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Integral polytope
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<table><tbody><tr><td></td><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td><td></td></tr><tr><td><a href="/facts/Cube/65lUaAqN">Cube</a></td><td><a href="/facts/Cuboctahedron/hD7p0Mem">Cuboctahedron</a></td><td><a href="/facts/Octahedron/G38q92xW">Octahedron</a></td><td><a href="/facts/Truncated_octahedron/GA98b4TH">Truncatedoctahedron</a></td></tr><tr><td>(±1, ±1, ±1)</td><td>(0, ±1, ±1)</td><td>(0, 0, ±1)</td><td>(0, ±1, ±2)</td></tr><tr><th colspan="4">Four integral polytopes in three dimensions</th></tr></tbody></table> <p>In geometry and <a href="/facts/Polyhedral_combinatorics/oVFESPjJ">polyhedral combinatorics</a>, an integral polytope is a <a href="/facts/Convex_polytope/2pRHcRgC">convex polytope</a> whose <a href="/facts/Vertex_(geometry)/ID3cea9z">vertices</a> all have <a href="/facts/Integer/PUwwrawx">integer</a> <a href="/facts/Cartesian_coordinates/lkTqvMmB">Cartesian coordinates</a>. That is, it is a polytope that equals the <a href="/facts/Convex_hull/D4Id3JDS">convex hull</a> of its <a href="/facts/Integer_point/h4FP4pjm">integer points</a>. Integral polytopes are also called lattice polytopes or Z-polytopes. The special cases of two- and three-dimensional integral polytopes may be called polygons or polyhedra instead of polytopes, respectively. </p>