In geometry, the Reeve tetrahedron is a polyhedron, named after John Reeve, in R3 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 Z3). 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 R3 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 r, but different volumes.
Notes and references
- J. E. Reeve, "On the Volume of Lattice Polyhedra", Proceedings of the London Mathematical Society, s3–7(1):378–395
- Kołodziejczyk, Krzysztof (1996). "An 'Odd' Formula for the Volume of Three-Dimensional Lattice Polyhedra", Geometriae Dedicata 61: 271–278.