Reeve tetrahedra

Reeve tetrahedra
Rank	3
Dimension	3
Type	Integral
Elements
Faces	1+1+1+1
Edges	1+1+1+1+1+1
Vertices	1+1+1+1
Measures (edge length 1)
Volume	r/6
Central density	1
Abstract & topological properties
Flag count	24
Euler characteristic	2
Surface	Sphere
Orientable	Yes
Genus	0
Properties
Symmetry	I×I×I, order 1
Flag orbits	24
Convex	Yes
Nature	Tame

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:

• $\left(0,\,0,\,0\right)$,
• $\left(1,\,0,\,0\right)$,
• $\left(0,\,1,\,0\right)$,
• $\left(1,\,1,\,r\right)$,

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.

External links

• Wikipedia contributors. "Reeve tetrahedra".

Bibliography

• Kiradjiev, Kristian (December 2018), "Connecting the Dots with Pick's Theorem" (PDF), Mathematics Today, Institute of mathematics and its applications, retrieved January 6, 2023