Log24

Today the World Wide Web turns 20.

See also Galois Memorial and Correspondences.

Continued from March 10, 2011 — A post that says

"If Galois geometry is thought of as a paradigm shift
from Euclidean geometry, both… the Kuhn cover
and the nine-point affine plane may be viewed…
as illustrating the shift."

Yesterday's posts The Fano Entity and Theology for Antichristmas,
together with this morning's New York Times  obituaries (below)—

—suggest a Sunday School review from last year's
    Devil's Night (October 30-31, 2010)—

See also Ash Wednesday Surprise and Geometry for Jews.

The late translator Helen Lane in Translation Review , Vol. 5, 1980—

"Among the awards, I submit, should be one for the entire oeuvre  of a lifetime "senior" translator— and  one for the best first  translation…. Similar organization, cooperation, and fund-finding for a first-rate replacement for the sorely missed Delos ."

This leads to one of the founders of Delos , the late Donald Carne-Ross, who died on January 9, 2010.

For one meditation on the date January 9, see Bridal Birthday (last Thursday).

Another meditation, from the date of Carne-Ross's death—

See also Delos in this journal.

Part I — Unity and Multiplicity
              (Continued from The Talented and Galois Cube)

Part II — "A feeling, an angel, the moon, and Italy"—

Click for details

Today's earlier post mentions one approach to the concepts of unity and multiplicity. Here is another.

Unity:
The 3×3×3 Galois Cube

Multiplicity:

One of a group, GL(3,3), of 11,232
natural transformations of the 3×3×3 Cube

See also the earlier 1985 3×3 version by Cullinane.

For the title, see Palm Sunday.

"There is a pleasantly discursive treatment of
Pontius Pilate's unanswered question 'What is truth?'" — H. S. M. Coxeter, 1987

From this date (April 22) last year—

Richard J. Trudeau in The Non-Euclidean Revolution , chapter on "Geometry and the Diamond Theory of Truth"– "… Plato and Kant, and most of the philosophers and scientists in the 2200-year interval between them, did share the following general presumptions: (1) Diamonds– informative, certain truths about the world– exist. (2) The theorems of Euclidean geometry are diamonds. Presumption (1) is what I referred to earlier as the 'Diamond Theory' of truth. It is far, far older than deductive geometry." Trudeau's book was published in 1987. The non-Euclidean* figures above illustrate concepts from a 1976 monograph, also called "Diamond Theory." Although non-Euclidean,* the theorems of the 1976 "Diamond Theory" are also, in Trudeau's terminology, diamonds. * "Non-Euclidean" here means merely "other than  Euclidean." No violation of Euclid's parallel postulate is implied.

Trudeau comes to reject what he calls the "Diamond Theory" of truth. The trouble with his argument is the phrase "about the world."

Geometry, a part of pure mathematics, is not  about the world. See G. H. Hardy, A Mathematician's Apology .

Today's news from Oslo suggests a review—

Click for further details.

The circular sculpture in the foreground
is called by the artist "The Omega Point."
This has been described as
"a portal that leads in or out of time and space."

Some related philosophical remarks—

Oslo Connection and some notes on Galois connections.

The following has rather mysteriously appeared in a search at Google Scholar for "Steven H. Cullinane."

This turns out to be a link to a search within this weblog. I do not know why Google Scholar attributes the resulting web page to a journal article by "AB Story" or why it drew the title from a post within the search and applied it to the entire list of posts found. I am, however, happy with the result— a Palm Sunday surprise with an eclectic mixture of styles that might please the late Robert de Marrais.

I hope the late George Temple would also be pleased. He appears in "Romancing" as a resident of Quarr Abbey, a Benedictine monastery.

The remarks by Martin Hyland quoted in connection with Temple's work are of particular interest in light of the Pope's Christmas remark on mathematics quoted here yesterday.

