Narrative

The image “http://www.log24.com/log/pix05B/050822-Narr.jpg” cannot be displayed, because it contains errors.

Apostolos Doxiadis on last month's conference on "mathematics and narrative"–

Doxiadis is describing how talks by two noted mathematicians were related to

    "… a sense of a 'general theory bubbling up' at the meeting… a general theory of the deeper relationship of mathematics to narrative…. "

Doxiadis says both talks had "a big hole in the middle."  

    "Both began by saying something like: 'I believe there is an important connection between story and mathematical thinking. So, my talk has two parts.  [In one part] I’ll tell you a few things about proofs.  [And in the other part] I’ll tell you about stories.' …. And in both talks it was in fact implied by a variation of the post hoc propter hoc, the principle of consecutiveness implying causality, that the two parts of the lectures were intimately related, the one somehow led directly to the other."
  "And the hole?"
  "This was exactly at the point of the link… [connecting math and narrative]… There is this very well-known Sidney Harris cartoon… where two huge arrays of formulas on a blackboard are connected by the sentence ‘THEN A MIRACLE OCCURS.’ And one of the two mathematicians standing before it points at this and tells the other: ‘I think you should be more explicit here at step two.’ Both… talks were one half fascinating expositions of lay narratology– in fact, I was exhilarated to hear the two most purely narratological talks at the meeting coming from number theorists!– and one half a discussion of a purely mathematical kind, the two parts separated by a conjunction roughly synonymous to ‘this is very similar to this.’  But the similarity was not clearly explained: the hole, you see, the ‘miracle.’  Of course, both [speakers]… are brilliant men, and honest too, and so they were very clear about the location of the hole, they did not try to fool us by saying that there was no hole where there was one."
 

Part II: Possible Worlds

"At times, bullshit can only be countered with superior bullshit."
Norman Mailer

Many Worlds and Possible Worlds in Literature and Art, in Wikipedia:

    "The concept of possible worlds dates back to a least Leibniz who in his Théodicée tries to justify the apparent imperfections of the world by claiming that it is optimal among all possible worlds.  Voltaire satirized this view in his picaresque novel Candide….
    Borges' seminal short story El jardín de senderos que se bifurcan ("The Garden of Forking Paths") is an early example of many worlds in fiction."

Background:

Modal Logic in Wikipedia

Possible Worlds in Wikipedia

Possible-Worlds Theory, by Marie-Laure Ryan
(entry for The Routledge Encyclopedia of Narrative Theory)

The God-Shaped Hole
 

Part III: Modal Theology

 
  "'What is this Stone?' Chloe asked….
  '…It is told that, when the Merciful One made the worlds, first of all He created that Stone and gave it to the Divine One whom the Jews call Shekinah, and as she gazed upon it the universes arose and had being.'"

  — Many Dimensions, by Charles Williams, 1931 (Eerdmans paperback, April 1979, pp. 43-44)


"The lapis was thought of as a unity and therefore often stands for the prima materia in general."

  — Aion, by C. G. Jung, 1951 (Princeton paperback, 1979, p. 236)

 

"Its discoverer was of the opinion that he had produced the equivalent of the primordial protomatter which exploded into the Universe."

  — The Stars My Destination, by Alfred Bester, 1956 (Vintage hardcover, July 1996, p. 216)
 
"We symbolize
logical necessity
with the box (box.gif (75 bytes))
and logical possibility
with the diamond (diamond.gif (82 bytes))."

 

 

Keith Allen Korcz 

The image “http://www.log24.com/log/pix05B/050802-Stone.gif” cannot be displayed, because it contains errors.

"The possibilia that exist,
and out of which
the Universe arose,
are located in
     a necessary being…."

Michael Sudduth,
Notes on
God, Chance, and Necessity
by Keith Ward,
Regius Professor of Divinity
at Christ Church College, Oxford
(the home of Lewis Carroll)

Wednesday, June 8, 2005

Wednesday June 8, 2005

Filed under: General — Tags: — m759 @ 11:11 am
Quest

“Mike Nichols, who oversaw Monty Python’s Spamalot, picked up the prize for directing a musical.

