Name |
Tetrahedron |
Hexahedron |
Octahedron |
Dodecahedron |
Icosahedron |
|
Picture |
|
|
|
|
|
|
Schläfli symbol |
{3, 3} | {4, 3} | {3, 4} | {5, 3} | {3, 5} | ![{\displaystyle \{p,q\}}](//wikimedia.org/api/rest_v1/media/math/render/svg/bbc4bf30345d33ef9dca8944f51ea15fbb73fb3c) |
Face polygon |
triangle | square | triangle | pentagon | triangle | ![{\displaystyle \{p\}}](//wikimedia.org/api/rest_v1/media/math/render/svg/4a51b4d14ebfbef2ccdd2c4f108fa8e5bfc31796) |
Vertex figure |
triangle | triangle | square | triangle | pentagon | ![{\displaystyle \{q\}}](//wikimedia.org/api/rest_v1/media/math/render/svg/776a93f310cc623c2dfecbe153dad9066c6c3712) |
Vertices ( ) |
4 | 8 | 6 | 20 | 12 | ![{\displaystyle {\frac {4p}{4-(p-2)(q-2)}}}](//wikimedia.org/api/rest_v1/media/math/render/svg/1818199da70db364128f28643c72ae0aa3df2dad) |
Edges ( ) |
6 |
12 |
30 |
![{\displaystyle {\frac {2pq}{4-(p-2)(q-2)}}}](//wikimedia.org/api/rest_v1/media/math/render/svg/9210829e0c7e602e423e2c473fea4579b87d375d) |
Faces ( ) |
4 | 6 | 8 | 12 | 20 | ![{\displaystyle {\frac {4q}{4-(p-2)(q-2)}}}](//wikimedia.org/api/rest_v1/media/math/render/svg/bfbdd55fdbd39bfae37ff76c5e8828a6657d5f44) |
Euler characteristic ( ) |
2 |
2 |
Coxeter number ( ) |
4 |
6 |
10 |
![{\displaystyle {\frac {2(p+q+2)}{10-p-q}}}](//wikimedia.org/api/rest_v1/media/math/render/svg/0e82b6b71b4118dccf41abb6fad5b4a2c3834298) |
Vertex configuration |
3.3.3 | 4.4.4 | 3.3.3.3 | 5.5.5 | 3.3.3.3.3 | ![{\displaystyle p^{q}}](//wikimedia.org/api/rest_v1/media/math/render/svg/75ead10b72d59f44fbc277dbd66b6797dcbbf634) |
Wythoff symbol |
3 | 3 2 | 3 | 4 2 | 4 | 3 2 | 3 | 5 2 | 5 | 3 2 | q | p 2 |
Coxeter diagram |
![](//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/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/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/b/bd/CDel_node_1.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/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/c/c3/CDel_3.png) ![](//upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png) |
![](//upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png) ![](//upload.wikimedia.org/wikipedia/commons/0/0e/CDel_p.png) ![](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png) ![](//upload.wikimedia.org/wikipedia/commons/0/0b/CDel_q.png) ![](//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.png) |
Dual polyhedron |
tetrahedron | octahedron | hexahedron | icosahedron | dodecahedron | ![{\displaystyle \{q,p\}}](//wikimedia.org/api/rest_v1/media/math/render/svg/00bec71d501201ac2f0d225dbaecb5966ed006c8) |
Symmetry group |
tetrahedral group [3, 3] *332 Td A3 |
octahedral group [4, 3] *432 Oh BC3 |
icosahedral group [5, 3] *532 Ih H3 |
![{\displaystyle [p,q]}](//wikimedia.org/api/rest_v1/media/math/render/svg/28e13d57ecb142eac51528510911a42941046fcc) |
Symmetries |
24 | 48 | 120 | ![{\displaystyle 4N_{1}}](//wikimedia.org/api/rest_v1/media/math/render/svg/ab14fc32d75c6f362c648a8fa8b2a9c5a87cb1eb) |
Rotation symmetry group |
[3, 3]+ 332 T |
[4, 3]+ 432 O |
[5, 3]+ 532 I |
![{\displaystyle [p,q]^{+}}](//wikimedia.org/api/rest_v1/media/math/render/svg/0cac33465a0601ba6a86beb9d3913f70575a9019) |
Rotational symmetries |
12 | 24 | 60 | ![{\displaystyle 2N_{1}}](//wikimedia.org/api/rest_v1/media/math/render/svg/2eef6af0dfcec81e7eafee868e4596c0a235adbb) |
Face solid angle |
![{\displaystyle {\frac {\tau }{2}}}](//wikimedia.org/api/rest_v1/media/math/render/svg/4d2fad634e47741d4cc0325120ca1a14ed44e5e4) | ![{\displaystyle {\frac {\tau }{3}}}](//wikimedia.org/api/rest_v1/media/math/render/svg/6b4f9b8436f8f2d853c25d48cd6b19baf08651b6) | ![{\displaystyle {\frac {\tau }{4}}}](//wikimedia.org/api/rest_v1/media/math/render/svg/9cc338339ad0a66209f8e053e9961bab1b0d9272) | ![{\displaystyle {\frac {\tau }{6}}}](//wikimedia.org/api/rest_v1/media/math/render/svg/4f8d1a9b45ed7cd475d80e9925aa7f493d0f1ccd) | ![{\displaystyle {\frac {\tau }{10}}}](//wikimedia.org/api/rest_v1/media/math/render/svg/e737d79a6a86aa67763ab66e0a7d9b4bedcf5533) | ![{\displaystyle {\frac {2\tau }{N_{2}}}}](//wikimedia.org/api/rest_v1/media/math/render/svg/477c32f06d8d1537c4ff7f47e46658581b364892) |
