Well-known folklore?

The seriocomical history of this paper—or rather fragment—goes as follows:-

(1) The five platonic solids, the crowning glory of Euclid’s Elements, are well-known to all mathematicians, as is the fact that the regular maps {p,q} that can live on a closed surface which is topologically equivalent to a round sphere are these five only, i.e., {3,3}, {3,4}, {4,3}, {3,5} and {5,3}. By October 10, 2007, the date on which I had finished Cacti and Mathematics – in which paper, using patterns on plants as my cue, I have given a popular exposition of the result just mentioned – I was wondering as to what happens if one removes this topological constraint: does there exist a finite regular map {p,q} for any pairs of numbers p ³ 3, q ³ 3? Surely, I reasoned, this natural question must have occurred to many since at least the time of Riemann, and its answer known by now, and if, per chance, such a sweeping generalization of the five platonic solids was true, it must surely be prominently mentioned in any of the three books devoted to such-like things that I happened to have in my library, viz., Coxeter’s Regular Polytopes, Coxeter and Moser’s Generators and Relations for Discrete Groups, and McMullen and Schulte’s relatively recent, Abstract Regular Polytopes. However a quick and cursory browsing of these three books revealed nothing of this nature. This apparent absence, especially from the second book, which is mostly about describing symmetry groups of finite regular maps, coupled with the fact that at the end of their initial recap of regular maps in the first chapter of the third book, the authors point out that dropping the requirement of symmetry gives many more maps, which they call equivelar – these were called uniform tilings in my exposition – led me to believe that there might be some group theoretic obstruction to building all regular {p,q}’s, but probably one did have all equivelar {p,q}’s? So I persisted with uniform tilings only, and soon had so many diverse constructions of the same in my kitty that they seemed sufficient, but after typing up some I realized that I was still falling short … anyway, what I had already typed up seemed interesting enough, and that is what you’ll find in On Equivelar Maps … but you’ll note that its concluding section is frozen in mid-stride in a “pre”-version of sorts … here you’ll see (p,q) instead of {p,q}, and the numbering of references is different … this is so because events had taken the following sudden turn at this point.

(2) For the planned bibliographical remarks in this section, I had decided to look at as many of the pertinent items in the extensive Bibliography of McMullen and Schulte as was feasible, and the title of [437], a 1997 paper of H.J. Voss, had soon caught my fancy. This particular paper I still have not seen, but I was readily able to obtain from the web a related paper by the same author, viz., Sachs triangulations and infinite sequences of regular maps on given type, Disc. Math. 191 (1998), pp. 223-240, which has the following on its very first page.

“In 1974, Grünbaum [5] conjectured that a finite regular map of type {p,q} exists for each p ³ 3 and q ³ 3. In 1983,

Vince [10] proved a generalization of Grünbaum’s conjecture using Mal’cev’s theorem about groups of matrices.”

That was that, I thought, and turned my attention towards A. Vince, Regular combinatorial maps, J. Combin. Theory B 35 (1983), pp. 256-157. It turns out that the result in question is an almost immediate corollary of this 1940 theorem of Malcev about finitely generated linear groups, and since everyone and his uncle knows that the symmetry groups of the infinite hyperbolic tilings are of this type, it seems fair to state that, Grünbaum was in 1974 conjecturing something that was already known to be true for decades (!), and which moreover is so strikingly sweeping that it obviously deserved to be well-known by 1974. That is, in the reasonable—not the mathematicians’ rather rarefied—sense of this hackneyed term : but as my own example shows, it was not well-known to me even in 2007, and I have a hunch it is not well-known to the majority of mathematicians even today in 2010. I remark that Malcev’s theorem – which, by the way, is also quite simple to prove – in fact gives us infinitely many finite regular maps for each non-platonic pair {p,q}, and what was really new in Vince’s paper was that he notices that it applies also to the symmetry groups of the elegant combinatorial generalization of hyberbolic tilings that he was considering. I had noticed then only that Vince’s paper was [436] in McMullen and Schulte’s Bibliography, and some further browsing of this book done now, while writing this covering note, has shown me that these authors make a parallel use of Malcev’s theorem for their own, and equally elegant combinatorial generalization of hyperbolic tilings. Moreover, this section 4C of their book begins: “It is folklore in Riemann surface theory that each regular tessellation {p,q} of the euclidean or hyperbolic plane is the universal 2-covering for an infinite number of finite regular maps of the same type {p,q} on closed surfaces.” So the result in question was there in this book, only it was not where – because of its striking and basic nature – I had naively presumed that it would be, viz., in their initial recap of regular maps itself! In the sentence just quoted we have an example of another usage common chez mathematicians: folklore! A catch-all for key facts or insights of uncertain provenance which are common knowledge to some insiders, and so remain unpublished as such because not many brownie points can be earned by their exposition; however, from the point of view of understanding, it is much easier for an outsider if somehow this oral tradition can be grasped first, complicated-looking things have then a way of becoming very simple indeed.

(3) Anyway, I have now done my bit to make this folklore more well-known : you can read from “213, 16A” and Mathematics, which discusses in everyday language the mathematics underlying some architectural motifs, an account, understandable to any bright high-school student, showing from scratch why one has a finite regular map {p,q} for any p ³ 3 and q ³ 3! This is in the eighth and final ‘lecture’ of this paper, entitled “Magic Carpet”, and represents a known alternative route to this result using permutations instead of matrices, my own inspirations being Poincaré, and a short 108-year old paper of G. A. Miller, Groups defined by the orders of two generators and the order of their product, Amer. J. Math. 24 (1902), pp. 96-100. If one has an infinite regular tiling by p-gons, q at each vertex, then a group generated by two elements of order p and q with product of order 2—and no other relations—is infinite, which is hailed on page 54 of Coxeter and Moser thus: “This is surely one of the most remarkable contributions of geometry to algebra. For, the algebraic proof (MILLER 1902) is excessively complicated, requiring separate considerations of many different cases.” This is a tad unfair, for, the finite groups of this type that Miller makes in droves by imposing more relations are, interpreted geometrically, nothing but finite regular maps {p,q}: one might say that algebra has made an equally remarkable contribution to geometry! This 1902 paper begins by emphasizing that an infinity of these groups is going to be made for each non-platonic {p,q}, which to my mind suggests, though Miller gives no reference to this effect, that the existence of at least one such group for each p ³ 3 and q ³ 3 was already known, maybe to even Cayley? Much more is of course known now in 2010 about regular maps and their groups, but their full classification is hopelessly difficult, indeed, any sporadic finite simple group can be realized as the group of orientation-preserving symmetries of a finite regular map! However, since this discussion shall be continued in the planned extensive NOTES to “213, 16A” and Mathematics, which I’ll be posting on this website in due time, I’ll conclude for now by simply pointing out that the photograph of Uccello’s inlay work occurs again in this newer paper, but now I prefer to call this intriguing regular solid a ‘dodecahedragram’ – my own coinage – which is definitely shorter, and possibly more mellifluous, than the ‘small stellated dodecahedron’ used in Coxeter’s Regular Polytopes.