On prismatic tiles (Q1280254)

A tiling \(\mathcal T\) of Euclidean \(d\)-space \(E^d\) is a countable family of closed subsets \(T\) of \(E^d\), the tiles of \(\mathcal T\), which cover \(E^d\) without gaps or overlaps. A monohedral tiling \(\mathcal T\) of \(E^d\) is a tiling in which all tiles are congruent to one fixed set \(P\), the prototile of \(\mathcal T\); we also say that \(P\) admits the tiling \(\mathcal T\). A tiling \(\mathcal T\) by polytopes is face-to-face if the intersection of any two tiles of \(\mathcal T\) is empty or a face of each tile. The authors prove: Theorem 1. If a (convex or non-convex) right prism admits a face-to-face tiling of \(E^3\), then its base admits a face-to-face tiling of \(E^2\).