From last night's note on finite geometry—

"The (8₃, 8₃) Möbius-Kantor configuration here described by Coxeter is of course part of the larger (9₄, 12₃) Hesse configuration. Simply add the center point of the 3×3 Galois affine plane and the four lines (1 horizontal, 1 vertical, 2 diagonal) through the center point." An illustration—

This suggests a search for "diamond+star."

Indiana Jones and the Magical Oracle

Mathematician Ken Ono in the December 2010 American Mathematical Society Notices—

The "dying genius" here is Ramanujan, not Galois. The story now continues at the AMS website—

      (Excerpt from Jan. 27 screenshot;
      the partitions story has been the top
      news item at the site all week.)

From a Jan. 20, 2011, Emory University press release —
"Finite formula found for partition numbers" —

"We found a function, that we call P, that is like a magical oracle," Ono says. "I can take any number, plug it into P, and instantly calculate the partitions of that number. P does not return gruesome numbers with infinitely many decimal places. It's the finite, algebraic formula that we have all been looking for."

For an introduction to the magical oracle, see a preprint, "Bruinier-Ono," at the American Institute of Mathematics website.

Ono also discussed the oracle in a video (see minute 25) recorded Jan. 21 and placed online today.

See as well "Exact formulas for the partition function?" at mathoverflow.net.

A Nov. 29, 2010, remark by Thomas Bloom on that page leads to a 2006 preprint by Ono and Kathrin Bringmann, "An Arithmetic Formula for the Partition Function*," that seems not unrelated to Ono's new "magical oracle" formula—

Click to enlarge

The Bruinier-Ono paper does not mention the earlier Bringmann-Ono work.

(Both the 2011 Bruinier-Ono paper and the 2006 Bringmann-Ono paper mention their debt to a 2002 work by Zagier—  Don Zagier, "Traces of singular moduli," in Motives, Polylogarithms and Hodge theory, Part II  (Irvine, CA, 1998), International Press Lecture Series 3 (International Press, Somerville, MA, 2002),   pages 211-244.)

Some background for those who prefer mathematics to narrative—
The Web of Modularity: Arithmetic of the Coefficients of Modular Forms and q-Series ,
by Ken Ono, American Mathematical Society CBMS Series, 2004.

* Proc. Amer. Math. Soc. 135 (2007), 3507-3514.

Galois Meets Doctor Faustus

Galois's theory of mathematical  ambiguity (see June 14) —

  My principal meditations for some time have been directed towards
  the application of the theory of ambiguity to transcendental
  analysis.  It was a question of seeing a priori in a relation
  between quantities or transcendent functions, what exchanges one
  could make, which quantities one could substitute for the given
  quantities without the original relation ceasing to hold.  That
  immediately made clear the impossibility of finding many expressions
  that one could look for.  But I do not have time and my ideas are
  not yet well developed on this ground which is immense.

 — Evariste Galois, testamentary letter, translated by James Dolan

Thomas Mann on musical  ambiguity in his novel Doctor Faustus —

Related material — Some context for the above and some remarks on the German original.

Julie Taymor in an interview published Dec. 12 —

“I’ve got two Broadway shows, a feature film, and Mozart,’’ she said. “It’s a very interesting place to be and to be able to move back and forth, but at a certain point you have to be able to step outside and see,’’ and here she dropped her voice to a tranquil whisper, “it’s just theater. It’s all theater. It’s all theater. The whole thing is theater.’’

Non-theater —

"The interplay between Euclidean and Galois  geometry" and
related remarks on interplay — Keats's Laws of Aesthetics.

Part theater, part non-theater —

Cubist crucifixion.

Apollo's 13: A Group Theory Narrative —

I. At Wikipedia —

II. Here —

See Cube Spaces and Cubist Geometries.

