Shrinking Lattice Polyhedra

Abstract

This paper treats the following geometric problem: Given vertices $x_1 , \cdots ,x_n $ of a polyhedron in the integer lattice in k dimensions, can that polyhedron be “shrunk” to a similar one (i.e., one whose sides remain in the same ratio as the original polyhedron) while still remaining on the integer lattice? A necessary condition is given for this to be possible, which depends on the parity of k; for $k\leqq 4$ it is shown that the condition is sufficient, and algorithms are also given to do the shrinking. In two dimensions, the algorithm only involves computing greatest common divisors over the Gaussian integers and is polynomial time. For $k = 3$ and 4 the algorithms involve computing g.c.d.s in the algebra of Hurwitz quaternions. This gives a polynomial time algorithm for $k = 4$, but because the algorithm in three dimensions relies on determining the square factors of an integer, it is at present exponential. The proofs are remarkably simple and are quite computational in nature.