Well-known
folklore?

This is a covering note for On
equivelar manifolds, the seriocomical history of which  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 limited myself to uniform
tilings only, and soon had so many diverse constructions of the same in my bag
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 listed
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 understanding  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 knows that
the symmetry groups of the infinite hyperbolic tilings are of this type, it
seems fair to state that, Grünbaum was conjecturing something that was
already known to be true for decades (!), and which moreover is so strikingly
sweeping that it 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
feeling 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 that Vince’s paper was listed as [436] in the above Bibliography, and some further browsing
of this book done now 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 indeed say
that algebra has made an equally remarkable contribution to geometry!  This paper begins by emphasizing that it
is an infinity of these groups which
is going to be exhibited for each non-platonic {p,q}, which to me 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 this covering note by pointing
out that the photograph of Uccello’s inlay work is there 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.

K. S. Sarkaria.