Putting convex d -polytopes inside frames

TLDR: In this article, a construction of large neighborly and nearly-neighborly families of d-polytopes in rods was introduced, obtained by putting smaller such families in rods, to form suitable frames for a larger collection.

Abstract: We introduce a construction of large neighborly and nearly-neighborly families of d-polytopes in $$E^{d}$$ , obtained by putting smaller such families in rods, to form suitable frames for a larger collection. We use it to exhibit, among general results, a nearly-neighborly family, consisting of $$9\times 4^{d-2}$$ combinatorially d-boxes in $$E^{d}$$ , for every $$d \ge 3$$ (like 36 3-boxes in $$E^{3})$$ and a neighborly family of $$3\times 2^{d-1}$$ of prisms over d-simplices (like 12 triangular prisms in $$E^{3})$$ .