Integral polytope

An integral polytope or lattice 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.

Not all abstract polytopes admit a realization with integer vertex coordinates. All convex simple and simplicial polytopes can be realized as integral polytopes, and all convex polytopes up to rank 3 unconditionally admit integral realizations. But for rank 4 and above, convex polytopes exist that do not have an integral realization. If an integral realization does exist, the magnitudes of coordinates appear to grow doubly exponential in the number of vertices.

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 $A=i+b/2-1$.

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".