Regular convex 4-polytopes
Symmetry group
A4
B4
F4
H4
Name
5-cell Hyper-tetrahedron
16-cell Hyper-octahedron
8-cell Hyper-cube
24-cell
600-cell Hyper-icosahedron
120-cell Hyper-dodecahedron
SchlΓ€fli symbol
{3, 3, 3}
{3, 3, 4}
{4, 3, 3}
{3, 4, 3}
{3, 3, 5}
{5, 3, 3}
Coxeter mirrors
File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 4.png File:CDel node.png File:CDel node 1.png File:CDel 4.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 4.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel node 1.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel node.png File:CDel node 1.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png
Mirror dihedrals
π
/ 3 π
/ 3 π
/ 3 π
/ 2 π
/ 2 π
/ 2 π
/ 3 π
/ 3 π
/ 4 π
/ 2 π
/ 2 π
/ 2 π
/ 4 π
/ 3 π
/ 3 π
/ 2 π
/ 2 π
/ 2 π
/ 3 π
/ 4 π
/ 3 π
/ 2 π
/ 2 π
/ 2 π
/ 3 π
/ 3 π
/ 5 π
/ 2 π
/ 2 π
/ 2 π
/ 5 π
/ 3 π
/ 3 π
/ 2 π
/ 2 π
/ 2
Graph
File:4-simplex t0.svg
File:4-cube t0.svg
File:24-cell t0 F4.svg
File:600-cell graph H4.svg
File:120-cell graph H4.svg
Vertices
5 tetrahedral
8 octahedral
16 tetrahedral
24 cubical
120 icosahedral
600 tetrahedral
Edges
10 triangular
24 square
32 triangular
96 triangular
720 pentagonal
1200 triangular
Faces
10 triangles
32 triangles
24 squares
96 triangles
1200 triangles
720 pentagons
Cells
5 tetrahedra
16 tetrahedra
8 cubes
24 octahedra
600 tetrahedra
120 dodecahedra
Tori
1 5-tetrahedron
2 8-tetrahedron
2 4-cube
4 6-octahedron
20 30-tetrahedron
12 10-dodecahedron
Inscribed
120 in 120-cell
675 in 120-cell
2 16-cells
3 8-cells
25 24-cells
10 600-cells
Great polygons
2 squares x 3
4 rectangles x 4
4 hexagons x 4
12 decagons x 6
100 irregular hexagons x 4
Petrie polygons
1 pentagon
1 octagon
2 octagons
2 dodecagons
4 30-gons
20 30-gons
Long radius
1 {\displaystyle 1} 1 {\displaystyle 1} 1 {\displaystyle 1} 1 {\displaystyle 1} 1 {\displaystyle 1} 1 {\displaystyle 1}
Edge length
5 2 β 1.581 {\displaystyle {\sqrt {\tfrac {5}{2}}}\approx 1.581} 2 β 1.414 {\displaystyle {\sqrt {2}}\approx 1.414} 1 {\displaystyle 1} 1 {\displaystyle 1} 1 Ο β 0.618 {\displaystyle {\tfrac {1}{\phi }}\approx 0.618} 1 Ο 2 β’ 2 β 0.270 {\displaystyle {\tfrac {1}{\phi ^{2}{\sqrt {2}}}}\approx 0.270}
Short radius
1 4 {\displaystyle {\tfrac {1}{4}}} 1 2 {\displaystyle {\tfrac {1}{2}}} 1 2 {\displaystyle {\tfrac {1}{2}}} 1 2 β 0.707 {\displaystyle {\sqrt {\tfrac {1}{2}}}\approx 0.707} Ο 4 8 β 0.926 {\displaystyle {\sqrt {\tfrac {\phi ^{4}}{8}}}\approx 0.926} Ο 4 8 β 0.926 {\displaystyle {\sqrt {\tfrac {\phi ^{4}}{8}}}\approx 0.926}
Area
10 β’ ( 5 β’ 3 8 ) β 10.825 {\displaystyle 10\left({\tfrac {5{\sqrt {3}}}{8}}\right)\approx 10.825} 32 β’ ( 3 4 ) β 27.713 {\displaystyle 32\left({\sqrt {\tfrac {3}{4}}}\right)\approx 27.713} 24 {\displaystyle 24} 96 β’ ( 3 16 ) β 41.569 {\displaystyle 96\left({\sqrt {\tfrac {3}{16}}}\right)\approx 41.569} 1200 β’ ( 3 4 β’ Ο 2 ) β 198.48 {\displaystyle 1200\left({\tfrac {\sqrt {3}}{4\phi ^{2}}}\right)\approx 198.48} 720 β’ ( 25 + 10 β’ 5 8 β’ Ο 4 ) β 90.366 {\displaystyle 720\left({\tfrac {\sqrt {25+10{\sqrt {5}}}}{8\phi ^{4}}}\right)\approx 90.366}
Volume
5 β’ ( 5 β’ 5 24 ) β 2.329 {\displaystyle 5\left({\tfrac {5{\sqrt {5}}}{24}}\right)\approx 2.329} 16 β’ ( 1 3 ) β 5.333 {\displaystyle 16\left({\tfrac {1}{3}}\right)\approx 5.333} 8 {\displaystyle 8} 24 β’ ( 2 3 ) β 11.314 {\displaystyle 24\left({\tfrac {\sqrt {2}}{3}}\right)\approx 11.314} 600 β’ ( 2 12 β’ Ο 3 ) β 16.693 {\displaystyle 600\left({\tfrac {\sqrt {2}}{12\phi ^{3}}}\right)\approx 16.693} 120 β’ ( 15 + 7 β’ 5 4 β’ Ο 6 β’ 8 ) β 18.118 {\displaystyle 120\left({\tfrac {15+7{\sqrt {5}}}{4\phi ^{6}{\sqrt {8}}}}\right)\approx 18.118}
4-Content
5 24 β’ ( 5 2 ) 4 β 0.146 {\displaystyle {\tfrac {\sqrt {5}}{24}}\left({\tfrac {\sqrt {5}}{2}}\right)^{4}\approx 0.146} 2 3 β 0.667 {\displaystyle {\tfrac {2}{3}}\approx 0.667} 1 {\displaystyle 1} 2 {\displaystyle 2} Short Γ Vol 4 β 3.863 {\displaystyle {\tfrac {{\text{Short}}\times {\text{Vol}}}{4}}\approx 3.863} Short Γ Vol 4 β 4.193 {\displaystyle {\tfrac {{\text{Short}}\times {\text{Vol}}}{4}}\approx 4.193} 
