**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.*