A somewhat flustered Nichols told the audience he had forgotten what he intended to say, but then went on to thank his company and Eric Idle, ‘from whom all blessings flow.'”

The Age, Monday, June 6

One of my
favorite books:

The image “http://www.log24.com/log/pix05A/050608-Quest.jpg” cannot be displayed, because it contains errors.

Excerpt from the chapter
“All Blessings Flow (Very Large Array)”–

“I started to cry.  My search was over.
In a home for the deranged I had found
the last of the holy Thirty-Six….

‘Beam me up, Scotty.'”

The image “http://www.log24.com/log/pix05A/050608-Contact.gif” cannot be displayed, because it contains errors.

Related material:

In memory of Anne Bancroft
and her work in
84 Charing Cross Road

entries of Dec. 11-13, 2002,
and entries of
All Souls’ Day, 2004,
and of June 8, 2003.

Friday, May 27, 2005

Friday May 27, 2005

Filed under: General,Geometry — m759 @ 12:25 pm
Drama of the Diagonal,
Part Deux

Wednesday’s entry The Turning discussed a work by Roger Cooke.  Cooke presents a

“fanciful story (based on Plato’s dialogue Meno).”

The History of Mathematics is the title of the Cooke book.

Associated Press thought for today:

“History is not, of course, a cookbook offering pretested recipes. It teaches by analogy, not by maxims. It can illuminate the consequences of actions in comparable situations, yet each generation must discover for itself what situations are in fact comparable.”
 — Henry Kissinger (whose birthday is today)

For Henry Kissinger on his birthday:
a link to Geometry for Jews.

This link suggests a search for material
on the art of Sol LeWitt, which leads to
an article by Barry Cipra,
The “Sol LeWitt” Puzzle:
A Problem in 16 Squares
(ps),
a discussion of a 4×4 array
of square linear designs.
  Cipra says that

“If you like, there are three symmetry groups lurking within the LeWitt puzzle:  the rotation/reflection group of order 8, a toroidal group of order 16, and an ‘existential’* group of order 16.  The first group is the most obvious.  The third, once you see it, is also obvious.”

* Jean-Paul Sartre,
  Being and Nothingness,
  Philosophical Library, 1956
  [reference by Cipra]

For another famous group lurking near, if not within, a 4×4 array, click on Kissinger’s birthday link above.

Kissinger’s remark (above) on analogy suggests the following analogy to the previous entry’s (Drama of the Diagonal) figure:
 

  The image “http://www.log24.com/log/pix05/021126-diagonH2.jpg” cannot be displayed, because it contains errors.

Logos Alogos II:
Horizon

This figure in turn, together with Cipra’s reference to Sartre, suggests the following excerpts (via Amazon.com)–

From Sartre’s Being and Nothingness, translated by Hazel E. Barnes, 1993 Washington Square Press reprint edition:

