4. Homologues

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 A be a geometrical configuration with symmetry group GA, and 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 GAB of GA consisting of those motions in the symmetry group GA of 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

H =
i = 1 m   Khi,       (4.1)

where it is understood that the union is disjoint. Notice that m, which appears in (4.1) and which we call the index of K in H, is just the ratio of the order of H to the order of K. An example was provided earlier with H = Sn, the symmetric group, and K = An, the alternating group. Then m = 2.

Reverting to our geometrical situation , we consider a coset of GAB in GA, that is, a set GABg, g GA. Every element in GABg sends B 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 GAB in GA.

Example 4.1: Consider the pentagonal dipyramid 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, GA is the dihedral group D5, with 10 elements, given by the following permutations sending (1 2 3 4 5 6 7), resepectively, to

(1 2 3 4 5 6 7) (Identity)
(2 3 4 5 6 7 1) (rotation through 2p/5 about axis 67)
(3 4 5 1 2 6 7)
(4 5 1 2 3 6 7)
(5 1 2 3 4 6 7)
(5 4 3 2 1 7 6) (interchanging top and bottom)
(4 3 2 1 5 6 7) (interchange plus rotation through 2p/5)
(3 2 1 5 4 6 7)
(2 1 5 4 3 7 6)
(1 5 4 3 2 7 6)

  •       First, let be the edge 16. Then GAB = {Id}, since only the identity sends the subset (1,6) to itself. Thus the index of GAB in GA 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).
  •       Second, let B be the edge 12. Then GAB has 2 elements, since there are two elements of GA, namely the identity and permutation
    æ
    ç
    è
    1 2 3 4 5 6 7
    2 1 5 4 3 7 6
    ö
    ÷
    ø

    which send the subset (1, 2) to itself. Thus the index of GAB in GA is 5, and there are 5 homologues of the 12; these are the 5 edges around the equator.

  •       Third, let B be the face (126). Then GAB = {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.


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