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 $r\geq 13$, 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.