# Heronian tetrahedra are lattice tetrahedra

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
**R**^{3} 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

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Heronian tetrahedra are lattice tetrahedra.

Last change: 26 March 2012.

Our agreement with the MAA gives us permission to post the article here.

Last change: 26 March 2012.

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Our agreement with the MAA gives us permission to post the article here.