Weyl on what he calls the relativity problem—

"The relativity problem is one of central significance throughout geometry and algebra and has been recognized as such by the mathematicians at an early time."

— Hermann Weyl, 1949, "Relativity Theory as a Stimulus in Mathematical Research"

"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, 1946, The Classical Groups, Princeton University Press, p. 16

Twenty-four years ago a note of Feb. 20, 1986, supplied an example of such coordinatizations in finite geometry. In that note, the group of mediating transformations acted directly on coordinates within a 4×4 array. When the 4×4 array is embedded in a 4×6 array, a larger and more interesting group, M₂₄ (containing the original group), acts on the larger array.  There is no obvious solution to Weyl's relativity problem for M₂₄.  That is, there is no obvious way to apply exactly 24 distinct transformable coordinates (or symbol-strings) to the 24 array elements in such a way that the natural group of mediating transformations of the 24 symbol-strings is M₂₄.

There is, however, an assignment of symbol-strings that yields a family of sets with automorphism group M₂₄.

R.D. Carmichael in 1931 on his construction of the Steiner system S(5,8,24)–

"The linear fractional group modulo 23 of order 24•23•11 is often represented as a doubly transitive group of degree 24 on the symbols ∞, 0, 1, 2,…, 22. This transitive group contains a subgroup of order 8 each element of which transforms into itself the set ∞, 0, 1, 3, 12, 15, 21, 22 of eight elements, while the whole group transforms this set into 3•23•11 sets of eight each. This configuration of octuples has the remarkable property that any given set of five of the 24 symbols occurs in one and just one of these octuples. The largest permutation group Γ on the 24 symbols, each element of which leaves this configuration invariant, is a five-fold transitive group of degree 24 and order 24•23•22•21•20•48. This is the Mathieu group of degree 24."

— R. D. Carmichael, 1931, "Tactical Configurations of Rank Two," in American Journal of Mathematics, Vol. 53, No. 1 (Jan., 1931), pp. 217-240

An alleged manifestation in physics, from Scientific American  —

Manifestations in pure mathematics, from Plato and R. T. Curtis  —

For some entertaining literary  manifestations, see Wrinkle.

Commentary —

Omega by Lux:

Omega by Curtis:

Today is the dies natalis  of group theorist  Walter Feit.

     "The Steiner systems S (5,6,12) and S (5,8,24) are remarkable combinatorial
configurations unlike any others. Their automorphism groups are the Mathieu
groups M₁₂ and M₂₄. These are the only 5-transitive permutation groups other
than symmetric and alternating groups: (a fact long conjectured but only
proved as a consequence of the classification). The Leech lattice is a blown up
version of S (5,8,24). It is the unique even unimodular lattice in 24 dimensions
with no vectors of weight 2. This uniqueness is an essential reason why it is a
geometric object of fundamental importance. The automorphism group Co.O
of the Leech lattice involves about half of the sporadic groups and generally it
is felt that these are well understood."

— Walter Feit, book review, Bulletin of the American Mathematical Society ,
     Vol. 8 (1983), 120-124, page 123

A search for some background on Dmitri Tymoczko, the subject of yesterday's evening entry on music theory, shows that his name and mine once both appeared in the same web page— "This Week's Finds in Mathematical Physics (Week 234)," by John Baez, June 12, 2006 (linked to by the Wikipedia article on transformational music theory).

In that page, Baez speculates on the possibility of a connection between music theory and Mathieu groups and says—

"For a pretty explanation of M₂₄, also try this:

Steven H. Cullinane, Geometry of the 4 × 4 square, http://finitegeometry.org/sc/16/geometry.html."

I know of no connection* between the groups I discussed there and music theory. For some background on Tymoczko's work, see the helpful survey "Exploring Musical Space," by Julian Hook (Science  magazine, 7 July 2006).

* Apart, that is, from the tesseract (see Geometry of the 4 × 4 Square) shown by Tymoczko in a 2010 lecture—

This is perhaps "Chopin's tesseract" from section 8.5 of Tymoczko's new book
A Geometry of Music  (Oxford University Press, 2011).

There is a remarkable correspondence between the 35 partitions of an eight-element set H into two four-element sets and the 35 partitions of the affine 4-space L over GF(2) into four parallel four-point planes. Under this correspondence, two of the H-partitions have a common refinement into 2-sets if and only if the same is true of the corresponding L-partitions (Peter J. Cameron, Parallelisms of Complete Designs, Cambridge U. Press, 1976, p. 60). The correspondence underlies the isomorphism* of the group A₈ with the projective general linear group PGL(4,2) and plays an important role in the structure of the large Mathieu group M₂₄.

A 1954 paper by W.L. Edge suggests the correspondence should be named after E.H. Moore. Hence the title of this note.

Edge says that

It is natural to ask what, if any, are the 8 objects which undergo
permutation. This question was discussed at length by Moore…**.
But, while there is no thought either of controverting Moore's claim to
have answered it or of disputing his priority, the question is primarily
a geometrical one….

Excerpts from the Edge paper—

Excerpts from the Moore paper—

Pages 432, 433, 434, and 435, as well as the section mentioned above by Edge— pp. 438 and 439

* J.W.P. Hirschfeld, Finite Projective Spaces of Three Dimensions, Oxford U. Press, 1985, p. 72

** Edge cited "E.H. Moore, Math. Annalen, 51 (1899), 417-44." A more complete citation from "The Scientific Work of Eliakim Hastings Moore," by G.A. Bliss,  Bull. Amer. Math. Soc. Volume 40, Number 7 (1934), 501-514— E.H. Moore, "Concerning the General Equations of the Seventh and Eighth Degrees," Annalen, vol. 51 (1899), pp. 417-444.

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.”