The 13 symmetry axes of the (Euclidean) cube–
exactly one axis for each pair of opposite
subcubes in the 27-part (Galois) 3×3×3 cube–

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

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

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

"Shifting Amid, and Asserting, His Own Cinema"

— Headline of an essay on Bertolucci in tormorrow's Sunday New York Times

This, together with yesterday's post on the Paris "Symmetry, Duality, and Cinema" conference last June, suggests a review of the phrase "blue diamond" in this journal. The search shows a link to the French art film "Duelle."

Some background for the word and concept from a French dictionary —

For examples of  duel  and duelle  see Evariste Galois
and Helen Mirren (the latter in The Tempest  and in 2010 ).

Image from stoneship.org

Continued from June 4, 2010

See also Jon Han's fanciful illustration in today's New York Times  and "Galois Cube" in this journal.

"Always keep a diamond in your mind."

— Tom Waits/Kathleen Brennan song performed by Solomon Burke at the Paradiso in Amsterdam

The Galois Quaternion "The text is a two-way mirror that allows me to look into the life and times of the reader. Who knows, someday i  may rise to a text that will compel me to push through to the other side." — The French Mathematician    (Galois), by Tom Petsinis

Yesterday's architectural entertainment coincided, more or less, with the New York Times  article "The Hand of a Master Architect" (Online Sunday, Aug. 8, and in the print edition Monday, Aug. 9).

A search for some background on that architect (Philip Johnson, not Howard Roark) showed that the Art Libraries Society of North America published a notable graphic logo in 2005—

See this journal on April 7, 2005, for a related graphic design.

The ARLIS/NA 2005 page cited above says about Houston, Texas, that 

"Just beyond the museum district lies Rice University, the city's most prestigious and oldest college….

Other campuses that contain significant architecture include St. Thomas University where Philip Johnson has made his mark for a period that extends more than forty years."

University of St. Thomas, Chapel of St. Basil

Applying Jungian synchronicity, we note that Johnson designed the Chapel of St. Basil at the University of St. Thomas, that the traditional date of the Feast of St. Basil is June 14, and that this journal on that date contained the following, from the aforementioned Rice University—                          

… a properly formulated Principle of Sufficient Reason plays
a fundamental role in scientific thought and, furthermore, is
to be regarded as of the greatest suggestiveness from the
philosophic point of view.²

… metaphysical reasoning always relies on the Principle of
Sufficient Reason, and… the true meaning of this Principle
is to be found in the “Theory of Ambiguity” and in the associated
mathematical “Theory of Groups.”

If I were a Leibnizian mystic, believing in his “preestablished
harmony,” and the “best possible world” so satirized by Voltaire
in “Candide,” I would say that the metaphysical importance of
the Principle of Sufficient Reason and the cognate Theory of Groups
arises from the fact that God thinks multi-dimensionally³
whereas men can only think in linear syllogistic series, and the
Theory of Groups is the appropriate instrument of thought to
remedy our deficiency in this respect.

The founder of the Theory of Groups was the mathematician
Evariste Galois….

² As far as I am aware, only Scholastic Philosophy has fully recognized
  and exploited this principle as one of basic importance for philosophic thought.

³ That is, uses multi-dimensional symbols beyond our grasp.

— George David Birkhoff, 1940

For more about Scholastic Philosophy, see the Center for Thomistic Studies at the University of St. Thomas.

For more about the graphic symbol shown (as above) by ARLIS and by Log24 in April 2005, see in this journal "rature sous rature ."

   For aficionados of mathematics and narrative —

Illustration from
"The Galois Quaternion— A Story"

This resembles an attempt by Coxeter in 1950 to represent
a Galois geometry in the Euclidean plane—

The quaternion illustration above shows a more natural way to picture this geometry—
not with dots representing points in the Euclidean  plane, but rather with unit squares
representing points in a finite Galois  affine plane. The use of unit squares to
represent points in Galois space allows, in at least some cases, the actions
of finite groups to be represented more naturally than in Euclidean space.

