Non-convex polytopes realizing kissing contact graphs in dimensions 5-7

Abstract

Each of the 12 known conjecturally optimal kissing configurations in dimensions 5, 6, and 7 carries a contact-graph polytope: a polytope whose vertices are the configuration points and whose edges are exactly its contacts. Eight of these polytopes are non-convex and distinct from their convex hulls, which include longer edges. Each polytope is radial about the sphere's center: its boundary meets every ray from the center exactly once. It is the unique radial polytope with those vertices and edges. Seven of the nine non-lattice configurations carry polytopes constructed from the lattice contact polytope in the same dimension by splitting some facets and reassembling the pieces. This split-and-reassemble construction describes the configurations by facets rather than by layers. The fold angles of the reassembled facets explain the inner products $1/(10-n)$, giving $1/5$, $1/4$, and $1/3$ in dimensions 5, 6, and 7, respectively. The contact graphs of all 12 configurations are infinitesimally rigid. We also prove that $Q_5$ and $R_5$ are infinitesimally jammed, extending the 10 previously established cases to all 12 configurations. All mathematical computations use exact arithmetic.