1. on Page 51:
“He makes himself known to himself from the other side of the world and he looks from the horizon toward himself to recover his inner being.  Man is ‘a being of distances.'”
2. on Page 154:
“… impossible, for the for-itself attained by the realization of the Possible will make itself be as for-itself–that is, with another horizon of possibilities.  Hence the constant disappointment which accompanies repletion, the famous: ‘Is it only this?’….”
3. on Page 155:
“… end of the desires.  But the possible repletion appears as a non-positional correlate of the non-thetic self-consciousness on the horizon of the  glass-in-the-midst-of-the-world.”
4. on Page 158:
“…  it is in time that my possibilities appear on the horizon of the world which they make mine.  If, then, human reality is itself apprehended as temporal….”
5. on Page 180:
“… else time is an illusion and chronology disguises a strictly logical order of  deducibility.  If the future is pre-outlined on the horizon of the world, this can be only by a being which is its own future; that is, which is to come….”
6. on Page 186:
“…  It appears on the horizon to announce to me what I am from the standpoint of what I shall be.”
7. on Page 332:
“… the boat or the yacht to be overtaken, and the entire world (spectators, performance, etc.) which is profiled on the horizon.  It is on the common ground of this co-existence that the abrupt revelation of my ‘being-unto-death’….”
8. on Page 359:
“… eyes as objects which manifest the look.  The Other can not even be the object aimed at emptily at the horizon of my being for the Other.”
9. on Page 392:
“… defending and against which he was leaning as against a wail, suddenly opens fan-wise and becomes the foreground, the welcoming horizon toward which he is fleeing for refuge.”
10.  on Page 502:
“… desires her in so far as this sleep appears on the ground of consciousness. Consciousness therefore remains always at the horizon of the desired body; it makes the meaning and the unity of the body.”
11.  on Page 506:
“… itself body in order to appropriate the Other’s body apprehended as an organic totality in situation with consciousness on the horizon— what then is the meaning of desire?”
12.  on Page 661:
“I was already outlining an interpretation of his reply; I transported myself already to the four corners of the horizon, ready to return from there to Pierre in order to understand him.”
13.  on Page 754:
“Thus to the extent that I appear to myself as creating objects by the sole relation of appropriation, these objects are myself.  The pen and the pipe, the clothing, the desk, the house– are myself.  The totality of my possessions reflects the totality of my being.  I am what I have.  It is I myself which I touch in this cup, in this trinket.  This mountain which I climb is myself to the extent that I conquer it; and when I am at its summit, which I have ‘achieved’ at the cost of this same effort, when I attain this magnificent view of the valley and the surrounding peaks, then I am the view; the panorama is myself dilated to the horizon, for it exists only through me, only for me.”

Illustration of the
last horizon remark:

The image “http://www.log24.com/log/pix05/050527-CipraLogo.gif” cannot be displayed, because it contains errors.

The image “http://www.log24.com/log/pix05/050527-CIPRAview.jpg” cannot be displayed, because it contains errors.
 
From CIPRA – Slovenia,
the Institute for the
Protection of the Alps

For more on the horizon, being, and nothingness, see

Sunday, November 21, 2004

Sunday November 21, 2004

Filed under: General — Tags: — m759 @ 3:00 pm

Trinity and Counterpoint

Today's Roman Catholic meditation is from Gerry Adams, leader of Sinn Fein, the political arm of the Irish Republican Army:

"I certainly regret what happened and I make no bones about that," Adams said on the 30th anniversary of pub bombings that killed 21 on Nov. 21, 1974, in Birmingham, England.

Those who care what Roman Catholics think of the Trinity may read the remarks of St. Bonaventure at math16.com.

That site also offers a less holy but more intelligible trinity based on the irrefutable fact that 3 x 8 = 24 and on a remarkable counterpoint between group actions on a 4×2 array and group actions on a 4×4 array.

For a Protestant view of this trinity, see a website at the University of Birmingham in England.

That site's home page links to Birmingham's City Evangelical Church.
 

Tuesday, November 16, 2004

Tuesday November 16, 2004

Filed under: General,Geometry — m759 @ 12:12 pm

Geometry, continued

Added a long footnote on symplectic properties of the 4×4 array to “Geometry of the 4×4 Square.”

Wednesday, November 12, 2003

Wednesday November 12, 2003

Filed under: General,Geometry — Tags: — m759 @ 9:58 am

The Silver Table

“And suddenly all was changed.  I saw a great assembly of gigantic forms all motionless, all in deepest silence, standing forever about a little silver table and looking upon it.  And on the table there were little figures like chessmen who went to and fro doing this and that.  And I knew that each chessman was the idolum or puppet representative of some one of the great presences that stood by.  And the acts and motions of each chessman were a moving portrait, a mimicry or pantomine, which delineated the inmost nature of his giant master.  And these chessmen are men and women as they appear to themselves and to one another in this world.  And the silver table is Time.  And those who stand and watch are the immortal souls of those same men and women.  Then vertigo and terror seized me and, clutching at my Teacher, I said, ‘Is that the truth?….’ ”

— C.S. Lewis, The Great Divorce, final chapter

Follow-up to the previous four entries:

St. Art Carney, whom we may imagine to be a passenger on the heavenly bus in The Great Divorce, died on Sunday, Nov. 9, 2003.