See Galois Geometry, Geometry Simplified, and
Finite Geometry of the Square and Cube.

or, Darkness and Brightness at Noon

"Human perception is a saga of created reality. But we were devising entities beyond the agreed-upon limits of recognition or interpretation…. We tried to create new realities overnight, careful sets of words that resemble advertising slogans in memorability and repeatability. These were words that would yield pictures eventually and then become three-dimensional."

— Don DeLillo, Point Omega

GF(4) = {0, 1, ω, ω²}

— Symbolic representation of a Galois field

"One two three  four,
  who are we  for?"

— Cheerleaders' chant

27

The 13 symmetry axes of the (Euclidean) cube–
exactly one axis for each pair of opposite
  subcubes in the 27-part (Galois) 3×3×3 cube–

Jamie James in The Music of the Spheres—

"The Pythagorean philosophy, like Zoroastrianism, Taoism, and every early system of higher thought, is based upon the concept of dualism. Pythagoras constructed a table of opposites from which he was able to derive every concept needed for a philosophy of the phenomenal world. As reconstructed by Aristotle in his Metaphysics, the table contains ten dualities (ten being a particularly important number in the Pythagorean system, as we shall see):

Of these dualities, the first is the most important; all the others may be seen as different aspects of this fundamental dichotomy. To establish a rational and consistent relationship between the limited (man, finite time, and so forth) and the unlimited (the cosmos, eternity, etc.) is not only the aim of Pythagoras's system but the central aim of all Western philosophy."

Théorie de l'Ambiguité

According to a 2008 paper by Yves André of the École Normale Supérieure  of Paris—

"Ambiguity theory was the name which Galois used
 when he referred to his own theory and its future developments."

The phrase "the theory of ambiguity" occurs in the testamentary letter Galois wrote to a friend, Auguste Chevalier, on the night before Galois was shot in a duel.

Hermann Weyl in Symmetry, Princeton University Press, 1952—

"This letter, if judged by the novelty and profundity of ideas it contains, is perhaps
  the most substantial piece of writing in the whole literature of mankind."

Conclusion of the Galois testamentary letter, according to
the 1897 Paris edition of Galois's collected works—

The original—

A transcription—

Évariste GALOIS, Lettre-testament, adressée à Auguste Chevalier—

Tu sais mon cher Auguste, que ces sujets ne sont pas les seuls que j'aie
explorés. Mes principales méditations, depuis quelques temps,
étaient dirigées sur l'application à l'analyse transcendante de la théorie de
l'ambiguité. Il s'agissait de voir a priori, dans une relation entre des quantités
ou fonctions transcendantes, quels échanges on pouvait faire, quelles
quantités on pouvait substituer aux quantités données, sans que la relation
put cesser d'avoir lieu. Cela fait reconnaitre de suite l'impossibilité de beaucoup
d'expressions que l'on pourrait chercher. Mais je n'ai pas le temps, et mes idées
ne sont pas encore bien développées sur ce terrain, qui est
immense.

Tu feras imprimer cette lettre dans la Revue encyclopédique.

Je me suis souvent hasardé dans ma vie à avancer des propositions dont je n'étais
pas sûr. Mais tout ce que j'ai écrit là est depuis bientôt un an dans ma
tête, et il est trop de mon intérêt de ne pas me tromper pour qu'on
me soupconne d'avoir énoncé des théorèmes dont je n'aurais pas la démonstration
complète.

Tu prieras publiquement Jacobi et Gauss de donner leur avis,
non sur la vérité, mais sur l'importance des théorèmes.

Après cela, il y aura, j'espère, des gens qui trouveront leur profit
à déchiffrer tout ce gachis.

Je t'embrasse avec effusion.

