Regular convex 4-polytopes
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Symmetry group
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A4
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B4
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F4
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H4
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Name
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5-cell
Hyper-tetrahedron
5-point
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16-cell
Hyper-octahedron
8-point
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8-cell
Hyper-cube
16-point
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24-cell
Hyper-cuboctahedron
24-point
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600-cell
Hyper-icosahedron
120-point
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137-cell
Hyper-rhombic dodecahedron
137-point
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120-cell
Hyper-dodecahedron
600-point
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SchlΓ€fli symbol
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{3, 3, 3}
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{3, 3, 4}
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{4, 3, 3}
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{3, 4, 3}
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{3, 3, 5}
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{3, 5, 3}
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{5, 3, 3}
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Coxeter mirrors
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Mirror dihedrals
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π
/3 π
/3 π
/3 π
/2 π
/2 π
/2
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π
/3 π
/3 π
/4 π
/2 π
/2 π
/2
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π
/4 π
/3 π
/3 π
/2 π
/2 π
/2
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π
/3 π
/4 π
/3 π
/2 π
/2 π
/2
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π
/3 π
/3 π
/5 π
/2 π
/2 π
/2
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π
/3 π
/5 π
/3 π
/2 π
/2 π
/2
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π
/5 π
/3 π
/3 π
/2 π
/2 π
/2
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Graph
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Vertices[a]
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5 tetrahedral
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8 octahedral
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16 tetrahedral
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24 cubical
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120 icosahedral
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137 cuboctahedral
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600 tetrahedral
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Edges
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10 triangular
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24 square
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32 triangular
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96 triangular
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720 pentagonal
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1090 triangular
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1200 triangular
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Faces
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10 triangles
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32 triangles
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24 squares
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96 triangles
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1200 triangles
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1090 triangles
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720 pentagons
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Cells
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5 tetrahedra
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16 tetrahedra
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8 cubes
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24 octahedra
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600 tetrahedra
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137 rhombic dodecahedra
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120 dodecahedra
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Tori
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1 5-tetrahedron
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2 8-tetrahedron
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2 4-cube
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4 6-octahedron
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20 30-tetrahedron
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11 11-rhombic dodecahedron
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12 10-dodecahedron
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Inscribed
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120 in 120-cell 96 in 137-cell
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6 5-cells 675 in 120-cell
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2 16-cells 12 5-cells
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3 8-cells 3 16-cells
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25 24-cells 75 8-cells
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9 600-cells 200 24-cells
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11 137-cells 10 600-cells
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Pentads
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1
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6
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12
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24
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48
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96
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120
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Hexads
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1
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2
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3
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75
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600
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675
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Heptads
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1
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11
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55
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120
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Great polygons
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2 squares x 3
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4 rectangles x 4
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4 hexagons x 4
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12 decagons x 6
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11 hendecagons x 11
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100 irregular hexagons x 4
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Petrie polygons
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1 pentagon x 2
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1 octagon x 3
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2 octagons x 4
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2 dodecagons x 4
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4 30-gons x 6
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11 14-gons x 11
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20 30-gons x 4
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Long radius
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Edge length[b]
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Short radius
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Area
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Volume
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4-Content
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