Reeve tetrahedron

In geometry, the Reeve tetrahedron is a polyhedron, named after John Reeve, in $\mathbb{R}^3$ with vertices at $(0, 0, 0)$, $(1, 0, 0)$, $(0, 1, 0)$, and $(1, 1, r)$ where $r$ is a positive integer. Each vertex lies on a fundamental lattice point (a point in $\mathbb{Z}^3$). 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 $\mathbb{R}^3$ or higher-dimensional spaces.^[1] This is seen by noticing that Reeve tetrahedra have the same number of interior and boundary points for any value of $r$, but different volumes.

The Ehrhart polynomial of the Reeve tetrahedron $\mathcal{T}_r$ of height $r$ is

$L(\mathcal{T}_r, t) = \frac{r}{6}t^3 + t^2 + \left(2 - \frac{r}{6}\right)t + 1.$^[2]

Thus, for $r \ge 13$, the Ehrhart polynomial of $\mathcal{T}_r$ has a negative coefficient.

Notes

1. J. E. Reeve, "On the Volume of Lattice Polyhedra", Proceedings of the London Mathematical Society, s3–7(1):378–395
2. Beck, Matthias; Robins, Sinai (2007). Computing the Continuous Discretely. New York: Springer. p. 75.

References

• 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.