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The complete graph on six points has fifteen edges and fifteen perfect matchings. Its six one-factorizations organize a dual incidence geometry exchanging points with factorizations and edges with matchings. The classical names are duads, synthemes and totals of synthemes. This geometry supports duad-syntheme duality on six points and a construction of the small Witt design.

A Steiner system $S(t,k,v)$ is a $v$-point set with $k$-point blocks such that every $t$-point subset is contained in exactly one block. Necessarily the number of blocks is $\left(\genfrac{}{}{0ex}{}{v}{t}\right)/\left(\genfrac{}{}{0ex}{}{k}{t}\right)$. This count is a necessary consistency check, not a substitute for proving exact containment. The small Witt design is $S(5,6,12)$.