List Of Regular Polytopes

Two dimensional regular polytopes

• The equilateral triangle, with Schlfli symbol {3}
• The square, with Schlfli symbol {4}
• The regular pentagon, with Schlfli symbol {5}
• The regular hexagon, with Schlfli symbol {6}
• The regular heptagon, with Schlfli symbol {7}
• The regular octagon, with Schlfli symbol {8}
• The regular enneagon, with Schlfli symbol {9}
• The regular decagon, with Schlfli symbol {10}
• and so on, ad infinitum.

• The pentagram (five-pointed star), with Schlfli symbol {5/2}
• Two different types of seven-pointed star, with Schlfli symbols {7/2} and {7/3}
• An eight-pointed star, with Schlfli symbol {8/3}
• Two different types of nine-pointed star, with Schlfli symbols {9/2} and {9/4}
• and so on, ad infinitum.

Three dimensional regular polytopes

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

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

Four dimensional regular polytopes

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

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

Higher dimensional regular polytopes

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

External links

• The Platonic Solids
• Kepler-Poinsot Polyhedra
• 4d Polytope Applet
• Regular 4d Polytope Foldouts
• Multidimensional Glossary (Look up Hexacosichoron and Hecatonicosachoron)

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