                                               E. Galois   Le 29 Mai 1832

A translation by Dr. Louis Weisner, Hunter College of the City of New York, from A Source Book in Mathematics, by David Eugene Smith, Dover Publications, 1959–

You know, my dear Auguste, that these subjects are not the only ones I have explored. My reflections, for some time, have been directed principally to the application of the theory of ambiguity to transcendental analysis. It is desired see a priori  in a relation among quantities or transcendental functions, what transformations one may make, what quantities one may substitute for the given quantities, without the relation ceasing to be valid. This enables us to recognize at once the impossibility of many expressions which we might seek. But I have no time, and my ideas are not developed in this field, which is immense.

Print this letter in the Revue Encyclopédique.

I have often in my life ventured to advance propositions of which I was uncertain; but all that I have written here has been in my head nearly a year, and it is too much to my interest not to deceive myself that I have been suspected of announcing theorems of which I had not the complete demonstration.

Ask Jacobi or Gauss publicly to give their opinion, not as to the truth, but as to the importance of the theorems.

Subsequently there will be, I hope, some people who will find it to their profit to decipher all this mess.

J t'embrasse avec effusion.
                        
                                                     E. Galois.   May 29, 1832.

Translation, in part, in The Unravelers: Mathematical Snapshots, by Jean Francois Dars, Annick Lesne, and Anne Papillaut (A.K. Peters, 2008)–

"You know, dear Auguste, that these subjects are not the only ones I have explored. For some time my main meditations have been directed on the application to transcendental analysis of the theory of ambiguity. The aim was to see in a relation between quantities or transcendental functions, what exchanges we could make, what quantities could be substituted to the given quantities without the relation ceasing to take place. In that way we see immediately that many expressions that we might look for are impossible. But I don't have the time and my ideas are not yet developed enough in this vast field."

Another translation, by James Dolan at the n-Category Café—

"My principal meditations for some time have been directed towards the application of the theory of ambiguity to transcendental analysis. It was a question of seeing a priori in a relation between quantities or transcendent functions, what exchanges one could make, which quantities one could substitute for the given quantities without the original relation ceasing to hold. That immediately made clear the impossibility of finding many expressions that one could look for. But I do not have time and my ideas are not yet well developed on this ground which is immense."

Related material—

"Renormalisation et Ambiguité Galoisienne," by Alain Connes, 2004

"La Théorie de l’Ambiguïté : De Galois aux Systèmes Dynamiques," by Jean-Pierre Ramis, 2006

"Ambiguity Theory, Old and New," preprint by Yves André, May 16, 2008,

"Ambiguity Theory," post by David Corfield at the n-Category Café, May 19, 2008

"Measuring Ambiguity," inaugural lecture at Utrecht University by Gunther Cornelissen, Jan. 16, 2009

Today is commencement day at Princeton.

Sunday's A Post for Galois was suggested, in part, by the fact that the founder and CEO of Amazon.com was that day's Princeton baccalaureate speaker. The Galois post linked to the Amazon reviews of one Christopher G. Robinson, a resident of Cambridge, Mass., whose Amazon book list titled "Step Right Up!" reflects a continuing libertine tradition at Harvard.

For Princeton's commencement day, it seems fitting to cite another Amazon document that reflects the more conservative values of that university.

I recommend the review Postmodern Pythagoras, by Matthew Milliner. Milliner is, in his own words, "an art history Ph.D. candidate at Princeton University."

See also Milliner's other reviews at Amazon.com.

"For every kind of libertine,
there is a kind of cross."

— Saying adapted from Pynchon

From yesterday's Seattle Times—

According to police, employees of a Second Avenue mission said the suspect, clad in black and covered in duct tape, had come into the mission "and threatened to blow the place up." He then told staffers "that he was a vampire and wanted to eat people."

The man… also called himself "a space cowboy"….

This suggests two film titles…

Plan 9 from Outer Space—

and Apollo's 13—

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

“Logic is all about the entertaining of possibilities.”

– Colin McGinn, Mindsight: Image, Dream, Meaning,
   Harvard University Press, 2004

Geometry of Language,
continued from St. George's Day, 2009—

Related material:

Prima Materia,
The Galois Quaternion,
and The Wake of Imagination.

