Regular convex 4-polytopes |
Symmetry group |
A4 |
B4 |
F4 |
H4 |
Name |
5-cell
Hyper-tetrahedron
5-point |
16-cell
Hyper-octahedron
8-point |
8-cell
Hyper-cube
16-point |
24-cell
24-point
|
600-cell
Hyper-icosahedron
120-point |
120-cell
Hyper-dodecahedron
600-point |
SchlΓ€fli symbol |
{3, 3, 3} |
{3, 3, 4} |
{4, 3, 3} |
{3, 4, 3} |
{3, 3, 5} |
{5, 3, 3} |
Coxeter mirrors |
![](//upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png) ![](//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png) ![](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png) ![](//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png) ![](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png) ![](//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png) ![](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png) |
![](//upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png) ![](//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png) ![](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png) ![](//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png) ![](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png) ![](//upload.wikimedia.org/wikipedia/commons/8/8c/CDel_4.png) ![](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png) |
![](//upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png) ![](//upload.wikimedia.org/wikipedia/commons/8/8c/CDel_4.png) ![](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png) ![](//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png) ![](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png) ![](//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png) ![](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png) |
![](//upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png) ![](//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png) ![](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png) ![](//upload.wikimedia.org/wikipedia/commons/8/8c/CDel_4.png) ![](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png) ![](//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png) ![](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png) |
![](//upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png) ![](//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png) ![](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png) ![](//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png) ![](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png) ![](//upload.wikimedia.org/wikipedia/commons/1/16/CDel_5.png) ![](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png) |
![](//upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png) ![](//upload.wikimedia.org/wikipedia/commons/1/16/CDel_5.png) ![](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png) ![](//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png) ![](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png) ![](//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.png) ![](//upload.wikimedia.org/wikipedia/commons/5/5e/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 |
![](//upload.wikimedia.org/wikipedia/commons/thumb/b/b9/4-simplex_t0.svg/120px-4-simplex_t0.svg.png) |
![](//upload.wikimedia.org/wikipedia/commons/thumb/4/4c/4-cube_t3.svg/120px-4-cube_t3.svg.png) |
![](//upload.wikimedia.org/wikipedia/commons/thumb/6/67/4-cube_t0.svg/120px-4-cube_t0.svg.png) |
![](//upload.wikimedia.org/wikipedia/commons/thumb/a/af/24-cell_t0_F4.svg/120px-24-cell_t0_F4.svg.png) |
![](//upload.wikimedia.org/wikipedia/commons/thumb/f/f4/600-cell_graph_H4.svg/120px-600-cell_graph_H4.svg.png) |
![](//upload.wikimedia.org/wikipedia/commons/thumb/c/c1/120-cell_graph_H4.svg/120px-120-cell_graph_H4.svg.png) |
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 x 2 |
1 octagon x 3 |
2 octagons x 4 |
2 dodecagons x 4 |
4 30-gons x 6 |
20 30-gons x 4 |
Long radius |
![{\displaystyle 1}](//wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf) |
![{\displaystyle 1}](//wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf) |
![{\displaystyle 1}](//wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf) |
![{\displaystyle 1}](//wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf) |
![{\displaystyle 1}](//wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf) |
![{\displaystyle 1}](//wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf) |
Edge length |