The entry for that date (Weyl’s birthday) asks for the order of the automorphism group of a 4×4 array.  For a generalization to an 8×8 array — i.e., a chessboard — see

Geometry of the I Ching.

Audrey Meadows, said to have been the youngest daughter of her family, was born in Wuchang, China.

Tui: The Youngest Daughter

“Tui means to ‘give joy.’  Tui leads the common folk and with joy they forget their toil and even their fear of death. She is sometimes also called a sorceress because of her association with the gathering yin energy of approaching winter.  She is a symbol of the West and autumn, the place and time of death.”

Paraphrase of Book III, Commentaries of Wilhelm/Baynes.

Tuesday, November 11, 2003

Tuesday November 11, 2003

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

11:11

“Why do we remember the past
but not the future?”

— Stephen Hawking,
A Brief History of Time,
Ch. 9, “The Arrow of Time”

For another look at
the arrow of time, see

Time Fold.

Imaginary Time: The Concept

The flow of imaginary time is at right angles to that of ordinary time.“Imaginary time is a relatively simple concept that is rather difficult to visualize or conceptualize. In essence, it is another direction of time moving at right angles to ordinary time. In the image at right, the light gray lines represent ordinary time flowing from left to right – past to future. The dark gray lines depict imaginary time, moving at right angles to ordinary time.”

Is Time Quantized?

Yes.

Maybe.

We don’t really know.

Let us suppose, for the sake of argument, that time is in fact quantized and two-dimensional.  Then the following picture,

from Time Fold, of “four quartets” time, of use in the study of poetry and myth, might, in fact, be of use also in theoretical physics.

In this event, last Sunday’s entry, on the symmetry group of a generic 4×4 array, might also have some physical significance.

At any rate, the Hawking quotation above suggests the following remarks from T. S. Eliot’s own brief history of time, Four Quartets:

“It seems, as one becomes older,
That the past has another pattern,
and ceases to be a mere sequence….

I sometimes wonder if that is
what Krishna meant—
Among other things—or one way
of putting the same thing:
That the future is a faded song,
a Royal Rose or a lavender spray
Of wistful regret for those who are
not yet here to regret,
Pressed between yellow leaves
of a book that has never been opened.
And the way up is the way down,
the way forward is the way back.”

Related reading:

The Wisdom of Old Age and

Poetry, Language, Thought.

Sunday, November 9, 2003

Sunday November 9, 2003

Filed under: General,Geometry — Tags: — m759 @ 5:00 pm

For Hermann Weyl's Birthday:

A Structure-Endowed Entity

"A guiding principle in modern mathematics is this lesson: Whenever you have to do with a structure-endowed entity S, try to determine its group of automorphisms, the group of those element-wise transformations which leave all structural relations undisturbed. You can expect to gain a deep insight into the constitution of S in this way."

— Hermann Weyl in Symmetry

Exercise:  Apply Weyl's lesson to the following "structure-endowed entity."

4x4 array of dots

What is the order of the resulting group of automorphisms? (The answer will, of course, depend on which aspects of the array's structure you choose to examine.  It could be in the hundreds, or in the hundreds of thousands.)

Wednesday, September 3, 2003

Wednesday September 3, 2003

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

Reciprocity

From my entry of Sept. 1, 2003:

"…the principle of taking and giving, of learning and teaching, of listening and storytelling, in a word: of reciprocity….

… E. M. Forster famously advised his readers, 'Only connect.' 'Reciprocity' would be Michael Kruger's succinct philosophy, with all that the word implies."

