Two rank six geometries for the Higman-Sims sporadic group

We construct two rank six geometries that are firm and residually connected and on which the Higman-Sims sporadic group HS acts flag-transitively. Both of these geometries have five rank five residues isomorphic to geometries of the Mathieu group M22. They continue chains of geometries obtained for A5, A6, L3(4) and M22. Moreover, they are the first rank six geometries constructed for HS.