![{\displaystyle {\sqrt {\tfrac {5}{2}}}\approx 1.581}](//wikimedia.org/api/rest_v1/media/math/render/svg/d7891ac9b4af750cc3e54f5f21486f4b396729a2) |
![{\displaystyle {\sqrt {2}}\approx 1.414}](//wikimedia.org/api/rest_v1/media/math/render/svg/f8fafaf1870e0527a631858dcc0c7579b34ff297) |
![{\displaystyle 1}](//wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf) |
![{\displaystyle 1}](//wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf) |
![{\displaystyle {\tfrac {1}{\phi }}\approx 0.618}](//wikimedia.org/api/rest_v1/media/math/render/svg/5b1e8a92bb277903a43ca3ae567ea64505fc32eb) |
![{\displaystyle {\tfrac {1}{\phi ^{2}{\sqrt {2}}}}\approx 0.270}](//wikimedia.org/api/rest_v1/media/math/render/svg/29afe3cfc54031633a59df0637d1a3f7f71bf684) |
Short radius |
![{\displaystyle {\tfrac {1}{4}}}](//wikimedia.org/api/rest_v1/media/math/render/svg/4825cd2a1ca51dfc4d53042434f6d3733370a57f) |
![{\displaystyle {\tfrac {1}{2}}}](//wikimedia.org/api/rest_v1/media/math/render/svg/edef8290613648790a8ac1a95c2fb7c3972aea2f) |
![{\displaystyle {\tfrac {1}{2}}}](//wikimedia.org/api/rest_v1/media/math/render/svg/edef8290613648790a8ac1a95c2fb7c3972aea2f) |
![{\displaystyle {\sqrt {\tfrac {1}{2}}}\approx 0.707}](//wikimedia.org/api/rest_v1/media/math/render/svg/8f4dc0f6527d5ac3f56ed3e68bc66e0b2f6a2bc3) |
![{\displaystyle {\sqrt {\tfrac {\phi ^{4}}{8}}}\approx 0.926}](//wikimedia.org/api/rest_v1/media/math/render/svg/b05c35ecaf7f1d828adb58f78720535d06a3c177) |
![{\displaystyle {\sqrt {\tfrac {\phi ^{4}}{8}}}\approx 0.926}](//wikimedia.org/api/rest_v1/media/math/render/svg/b05c35ecaf7f1d828adb58f78720535d06a3c177) |
Area |
![{\displaystyle 10\left({\tfrac {5{\sqrt {3}}}{8}}\right)\approx 10.825}](//wikimedia.org/api/rest_v1/media/math/render/svg/3eaa7c7d04842630db48b33b9d74c338be70c6b2) |
![{\displaystyle 32\left({\sqrt {\tfrac {3}{4}}}\right)\approx 27.713}](//wikimedia.org/api/rest_v1/media/math/render/svg/14b95c593579e89ea2c8e3ebb2207bed8433a02c) |
![{\displaystyle 24}](//wikimedia.org/api/rest_v1/media/math/render/svg/0be92101c8b0277e66fdefeef1ccdd7788e88ef5) |
![{\displaystyle 96\left({\sqrt {\tfrac {3}{16}}}\right)\approx 41.569}](//wikimedia.org/api/rest_v1/media/math/render/svg/d7b99126f57cc5b6592e4ef72670abbdf84068c0) |
![{\displaystyle 1200\left({\tfrac {\sqrt {3}}{4\phi ^{2}}}\right)\approx 198.48}](//wikimedia.org/api/rest_v1/media/math/render/svg/5688be271d95c5759354e1e5459c0fc1f8cbfc64) |
![{\displaystyle 720\left({\tfrac {\sqrt {25+10{\sqrt {5}}}}{8\phi ^{4}}}\right)\approx 90.366}](//wikimedia.org/api/rest_v1/media/math/render/svg/f14322accb342db5a422b3b14329f5e4576ef6e5) |
Volume |
![{\displaystyle 5\left({\tfrac {5{\sqrt {5}}}{24}}\right)\approx 2.329}](//wikimedia.org/api/rest_v1/media/math/render/svg/8253a425495bc8bc6c197e38e71d1eb1515a37d7) |
![{\displaystyle 16\left({\tfrac {1}{3}}\right)\approx 5.333}](//wikimedia.org/api/rest_v1/media/math/render/svg/6d4c8bcc6a19368c79cc7db6ebad8bb5b10d5cfb) |
![{\displaystyle 8}](//wikimedia.org/api/rest_v1/media/math/render/svg/1aaa997e6ad67716cfaa9a02c4df860bf60a95b5) |
![{\displaystyle 24\left({\tfrac {\sqrt {2}}{3}}\right)\approx 11.314}](//wikimedia.org/api/rest_v1/media/math/render/svg/dbd86a479242f86673a77d371d02474fd1211994) |
![{\displaystyle 600\left({\tfrac {\sqrt {2}}{12\phi ^{3}}}\right)\approx 16.693}](//wikimedia.org/api/rest_v1/media/math/render/svg/6510bf43702967fc6602d15435caea9989169b09) |
![{\displaystyle 120\left({\tfrac {15+7{\sqrt {5}}}{4\phi ^{6}{\sqrt {8}}}}\right)\approx 18.118}](//wikimedia.org/api/rest_v1/media/math/render/svg/8e90aaf6322085e2c0dd2bf8ca33e054d4c46592) |
4-Content |
![{\displaystyle {\tfrac {\sqrt {5}}{24}}\left({\tfrac {\sqrt {5}}{2}}\right)^{4}\approx 0.146}](//wikimedia.org/api/rest_v1/media/math/render/svg/fdd8259f9d61b844367d627adf4da0edae4aef88) |
![{\displaystyle {\tfrac {2}{3}}\approx 0.667}](//wikimedia.org/api/rest_v1/media/math/render/svg/84f729b57d66245544a3f57b8d2b0a31a0a9a883) |
![{\displaystyle 1}](//wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf) |
![{\displaystyle 2}](//wikimedia.org/api/rest_v1/media/math/render/svg/901fc910c19990d0dbaaefe4726ceb1a4e217a0f) |
![{\displaystyle {\tfrac {{\text{Short}}\times {\text{Vol}}}{4}}\approx 3.863}](//wikimedia.org/api/rest_v1/media/math/render/svg/04209037c0378ec46376aba5e33615134ace5ddf) |
![{\displaystyle {\tfrac {{\text{Short}}\times {\text{Vol}}}{4}}\approx 4.193}](//wikimedia.org/api/rest_v1/media/math/render/svg/c5f33beed2b51250ff3c61ead8b4214092c74e0e) |