# Reeve tetrahedra

Reeve tetrahedra
Rank3
Dimension3
TypeIntegral
Elements
Faces1+1+1+1
Edges1+1+1+1+1+1
Vertices1+1+1+1
Measures (edge length 1)
Volumer/6
Central density1
Abstract & topological properties
Flag count24
Euler characteristic2
SurfaceSphere
OrientableYes
Genus0
Properties
SymmetryI×I×I, order 1
Flag orbits24
ConvexYes
NatureTame

The Reeve tetrahedra are a family of integral polyhedra. Their existence demonstrates that there is no direct generalization of Pick's theorem to 3 dimensions.

## Vertex coordinates

A Reeve tetrahedron is a tetrahedron with vertex coordinates:

• ${\displaystyle \left(0,\,0,\,0\right)}$,
• ${\displaystyle \left(1,\,0,\,0\right)}$,
• ${\displaystyle \left(0,\,1,\,0\right)}$,
• ${\displaystyle \left(1,\,1,\,r\right)}$,

where r  is a positive integer.

## Pick's theorem

Although no Reeve tetrahedron contains no interior integral points, and no boundary points other than its 4 vertices, its volume r/6 is unbounded. Thus there is no way to directly generalize Pick's theorem so that it correctly gives the volume of the Reeve tetrahedra.

However generalizations of Pick's theorem involving Ehrhart polynomials do correctly calculate the volumes of the Reeve tetrahedra.