List of regular polytopes

This page lists the regular polytopes in Euclidean space.

Two dimensional regular polytopes

The two dimensional convex regular polytopes are regular polygons.

• The equilateral triangle, with Schläfli symbol {3}
• The square, with Schläfli symbol {4}
• The regular pentagon, with Schläfli symbol {5}
• The regular hexagon, with Schläfli symbol {6}
• The regular heptagon, with Schläfli symbol {7}
• The regular octagon, with Schläfli symbol {8}
• The regular enneagon, with Schläfli symbol {9}
• The regular decagon, with Schläfli symbol {10}
• and so on, pentagram (five-pointed star), with Schläfli symbol {5/2}
• Two different types of seven-pointed star, with Schläfli symbols {7/2} and {7/3}
• An eight-pointed star, with Schläfli symbol {8/3}
• Two different types of nine-pointed star, with Schläfli symbols {9/2} and {9/4}
• and so on, ad infinitum.

Three dimensional regular polytopes

In three dimensions, the convex regular polytopes (or polyhedra) are the Platonic solids.

• The tetrahedron, with Schläfli symbol {3,3}, faces are triangles, vertex figures are also triangles.
• The cube, with Schläfli symbol {4,3}, faces are squares, vertex figures are triangles.
• The octahedron, with Schläfli symbol {3,4}, faces are triangles, vertex figures are squares.
• The dodecahedron, with Schläfli symbol {5,3}, faces are pentagons, vertex figures are triangles.
• The icosahedron, with Schläfli symbol {3,5}, faces are triangles, vertex figures are pentagons.

There exist also non-convex regular polyhedra. These are the Kepler-Poinsot polyhedra.

• The great stellated dodecahedron, with Schläfli symbol {5/2,3}, faces are pentagrams, vertex figures are triangles.
• The small stellated dodecahedron, with Schläfli symbol {5/2,5}, faces are pentagrams, vertex figures are pentagons.
• The great icosahedron, with Schläfli symbol {3,5/2}, faces are triangles, vertex figures are pentagrams.
• The great dodecahedron, with Schläfli symbol {5,5/2}, faces are pentagons, vertex figures are pentagrams.

Four dimensional regular polytopes

In four dimensions, the convex regular polytopes are as follows.

• The 4-dimensional simplex, with Schläfli symbol {3,3,3}, faces and vertex figures are tetrahedra.
• The 24-cell, with Schläfli symbol {3,4,3}, faces are octahedra, vertex figures are cubes.
• The 4-dimensional cube, also called a hypercube or tesseract, with Schläfli symbol {4,3,3}, faces are cubes, vertex figures are tetrahedra.
• The 4-dimensional cross-polytope, with Schläfli symbol {3,3,4}, faces are tetrahedra, vertex figures are octahedra.
• The 120-cell, with Schläfli symbol {5,3,3}, faces are dodecahedra, vertex figures are tetrahedra.
• The 600-cell, with Schläfli symbol {3,3,5}, faces are tetrahedra, vertex figures are icosahedra.

There exist also ten non-convex regular polytopes in four dimensions.

• The stellated 120-cell, with Schläfli symbol {5/2,5,3}
• The great 120-cell, with Schläfli symbol {5,5/2,5}
• The icosahedral 120-cell, with Schläfli symbol {3,5,5/2}
• The great stellated 120-cell, with Schläfli symbol {5/2,3,5}
• The grand 120-cell, with Schläfli symbol {5,3,5/2}
• The grand stellated 120-cell, with Schläfli symbol {5/2,5,5/2}
• The great icosahderal 120-cell, with Schläfli symbol {3,5/2,5}
• The great grand 120-cell, with Schläfli symbol {5,5/2,3}
• The great grand stellated 120-cell, with Schläfli symbol {5/2,3,3}
• The grand 600-cell, with Schläfli symbol {3,3,5/2}

Higher dimensional regular polytopes

In dimensions higher than 4, there are only three kinds of convex regular polytopes.

• n-dimensional simplex, with Schläfli symbol {3,...,3}
• n-dimensional cube, also called a hypercube or tesseract, with Schläfli symbol {4,3,...,3}
• n-dimensional cross-polytope, with Schläfli symbol {3,...,3,4}

There are no non-convex regular polytopes in dimensions higher than 4.