— William Boyd, review of Himmelfarb, New York Times Book Review, October 30, 1994

Last year's entry on this date: 

Today's birthday:
James Joseph Sylvester

"Mathematics is the music of reason."
— J. J. Sylvester

Sylvester, a nineteenth-century mathematician, coined the phrase "synthematic totals" to describe some structures based on 6-element sets that R. T. Curtis has called "rather unwieldy objects." See Curtis's abstract, Symmetric Generation of Finite Groups, John Baez's essay, Some Thoughts on the Number 6, and my website, Diamond Theory.

The picture above is of the complete graph K6  Six points with an edge connecting every pair of points… Fifteen edges in all.

Diamond theory describes how the 15 two-element subsets of a six-element set (represented by edges in the picture above) may be arranged as 15 of the 16 parts of a 4×4 array, and how such an array relates to group-theoretic concepts, including Sylvester's synthematic totals as they relate to constructions of the Mathieu group M24.

If diamond theory illustrates any general philosophical principle, it is probably the interplay of opposites….  "Reciprocity" in the sense of Lao Tzu.  See

Reciprocity and Reversal in Lao Tzu.

For a sense of "reciprocity" more closely related to Michael Kruger's alleged philosophy, see the Confucian concept of Shu (Analects 15:23 or 24) described in

Shu: Reciprocity.

Kruger's novel is in part about a Jew: the quintessential Jewish symbol, the star of David, embedded in the K6 graph above, expresses the reciprocity of male and female, as my May 2003 archives illustrate.  The star of David also appears as part of a graphic design for cubes that illustrate the concepts of diamond theory:

Click on the design for details.

Those who prefer a Jewish approach to physics can find the star of David, in the form of K6, applied to the sixteen 4×4 Dirac matrices, in

A Graphical Representation
of the Dirac Algebra
.

The star of David also appears, if only as a heuristic arrangement, in a note that shows generating partitions of the affine group on 64 points arranged in two opposing triplets.

Having thus, as the New York Times advises, paid tribute to a Jewish symbol, we may note, in closing, a much more sophisticated and subtle concept of reciprocity due to Euler, Legendre, and Gauss.  See

The Jewel of Arithmetic and

The Golden Theorem.

Sunday, August 17, 2003

Sunday August 17, 2003

Filed under: General,Geometry — Tags: , — m759 @ 6:21 pm

Diamond theory is the theory of affine groups over GF(2) acting on small square and cubic arrays. In the simplest case, the symmetric group of degree 4 acts on a two-colored diamond figure like that in Plato's Meno dialogue, yielding 24 distinct patterns, each of which has some ordinary or color-interchange symmetry .

This symmetry invariance can be generalized to (at least) a group of order approximately 1.3 trillion acting on a 4x4x4 array of cubes.

The theory has applications to finite geometry and to the construction of the large Witt design underlying the Mathieu group of degree 24.

Further Reading:

Wednesday, July 30, 2003

Wednesday July 30, 2003

Filed under: General — m759 @ 2:45 am

Transcendental Meditation

This week’s
 New Yorker
:

Transcendental Man
New books on
Ralph Waldo Emerson
for his bicentennial.
by John Updike

This week’s
 Time cover
:

The bicentennial of Ralph Waldo Emerson was on May 25, 2003.  For a commemoration of Emerson on that date, click on the picture below of Harvard University’s Room 305, Emerson Hall.

 

This will lead you to a discussion of the properties of a 5×5 array, or matrix, with a symbol of mystical unity at its center.  Although this symbol of mystical unity, the number “1,” is not, pace the Shema, a transcendental number, the matrix is, as perhaps a sort of Emersonian compensation, what postmodernists would call phallologocentric.  It is possible that Emerson is a saint; if so, his feast day (i.e., date of death), April 27, might reveal to us the sort of miraculous fact hoped for by Fritz Leiber in my previous entry.  A check of my April 27 notes shows us, lo and behold, another phallologocentric 5×5 array, this one starring Warren Beatty.  This rather peculiar coincidence is, perhaps, the sort of miracle appropriate to a saint who is, as this week’s politically correct New Yorker calls him, a Big Dead White Male.

