Embeddings of Steiner Triple Systems and 2-Folds

There are eighty (80) non-isomorphic Steiner Triple Systems of order 15 (STS(15)), (They are enumerated in Handbook of Combinatorial Designs by Colbourn and Dinitz, ), There are 3200 pairs of STS(15)'s.  For a fixed pair (m,n) (m and n refer to the enumeration in Handbook of Combinatorial Designs).   It is a priori not clear that they yield triangulation embeddings of 2-folds.  One embedding of every pair (m,n) is  exhibited HERE.

These embeddings are not necessarily on orientable surfaces.  The algorithm often leads to embeddings on non-orientable surfaces and some new examples of embeddings of pairs (m,n) on orientable surfaces are HERE.

The algorithm can generate all embeddings of a given pair.  One thousand non-isomorphic embeddings of the pair (10,10) is exhibited HERE.  This list is not exhaustive, and the CPU time required for their generation was of the order of minutes.  It is shown here as an example of the efficacy of the algorithm.

For two hundred embeddings of two pairs of randomly chosen STS(19)'s click  HERE. The CPU time for generating these embeddings was under ten minutes.

For some embeddings STS(n)'s for n=21,25 and 27 click HERE.

A new site with a large data base of embeddings is currently under construction.