Quarter-regular biembeddings of Latin squares

In this talk I will review the concept of the biembedding of two latin squares. Of particular interest will be the regular biemedding of two isomorphic copies of the latin square corresponding to the cyclic group of order n, denoted Cn. Grannell and Griggs have shown that, for all n, a regular biembedding exists, and in addition, that the automorphism group of the regular biembedding has order 12n. Grannell and Griggs have also developed a doubling construction in which the latin squares of order n can be used to construct a biembedding of latin squares of order 2n. In this talk I will apply this construction to the regular biembedding of Cn. The result is surprising in that the doubling construction produces a biembedding of two copies of C2n, however the automorphism group of this biembedding has order 12(2n)/4 = 12n.