 Leiber’s fiction furnishes “a behind-the-scenes view of the time change wars.”

“It’s quarter to three…” — St. Frank Sinatra

Monday, April 28, 2003

Monday April 28, 2003

Filed under: General,Geometry — Tags: , , — m759 @ 12:07 am

ART WARS:

Toward Eternity

April is Poetry Month, according to the Academy of American Poets.  It is also Mathematics Awareness Month, funded by the National Security Agency; this year's theme is "Mathematics and Art."

Some previous journal entries for this month seem to be summarized by Emily Dickinson's remarks:

"Because I could not stop for Death–
He kindly stopped for me–
The Carriage held but just Ourselves–
And Immortality.

………………………
Since then–'tis Centuries–and yet
Feels shorter than the Day
I first surmised the Horses' Heads
Were toward Eternity– "

 

Consider the following journal entries from April 7, 2003:
 

Math Awareness Month

April is Math Awareness Month.
This year's theme is "mathematics and art."


 

An Offer He Couldn't Refuse

Today's birthday:  Francis Ford Coppola is 64.

"There is a pleasantly discursive treatment
of Pontius Pilate's unanswered question
'What is truth?'."


H. S. M. Coxeter, 1987, introduction to Richard J. Trudeau's remarks on the "Story Theory" of truth as opposed to the "Diamond Theory" of truth in The Non-Euclidean Revolution

 

From a website titled simply Sinatra:

"Then came From Here to Eternity. Sinatra lobbied hard for the role, practically getting on his knees to secure the role of the street smart punk G.I. Maggio. He sensed this was a role that could revive his career, and his instincts were right. There are lots of stories about how Columbia Studio head Harry Cohn was convinced to give the role to Sinatra, the most famous of which is expanded upon in the horse's head sequence in The Godfather. Maybe no one will know the truth about that. The one truth we do know is that the feisty New Jersey actor won the Academy Award as Best Supporting Actor for his work in From Here to Eternity. It was no looking back from then on."

From a note on geometry of April 28, 1985:

 
The "horse's head" figure above is from a note I wrote on this date 18 years ago.  The following journal entry from April 4, 2003, gives some details:
 

The Eight

Today, the fourth day of the fourth month, plays an important part in Katherine Neville's The Eight.  Let us honor this work, perhaps the greatest bad novel of the twentieth century, by reflecting on some properties of the number eight.  Consider eight rectangular cells arranged in an array of four rows and two columns.  Let us label these cells with coordinates, then apply a permutation.

 


 Decimal 
labeling

 
Binary
labeling


Algebraic
labeling


Permutation
labeling

 

The resulting set of arrows that indicate the movement of cells in a permutation (known as a Singer 7-cycle) outlines rather neatly, in view of the chess theme of The Eight, a knight.  This makes as much sense as anything in Neville's fiction, and has the merit of being based on fact.  It also, albeit rather crudely, illustrates the "Mathematics and Art" theme of this year's Mathematics Awareness Month.

The visual appearance of the "knight" permutation is less important than the fact that it leads to a construction (due to R. T. Curtis) of the Mathieu group M24 (via the Curtis Miracle Octad Generator), which in turn leads logically to the Monster group and to related "moonshine" investigations in the theory of modular functions.   See also "Pieces of Eight," by Robert L. Griess.

Friday, April 4, 2003

Friday April 4, 2003

Filed under: General,Geometry — Tags: , , — m759 @ 3:33 pm

The Eight

Today, the fourth day of the fourth month, plays an important part in Katherine Neville's The Eight.  Let us honor this work, perhaps the greatest bad novel of the twentieth century, by reflecting on some properties of the number eight.  Consider eight rectangular cells arranged in an array of four rows and two columns.  Let us label these cells with coordinates, then apply a permutation.


