Symmetric Geometric Embeddings of Cyclic 3-configurations: preliminary report

Minisymposium: POLYTOPES AND GRAPHS

Content: A \emph{cyclic configuration} is a combinatorial configuration in which all the lines (or blocks) of the configuration are formed via a cyclic permutation of an initial block. Moreover, the fact that the block [0,1,3] forms an initial block for a cyclic configuration $\mathcal{C}_3(n)$ for any $n \geq 7$ has been used to show that combinatorial 3-configurations exist for all possible $n$. There are well-known methods for geometrically embedding cyclic 3-configurations in the plane, but the embeddings these produce are typically without any geometric symmetry, which is undesirable for configurations with such high combinatorial symmetry (for example, all cyclic configurations are vertex transitive). This talk will present preliminary results on embeddings of cyclic 3-configurations that possess high degrees of rotational symmetry.

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