Two neighborly families of 3-pyramids and of 3-boxes in \(E^{3}\) (Q2413562)

A family \(\mathcal{P} = \{P_1,\dots,P_n\}\) of \(d\)-dimensional convex polytopes in Euclidean \(d\)-space is called \textit{neighborly} if, for each \(P_i\) and \(P_j\) with \(i \neq j\), the intersection \(P_i \cap P_j\) is a facet of both \(P_i\) and \(P_j\). It is known that a neighborly family of tetrahedra in \(E^3\) has at most eight tetrahedra, and that a neighborly family of \(3\)-pyramids has at most \(31\) pyramids. In this article, the author produces a neighborly family of sixteen \(3\)-pyramids, and a neighborly family of twenty-four combinatorial \(3\)-boxes.