Decimal 
labeling


Binary
labeling


Algebraic
labeling

IMAGE- Knight figure for April 4
Permutation
labeling

 

The resulting set of arrows that indicate the movement of cells in a permutation (known as a Singer 7-cycle) outlines rather neatly, in view of the chess theme of The Eight, a knight.  This makes as much sense as anything in Neville's fiction, and has the merit of being based on fact.  It also, albeit rather crudely, illustrates the "Mathematics and Art" theme of this year's Mathematics Awareness Month.  (See the 4:36 PM entry.)

 

 

The visual appearance of the "knight" permutation is less important than the fact that it leads to a construction (due to R. T. Curtis) of the Mathieu group M24 (via the Curtis Miracle Octad Generator), which in turn leads logically to the Monster group and to related "moonshine" investigations in the theory of modular functions.   See also "Pieces of Eight," by Robert L. Griess.
 

Friday, November 8, 2002

Friday November 8, 2002

Filed under: General,Geometry — m759 @ 3:33 am

Religious Symbolism
at Princeton

In memory of Steve McQueen (“The Great Escape” and “The Thomas Crown Affair”… see preceding entry) and of Rudolf Augstein (publisher of Der Spiegel), both of whom died on November 7 (in 1980 and 2002, respectively), in memory of the following residents of

The Princeton Cemetery
of the Nassau Presbyterian Church
Established 1757

SYLVIA BEACH (1887-1962), whose father was pastor of the First Presbyterian Church, founded Shakespeare & Company, a Paris bookshop which became a focus for struggling expatriate writers. In 1922 she published James Joyce’s Ulysses when others considered it obscene, and she defiantly closed her shop in 1941 in protest against the Nazi occupation.

KURT GÖDEL (1906-1978), a world-class mathematician famous for a vast array of major contributions to logic, was a longtime professor at the Institute for Advanced Study, founded in 1930. He was a corecipient of the Einstein Award in 1951.

JOHN (HENRY) O’HARA (1905-1970) was a voluminous and much-honored writer. His novels, Appointment in Samarra (1934) and Ten North Frederick (1955), and his collection of short stories, Pal Joey (1940), are among his best-known works.

and of the long and powerful association of Princeton University with the Presbyterian Church, as well as the theological perspective of Carl Jung in Man and His Symbols, I offer the following “windmill,” taken from the Presbyterian Creedal Standards website, as a memorial:

The background music Les Moulins de Mon Coeur, selected yesterday morning in memory of Steve McQueen, continues to be appropriate.

“A is for Anna.”
— James Joyce

Saturday, July 20, 2002

Saturday July 20, 2002

 

ABSTRACT: Finite projective geometry explains the surprising symmetry properties of some simple graphic designs– found, for instance, in quilts. Links are provided for applications to sporadic simple groups (via the "Miracle Octad Generator" of R. T. Curtis), to the connection between orthogonal Latin squares and projective spreads, and to symmetry of Walsh functions.

We regard the 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

Walsh Functions

Inscapes

The Diamond Theory of Truth

Geometry of the I Ching

Solomon's Cube and The Eightfold Way

Crystal and Dragon in Diamond Theory

The Form, the Pattern

The Grid of Time

Block Designs

Finite Relativity

Theme and Variations

Models of Finite Geometries

Quilt Geometry

Pattern Groups

The Fano Plane Revisualized,
or the Eightfold Cube

The Miracle Octad Generator

Kaleidoscope

Visualizing GL(2,p)

Jung's Imago

Author's home page

AMS Mathematics Subject Classification:

20B25 (Group theory and generalizations :: Permutation groups :: Finite automorphism groups of algebraic, geometric, or combinatorial structures)

05B25 (Combinatorics :: Designs and configurations :: Finite geometries)

51E20 (Geometry :: Finite geometry and special incidence structures :: Combinatorial structures in finite projective spaces)



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