# Integral polytope

An **integral polytope** is a polytope where all vertices can be placed at integer coordinates in Euclidean space. The term is ambiguous without specifying the dimension of Euclidean space; for example, an equilateral triangle is not an integral polytope in 2D, but it is in 3D and above.

If a finite polytope can be represented with rational coordinates then it is an integral polytope as well. Thus some theorems that reference **rational polytopes** apply equally to integral polytopes.

## Examples

Examples of integral polytopes include

- hypercubes
- orthoplexes
- demihypercubes
- the n -simplex (in
*n*+ 1 dimensions) - the n -permutohedron (in
*n*+ 1 dimensions) - the n -Birkhoff polytope (in n 2 dimensions)
- the truncated n -simplex (in
*n*+ 1 dimensions) - Farey sunbursts
- Jessen's icosahedron
- Golygons.

## Pick's theorem and Ehrhart polynomials

Integral polygons in 2D that lack self-intersections or holes are subject to a simple formula known as Pick's theorem that relate the area of the polygon *A*, the number of lattice points on the boundary *b*, and the number of lattice points in the interior *i*. The theorem states that .

Attempting to generalize Pick's theorem to three dimensions or more requires extra work, as shown by John Reeve in 1957. Consider a tetrahedron whose vertices are located at (0, 0, 0), (1, 0, 0), (0, 1, 0), and (1, 1, *r*) for integer *r* > 0. The volume of this tetrahedron is *r*/6, but the number of interior and boundary lattice points does not change with *r*, so the tetrahedron's volume cannot be a function of *i* and *b*. These tetrahedra are known as Reeve tetrahedra.

The proper generalization to any number of dimensions is given by Ehrhart polynomials. For an *n*-polytope *P* integral in *n* dimensions and positive integer *t*, let *L*(*t*) be the number of lattice points in the boundary and interior of *P* if each vertex in *P* is uniformly scaled from the origin by a factor of *t*. Then *L*(*t*) is a polynomial in *t* with rational coefficients.

## External links

- Wikipedia contributors. "Integral polytope".