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We call a convex polytope P of dimension 3 admissible if it has the following two properties: (1) for each vertex of P the set of its first-neighbours is coplanar; (2) all planes determined by the first-neighbours are distinct. It is shown that the Levi graph of a point-plane configuration obtained by V-construction from an admissible polytope P is the Kronecker cover of the 1-skeleton of P. We investigate the combinatorial nature of the V-construction and use it on unit-distance graphs to construct novel isometric point-circle configurations. In particular, we present an infinite series all of whose members are subconfigurations of the renowned Clifford configurations.

A. Matroid polytopes form an intermediate structure useful in searching for realizable convex spheres. In this article we present a class of self-polar 3-spheres that motivated research in the inductive generation of matroid polytopes, along with two new methods of generation.

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In memory of Branko Grünbaum.