See also the following from a physicist
(not of the most orthodox sort, but his remarks
  here on Heisenberg seem quite respectable)–

"But wait, there's more!"
– Stanley Fish, NY Times Jan. 28

From the editors at The New York Times who, left to their own devices, would produce yet another generation of leftist morons who don't know the difference between education and entertainment–

A new Times column starts today–

The quality of the column's logo speaks for itself. It pictures a cone with dashed lines indicating height and base radius, but unlabeled except for a large italic x to the right of the cone. This enigmatic variable may indicate the cone's height or slant height– or, possibly, its surface area or volume.

Instead of the column's opening load of crap about numbers and Sesame Street, a discussion of its logo might be helpful.

The cone plays a major role in the historical development of mathematics.

Some background from an online edition of Euclid—

"Euclid proved in proposition XII.10 that the cone with the same base and height as a cylinder was one third of the cylinder, but he could not find the ratio of a sphere to the circumscribed cylinder. In the century after Euclid, Archimedes solved this problem as well as the much more difficult problem of the surface area of a sphere."

For Archimedes and the surface area of a sphere, see (for instance) a discussion by Kevin Brown. For more material on Archimedes, see "Archimedes: Volume of a Sphere," by Doug Faires (2001)– Archimedes' heuristic argument from mechanics that involves the volume of a cone– and Archimedes' more rigorous approach in The Works of Archimedes, edited by T. L. Heath (1897).

The work of Euclid and Archimedes on volumes was, of course, long before the discovery of calculus.  For a helpful discussion of cone volumes involving high-school-level calculus, see, for instance,  the following–

The Times editors apparently feel that
few of their readers are capable of
such high-school-level sophistication.

For some other geometric illustrations
perhaps more appealing than the Times's

dunce cap, see the symbol of
  today's saint– a Bridget Cross—
and a web page on
visualized quaternions.

"The positional meaning of a symbol derives from its relationship to other symbols in a totality, a Gestalt, whose elements acquire their significance from the system as a whole."

— Victor Turner, The Forest of Symbols, Ithaca, NY, Cornell University Press, 1967, p. 51, quoted by Beth Barrie in "Victor Turner."

To everything, turn, turn, turn…
— Peter Seeger

The Galois Quaternion:

Click for context.

The Galois Quaternion

Related material:
The Galois Quaternion

Click for context.
(See also Nativity and the end
of this morning's post.)

Recent Wikipedia activity in the area of finite geometry–

A list, complete up to now, of all Wikipedia changes made by anonymous user Marconet:

• 15:27, 13 December 2009 (hist | diff) m Robert Daniel Carmichael ‎ (top) (Tag: references removed)
• 14:46, 21 November 2009 (hist | diff) General linear group ‎ (External Links)
• 10:41, 8 October 2008 (hist | diff) User talk:Barkeep ‎ (Translation Plane Link)
• 17:36, 7 October 2008 (hist | diff) Multiple time dimensions ‎ (Move to External links)
• 17:26, 7 October 2008 (hist | diff) Mathieu group ‎ (Move unpublished material/original research to External links)
• 17:12, 7 October 2008 (hist | diff) Translation plane ‎ (External link instead of internal)

Note that all these items are related to changes in links that lead to my own web pages– with one exception, rather technical pages on finite geometry.

