In geometry, the Reeve tetrahedron is a polyhedron, named after John Reeve, in with vertices at , , , and where is a positive integer. Each vertex lies on a fundamental lattice point (a point in ). No other fundamental lattice points lie on the surface or in the interior of the tetrahedron. In 1957 Reeve used this tetrahedron as a counterexample to show that there is no simple equivalent of Pick's theorem in or higher-dimensional spaces. This is seen by noticing that Reeve tetrahedra have the same number of interior and boundary points for any value of , but different volumes.
The Ehrhart polynomial of the Reeve tetrahedron of height is
Thus, for , the Ehrhart polynomial of has a negative coefficient.
- J. E. Reeve, "On the Volume of Lattice Polyhedra", Proceedings of the London Mathematical Society, s3–7(1):378–395
- Beck, Matthias; Robins, Sinai (2007). Computing the Continuous Discretely. New York: Springer. p. 75.
- Kołodziejczyk, Krzysztof (1996). "An 'Odd' Formula for the Volume of Three-Dimensional Lattice Polyhedra", Geometriae Dedicata 61: 271–278.
- Beck, Matthias; Robins, Sinai (2007), Computing the Continuous Discretely, Integer-point enumeration in polyhedra, Undergraduate Texts in Mathematics, New York: Springer-Verlag, ISBN 978-0-387-29139-0, MR 2271992.