George Pólya, who made great contributions not only to mathematics itself, but also to the understanding of how and why we do mathematics - or perhaps one should say 'how and why we should do mathematics' - was particularly fascinated by the Platonic solids and first introduced his notion of homologues in connection with the study of their symmetry; they later played an important role in one of his most important contributions to the branch of mathematics know as combinatorics, namely, the Pólya Enumeration Theorem (see [P1] for an intuitive account). Let us describe this notion of homologues in terms of symmetry groups. We believe that we are thereby increasing the scope of the notion and entirely maintaining the spirit.
Let B be a subset of A.
Thus, for example, A may be a
polyhedron and B a face of that polyhedron. We consider the subgroup
G of _{AB}G consisting of those motions in the
symmetry group _{A}G of _{A}A which map B to itself.
Now subgroups partition a group into cosets: If K is a subgroup of H,
we define a (right) coset of K in H as a collection of elements kh,
with h fixed and k varying over K. We call h a representative
of this coset, which we write Kh. Any two cosets Kh,
Kh¢ are
either disjoint or identical (this is easy to prove), so we may imagine that
we have picked a set of coset representatives, one for each coset. In the
case in which we are interested the group H is finite so we may write,
for some m
where it is understood that the union is disjoint. Notice that K = A, the alternating group. Then _{n}m = 2.
Reverting to our geometrical situation , we consider a coset of
G, that is, a set
_{A}G, _{AB}gg Î G.
Every element in _{A}G sends _{AB}gB to the same subset
Bg of A. The collection of these subsets is what Pólya called
the collection of homologues of B in A. We see that the set of
homologues of B is in one-one correspondence with the set of cosets of
G in _{AB}G.
_{A}
A of
Figure 5(b) of [Rec].
We may specify any motion in
the symmetry group of A by the resulting permutation of its vertices
1,2,3,4,5,6,7. In fact, G is the dihedral group
_{A}D_{5}, with 10 elements, given by the following permutations
sending (1 2 3 4 5 6 7), resepectively, to
G = {_{AB}Id}, since
only the identity sends the subset (1,6) to itself. Thus the
index of G in _{AB}G is 10, and there are
10 homologues of the edge 16; these are the 10 'spines' of the dipyramid (i.e., we
exclude the edges around the equator).
_{A}B be the edge 12. Then G has 2 elements,
since there are two elements of _{AB}G, namely the identity and
permutation
_{A}
which send the subset
(1, 2) to itself. Thus the index of G
is 5, and there are 5 homologues of the 12; these are the 5 edges around the equator.
_{A}B be the face (126). Then G = {_{AB}Id},
so that, as in the first case, there are 10 homologues of the face (126);
in other words all the (triangular) faces are homologues.
Let us now explain the Pólya Enumeration Theorem - actually, there are two theorems - and see how the notion of homologue fits into the story. |