A list, complete up to now, of all Wikipedia changes made by anonymous user Greenfernglade:

• 14:00, 14 December 2009 (hist | diff) Multiple time dimensions ‎ (WP:Self-Promotion) (top)
• 13:57, 14 December 2009 (hist | diff) Mathieu group ‎ (WP:Self-Promotion) (top)
• 13:56, 14 December 2009 (hist | diff) Logical connective ‎ (WP:Self-Promotion) (top)
• 13:55, 14 December 2009 (hist | diff) General linear group ‎ (WP:Self-Promotion)
• 13:54, 14 December 2009 (hist | diff) Galois geometry ‎ (WP:Self-Promotion) (top)
• 13:51, 14 December 2009 (hist | diff) Finite geometry ‎ (WP:Self-Promotion) (top)

Again, all these items are related to changes (in this case, deletions) in links that lead to my own web pages. Greenfernglade may or may not be the same person as Marconet. Neither one has a user home page at Wikipedia, but use of the pseudonyms has apparently served to cover up the IP address(es?) of the changes’ originator(s?).

For similar changes in the past, see my “user talk” page at Wikipedia. As I noted there on May 31, 2007, “There seems little point in protesting the deletions while Wikipedia still allows any anonymous user to change their articles.”

Recent abstracts of interest:

Kuwait Foundation Lectures —

Jan. 29, 2008: J. P. Wintenberger, “On the Proof of Serre’s Conjecture“

Oct. 28, 2008: Chandrashekhar Khare, “Modular Forms and Galois Representations“

Background:

The Last Theorem, a novel by Arthur C. Clarke and Frederik Pohl published Aug. 5, 2008

“Going Beyond Fermat’s Last Theorem,” a news article in The Hindu published April 25, 2005

Wikipedia: Serre Conjecture (Number Theory)

Henri Darmon, “Serre’s Conjectures“

Today is the 21st birthday of my note “The Relativity Problem in Finite Geometry.”

Some relevant quotations:

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

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

Describing the branch of mathematics known as Galois theory, Weyl says that it

“… is nothing else but the relativity theory for the set Sigma, a set which, by its discrete and finite character, is conceptually so much simpler than the infinite set of points in space or space-time dealt with by ordinary relativity theory.”

— Weyl, Symmetry, Princeton University Press, 1952, p. 138

Weyl’s set Sigma is a finite set of complex numbers.   Some other sets with “discrete and finite character” are those of 4, 8, 16, or 64 points, arranged in squares and cubes.  For illustrations, see Finite Geometry of the Square and Cube.  What Weyl calls “the relativity problem” for these sets involves fixing “objectively” a class of equivalent coordinatizations.  For what Weyl’s “objectively” means, see the article “Symmetry and Symmetry  Breaking,” by Katherine Brading and Elena Castellani, in the Stanford Encyclopedia of Philosophy:

“The old and natural idea that what is objective should not depend upon the particular perspective under which it is taken into consideration is thus reformulated in the following group-theoretical terms: what is objective is what is invariant with respect to the transformation group of reference frames, or, quoting Hermann Weyl (1952, p. 132), ‘objectivity means invariance with respect to the group of automorphisms [of space-time].^‘[22]

22. The significance of the notion of invariance and its group-theoretic treatment for the issue of objectivity is explored in Born (1953), for example. For more recent discussions see Kosso (2003) and Earman (2002, Sections 6 and 7).

References:

Born, M., 1953, “Physical Reality,” Philosophical Quarterly, 3, 139-149. Reprinted in E. Castellani (ed.), Interpreting Bodies: Classical and Quantum Objects in Modern Physics, Princeton, NJ: Princeton University Press, 1998, pp. 155-167.

Earman, J., 2002, “Laws, Symmetry, and Symmetry Breaking; Invariance, Conservation Principles, and Objectivity,’ PSA 2002, Proceedings of the Biennial Meeting of the Philosophy of Science Association 2002, forthcoming [Abstract/Preprint available online]

Kosso, P., 2003, “Symmetry, objectivity, and design,” in K. Brading and E. Castellani (eds.), Symmetries in Physics: Philosophical Reflections, Cambridge: Cambridge University Press, pp. 410-421.

Weyl, H., 1952, Symmetry, Princeton, NJ: Princeton University Press.

