Smallest Integral Combinatorial Box

Abstract

To prove or disprove the existence of a perfect box is an old unsolved problem (see [2,4]). Here a perfect box or cuboid means a rectangular parallelepiped with all edges, face diagonals and space diagonals of integer lengths. More general, in [5] combinatorial boxes are considered, that are, convex bodies with six quadrilaterals as faces. These bodies then also have 8 vertices and 12 edges. In [5] the existence of integral combinatorial boxes is proved by the presentation of 20 examples where all edges, face diagonals and space diagonals are of integer lengths. It is conjectured in [5] that one of these examples (see Figure 1) is the smallest integral combinatorial box, that means, that the largest distance between any two vertices, 17 in this case, is the minimum of any largest integer distance between vertices of all integral combinatorial boxes. Here we will prove that up to this largest distance 17 other integral combinatorial boxes do not exist.