Polytopes

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Polytopes are the n-dimensional "solids." 2-polytopes are polygons, 3-polytopes are polyhedra (the so called solids) and 4-polytopes are polychora. Regular polytopes are those with all sides, faces, cells, ... congruent.

Regular polytopes

There are four infinite families of regular polytopes.

For any number of sides n ≥ 3 there is a regular 2-polytope with that number of sides. These are the equilateral triangle, square, regular pentagon, ....

For any dimension d ≥ 1 there is a measure polytope formed by creating two copies of a d−1–measure polytope at unit distance and connecting corresponding points. These are a line segment, square, cube, tesseract, ....

For any dimension d ≥ 1 there is a simplex formed by connecting all points of a d−1–simplex to a new point at unit distance from the center of the old simplex. These are a line segment, equilateral triangle, regular tetrahedron, 5-cell, ....

For any dimension d ≥ 1 there is a cross polytope formed by connecting all points of a d−1–simplex to each of two new points at unit distance from the center of the old cross polytope. These are a square, regular octahedron, 16-cell, ....

In addition, there are five more regular polytopes: the dodecahedron (3D), icosahedron (3D), 24-cell (4D), 120-cell (4D), and 600-cell (4D). In particular, in dimension greater than four, there are precisely three regular polytopes: the measure polytope, the cross polytope, and the simplex. See A060296.

External links

• I. Hubard, Maths 782 Discrete Geometry Course notes.