Reeve tetrahedra

In geometry, the Reeve tetrahedra are a family of polyhedra with vertices at

They are named after John Reeve, who in 1957 used them to show that higher-dimensional generalizations of Pick's theorem do not exist.

No other lattice points lie on the surface or in the interior of the tetrahedron.

In 1957 Reeve used this tetrahedron to show that there exist tetrahedra with four lattice points as vertices, and containing no other lattice points, but with arbitrarily large volume.

[2] In two dimensions, the area of every polyhedron with lattice vertices is determined as a formula of the number of lattice points at its vertices, on its boundary, and in its interior, according to Pick's theorem.

The Reeve tetrahedra imply that there can be no corresponding formula for the volume in three or more dimensions.

Any such formula would be unable to distinguish the Reeve tetrahedra with different choices of r from each other, but their volumes are all different.

The Ehrhart polynomial of the Reeve tetrahedron Tr of height r is[4]

Thus, for r ≥ 13, the coefficient of t in the Ehrhart polynomial of Tr is negative.