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Heronian Tetrahedra Are Lattice Tetrahedra

Susan H. Marshall and Alexander R. Perlis
The American Mathematical Monthly
Vol. 120, No. 2 (February 2013), pp. 140-149
DOI: 10.4169/amer.math.monthly.120.02.140
Stable URL: http://www.jstor.org/stable/10.4169/amer.math.monthly.120.02.140
Page Count: 10
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Heronian Tetrahedra Are Lattice Tetrahedra
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Abstract

Abstract Extending a similar result about triangles, we show that each Heronian tetrahedron may be positioned with integer coordinates. More generally, we show the following: if an integral distance set in R3 can be positioned with rational coordinates, then it can in fact be positioned with integer coordinates. The proof, which uses the arithmetic of quaternions, is tantamount to an algorithm.

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