See also

Archives Henri Poincaré (research unit UMR 7117, at Université Nancy 2, of the CNRS)–

“Minkowski, Mathematicians, and the Mathematical Theory of Relativity,” by Scott Walter, in The Expanding Worlds of General Relativity (Einstein Studies, volume 7), H. Goenner, J. Renn, J. Ritter and T. Sauer, editors, Boston/Basel: Birkhäuser, 1999, pp. 45-86–

“Developing his ideas before Göttingen mathematicians in April 1909, Klein pointed out that the new theory based on the Lorentz group (which he preferred to call ‘Invariantentheorie’) could have come from pure mathematics (1910: 19). He felt that the new theory was anticipated by the ideas on geometry and groups that he had introduced in 1872, otherwise known as the Erlangen program (see Gray 1989: 229).”

References:

Gray, Jeremy J. (1989). Ideas of Space. 2d ed. Oxford: Oxford University Press.

Klein, Felix. (1910). “Über die geometrischen Grundlagen der Lorentzgruppe.” Jahresbericht der deutschen Mathematiker-Vereinigung 19: 281-300. [Reprinted: Physikalische Zeitschrift 12 (1911): 17-27].

Related material: A pathetically garbled version of the above concepts was published in 2001 by Harvard University Press.  See Invariances: The Structure of the Objective World, by Robert Nozick.

— From Pedagogy, Praxis, Ulysses
 

A quotation omitted from the above excerpt:

In Ulysses, there is "… the same quality of simultaneity as in cubist collage. Thus, for example, Bloom surveys the tombstones at Paddy Dignam's funeral and, in the midst of platitudinous and humorous thoughts, remembers Molly 'wanting to do it at the window'…."

Related material from quotations at the poetry journal eratio:

"The guiding law of the great variations in painting is one of disturbing simplicity.  First things are painted; then, sensations; finally, ideas.  This means that in the beginning the artist's attention was fixed on external reality; then, on the subjective; finally, on the intrasubjective.  These three stages are three points on a straight line."

— Jose Ortega y Gasset ("On Point of View in the Arts," an essay on the development of cubism)

Related material on
tombstones and windows:

Geometry's Tombstones,
Galois's Window, and
Architecture of Eternity.

See also the following part
of the eratio quotations:

Quotations arranged by
Gregory Vincent St. Thomasino

¹ Or hypercubism: See 10/31/06.

² Or "Wake" speech: See 10/31/05.
 

Today’s birthday:
Tom Hanks, star of
“The Da Vinci Code”

“Kufi Blocks“*

by Ben Nicholson,
Illinois Institute of Technology

Part II:
Some Background

Oslo Connection

Today is the birthday of Oystein Ore (1899-1968), Sterling Professor of Mathematics at Yale for 37 years, who was born and died in Oslo, Norway.  Ore is said to have coined the term “Galois connection.”  In his honor, an excerpt dealing with such connections:

Click to enlarge.

(Excerpt was added to Pattern Groups.)

New Web Page:

Galois Geometry

Mental Health Month, Day 26:

Many Dimensions,
Part II — The Blue Matrix 

But seriously…

John Baez in July 1999:

"…it's really the fact that the Leech lattice is 24-dimensional that lets us compactify 26-dimensional spacetime in such a way as to get a bosonic string theory with the Monster group as symmetries."

Well, maybe.  I certainly hope so.  If the Leech lattice and the Monster group turn out to have some significance in theoretical physics, then my own work, which deals with symmetries of substructures of the Leech lattice and the Monster, might be viewed in a different light.  Meanwhile, I take (cold) comfort from some writers who pursue the "story" theory of truth, as opposed to the "diamond" theory.  See the following from my journal:

Evariste Galois and the Rock that Changed Things, and

A Time to Gather Stones Together: Readings for Yom Kippur.

See, too, this web page on Marion Zimmer Bradley's fictional

Matrices, or Blue Star-Stones, and

the purely mathematical site Diamond Theory, which deals with properties of the above "blue matrix" and its larger relatives.