Heronian tetrahedra are lattice tetrahedra

American Mathematical Monthly Cover Susan H. Marshall, Alexander R. Perlis.
Heronian tetrahedra are lattice tetrahedra.
American Mathematical Monthly 120:2 (2013), 140-149.

Received the 2014 MAA Paul Halmos–Lester Ford award.

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.

2010 Mathematics Subject Classification (MSC):
  • 52C10 Convex and discrete geometry | Discrete geometry | Erdos problems and related topics of discrete geometry
  • 11R52 Number theory | Algebraic number theory: global fields | Quaternion and other division algebras: arithmetic, zeta functions
  • 52B20 Convex and discrete geometry | Polytopes and polyhedra | Lattice polytopes
Key words and phrases:
  • integral distance set
  • Heronian
  • tetrahedron
  • quaternion
  • Lipschitz integer

Downloads:

Image of page 1 Heronian tetrahedra are lattice tetrahedra.
Last change: 26 March 2012.
PDF385K
Our agreement with the MAA gives us permission to post the article here.