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Symmetry from Plato to the
Four-Color Conjecture

by Steven H. Cullinane

"While the reader may draw many a moral from our tale,
I hope that the story is of interest for its own sake.
Moreover, I hope that it may inspire others, participants
or observers, to preserve the true and complete record of
our mathematical times."
-- Thomas M. Thompson [TH]

The historical road from the Platonic solids to the finite simple groups is well known. This website sketches, in accordance with Thompson's hope expressed above, a road less traveled. This road begins more than 24 centuries ago, with Plato's diamond figure [PL, also HE]. It leads, through the study of symmetry, to the powerful combinatorial structure S(5,8,24) [MA, also CA] used in studies of finite simple groups [TH, also CO] and also in a recent attack [DH] on the four-color conjecture.

In 1975, the author discovered symmetry properties of Plato's diamond that he described in a preprint [CUL1] distributed at the 1976 American Mathematical Society summer meeting in Toronto. This later resulted in the following citation:

"In every civilization and culture, colored tilings and patterns appear among the earliest decorations.... In particular, 2-color patterns arose -- early and frequently -- through a device known as 'counterchange'.... An early paper with remarkable counterchange designs formed by diagonally divided squares -- one-half black, one-half white -- was published by Truchet (1704). For a more recent treatment, with many illustrations, see Cullinane (1976). However, all these were more or less 'accidental' occurrences, independently reinvented many times, and passed from generation to generation by artists and artisans. The only artist who deliberately and consistently tried to investigate colored patterns (more specifically, colored tilings) was M. C. Escher...."
-- Branko Grunbaum and G. C. Shephard [GR, 1987]

Compare these words with those of G. H. Hardy:

"A mathematician, like a painter or a poet, is a maker of patterns." [HA]

Compare also the quotations in the website A Mathematician's Aesthetics. [CUL4]

Later, in 1978, the author came across the following:

"...A guiding principle in modern mathematics is this lesson: Whenever you have to do with a structure-endowed entity S [capital sigma in the original] 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 [WE, p. 144]

Inspired by Weyl's lesson, the author determined in 1978 the group of automorphisms of the patterns described in the preprint [CUL1, updated in CUL3], and published this information in an abstract [CUL2, February 1979]. In March 1979, R. T. Curtis sent the author a copy of his paper [CUR]. This paper describes the action of the Mathieu group M24 on the Steiner system S=S(5,8,24). Part of this group action is identical to the action described in the abstract [CUL2]. Thus a trail may be traced from Plato's diamond in the Meno dialogue, via S, to recent studies of finite simple groups. For more on the relation of these studies to S, see (for instance) Feit [FE], who notes that the Leech lattice underlying various new sporadic groups "is a blown up version of S(5,8,24)."

Also discussed at the 1976 A.M.S. meeting in Toronto was an attempted computer proof of the four-color conjecture. For a recent human proof attempt, see Dharwadker's website [DH]. This new attack (as yet unrefereed) on the four-color conjecture uses the same Steiner system S discussed above. The road that led to Dharwadker's use of S is a story yet to be told.

For the author's rough critique (as of November 28, 2000) of Dharwadker's proof attempt, click here.

For other critiques, enter Dharwadker as the search term after you have clicked here.

"Symmetry is a vast subject, significant in art and nature.
Mathematics lies at its root, and it would be hard to find
a better one on which to demonstrate the working of
the mathematical intellect."
-- Hermann Weyl [WE, p. 145]


[CA] Book. Parallelisms of Complete Designs, by Peter J. Cameron, Cambridge University Press, 1976
[CO] Paper. "Three Lectures on Exceptonal Groups," by J. H. Conway, in Finite Simple Groups, ed. by M. B. Powell and G. Higman, Academic Press, 1971
[CUL1] Preprint. "Diamond Theory," by S. H. Cullinane, 1976
[CUL2] Abstract. "Symmetry invariance in a diamond ring," by S. H. Cullinane, A.M.S. Notices, February 1979, pages A-193,194
[CUL3] Website. "Diamond Theory," by S. H. Cullinane, 2000
[CUL4] Website. "A Mathematician's Aesthetics," by S. H. Cullinane, 2000
[CUR] Paper. "A new combinatorial approach to M24," by R. T. Curtis, Math. Proc. Camb. Phil. Soc. 79 (1976), pp. 25-42
[DH] Website. "A New Proof of the Four Colour Theorem," by Ashay Dharwadker, 2000
[FE] Review. Walter Feit's review of Finite Simple Groups, by Daniel Gorenstein. Bulletin of the A.M.S., January 1983, pp. 120-124
[GR] Book. Tilings and Patterns, by Branko Grunbaum and G. C. Shephard, W. H. Freeman and Company, 1987
[HA] Book. A Mathematician's Apology, by G. H. Hardy, Cambridge University Press, 1940
[HE] Book. The Thirteen Books of Euclid's Elements, Volume I, by Sir Thomas L. Heath, Dover Publications, 1956. Plato's diamond is on page 352.
[MA] Book. The Theory of Error-Correcting Codes, by F. J. MacWilliams and N. J. A. Sloane, North-Holland Publishing, 1977
[PL] Classic. Plato's Meno, 85 a,b
[TH] Book. From Error-Correcting Codes through Sphere Packings to Simple Groups, by Thomas M. Thompson, Mathematical Association of America, 1983
[WE] Book. Symmetry, by Hermann Weyl, Princeton University Press, 1952

Page last updated Nov. 30, 2000; created Nov. 19, 2000.