On Certain Semi-Perfect Cuboids
W. J. A. Colman · The Fibonacci Quarterly · 1988
1. The classical cuboid is a rectangular block with integral edges and face diagonals. If we consider the internal diagonal as well, then there are seven lengths in all * It is known [3] that any six of the seven lengths can be integral. We can call such cuboids semi-perfect. Semi-perfect cuboids fall into three categories such that there is no integral specification for: (1) the internal diagonal, (2) one face diagonal, (3) one edge. If all seven lengths were integral, then we would have what is known as a perfect cuboid. No such perfect cuboids are known; indeed, their existence is a classical open question. It is known [3] that there are an infinity of semiperfect cuboids in all categories, as certain parametric solutions are known. Unfortunately, none of these solutions is complete. Clearly, if perfect cuboids exist, they must fall into all three categories and so the complete determination of all semi-perfect cuboids in any one category would reduce the problem of perfect cuboids to the consideration of the seventh nonspecified length. It has been shown that some of these partial parametric solutions cannot be perfect (see [2], [3], and [4]). In this paper we shall determine a two-parameter solution for category (3) which is the generalization of a solution first given by Bromhead [1], and then show in a simple manner that this too can never give a perfect cuboid. 2. It is instructive first to consider the smallest real solutions (with c?>0) in category (3). If we measure the size of the cuboid by the length of the internal diagonal d (say) with edges a, b9 and vc, then Leech [3] has given the smallest solutions. The first four being a b