Angular deficiency ( ) |
![{\displaystyle {\frac {\tau }{2}}}](//wikimedia.org/api/rest_v1/media/math/render/svg/4d2fad634e47741d4cc0325120ca1a14ed44e5e4) | ![{\displaystyle {\frac {\tau }{4}}}](//wikimedia.org/api/rest_v1/media/math/render/svg/9cc338339ad0a66209f8e053e9961bab1b0d9272) | ![{\displaystyle {\frac {\tau }{3}}}](//wikimedia.org/api/rest_v1/media/math/render/svg/6b4f9b8436f8f2d853c25d48cd6b19baf08651b6) | ![{\displaystyle {\frac {\tau }{10}}}](//wikimedia.org/api/rest_v1/media/math/render/svg/e737d79a6a86aa67763ab66e0a7d9b4bedcf5533) | ![{\displaystyle {\frac {\tau }{6}}}](//wikimedia.org/api/rest_v1/media/math/render/svg/4f8d1a9b45ed7cd475d80e9925aa7f493d0f1ccd) |
![{\displaystyle {\frac {2\tau }{N_{0}}}=\tau -q\tau \left({1 \over 2}-{1 \over p}\right)}](//wikimedia.org/api/rest_v1/media/math/render/svg/e0449e3cb35bfdb88cad0c5d371577fc545f9f9b) |
Dihedral angle ( ) |
≈ 70.53° | ![{\displaystyle {\frac {\tau }{4}}}](//wikimedia.org/api/rest_v1/media/math/render/svg/9cc338339ad0a66209f8e053e9961bab1b0d9272) | ≈ 109.47° | ≈ 116.56° | ≈ 138.19° | |
![{\displaystyle \tan {\theta \over 2}}](//wikimedia.org/api/rest_v1/media/math/render/svg/ec0726cfe5f63059da82a1f8db87a363ca6674a4) |
![{\displaystyle 1 \over {\sqrt {2}}}](//wikimedia.org/api/rest_v1/media/math/render/svg/290f5df76f84660af20a0e20d8c3527d4480f0e5) | ![{\displaystyle 1\,}](//wikimedia.org/api/rest_v1/media/math/render/svg/bfd1e7984fe6e1b79a26404a8138a6c6ee41a476) | ![{\displaystyle {\sqrt {2}}}](//wikimedia.org/api/rest_v1/media/math/render/svg/b4afc1e27d418021bf10898eb44a7f5f315735ff) | ![{\displaystyle \varphi }](//wikimedia.org/api/rest_v1/media/math/render/svg/33ee699558d09cf9d653f6351f9fda0b2f4aaa3e) | ![{\displaystyle \varphi ^{2}}](//wikimedia.org/api/rest_v1/media/math/render/svg/0ddf78fcdf8a1800bcf11694e2bb57d1f1cc735e) |
![{\displaystyle {\frac {\cos \left({\frac {\tau }{2q}}\right)}{\sin \left({\frac {\tau }{2h}}\right)}}}](//wikimedia.org/api/rest_v1/media/math/render/svg/309015291a8909dc734a6430df67c140b434a271) |
![{\displaystyle \cos \theta \,}](//wikimedia.org/api/rest_v1/media/math/render/svg/4082f787030c0c48e20d9270415c9e6208aa6b07) |
![{\displaystyle 1 \over 3}](//wikimedia.org/api/rest_v1/media/math/render/svg/0dd4476fddc9a8ba0c0b2c4e0b98b59417698da6) | ![{\displaystyle 0\,}](//wikimedia.org/api/rest_v1/media/math/render/svg/db4b06f9315849466a0502680377e30a9da8a1b5) | ![{\displaystyle -{1 \over 3}}](//wikimedia.org/api/rest_v1/media/math/render/svg/5f8e174390eb2743afa0ff5e7c83e5f62523363c) | ![{\displaystyle -{1 \over {\sqrt {5}}}}](//wikimedia.org/api/rest_v1/media/math/render/svg/67d25870c31bcf60349a83ecbb2fd1ede06193ef) | ![{\displaystyle -{{\sqrt {5}} \over 3}}](//wikimedia.org/api/rest_v1/media/math/render/svg/4f3ea025c3e64d72e1de3575e709bddf984617a7) | |
Solid angle ( ) |
≈ 0.551286 | ![{\displaystyle {\frac {\tau }{4}}}](//wikimedia.org/api/rest_v1/media/math/render/svg/9cc338339ad0a66209f8e053e9961bab1b0d9272) | ≈ 1.35935 | ≈ 2.96174 | ≈ 2.63455 | ![{\displaystyle q\theta -(q-2){\tau \over 2}}](//wikimedia.org/api/rest_v1/media/math/render/svg/6832a0beb3e35039a9995c63490e3bf4b304e86b) |
Standard vertex coordinates |
(1,1,−1) (1,−1,1) (−1,1,1) (−1,−1,−1) |
(±1,±1,±1) |
(±1,0,0) (0,±1,0) (0,0,±1) |
(±1,±1,±1) (0,±1/φ,±φ) (±1/φ,±φ,0) (±φ,0,±1/φ) |
(0,±1,±φ) (±1,±φ,0) (±φ,0,±1) |
Edge length ( ) |
![{\displaystyle 2{\sqrt {2}}}](//wikimedia.org/api/rest_v1/media/math/render/svg/c7e70f0d473d5c0f14cd3c10c139f51295788fb5) | ![{\displaystyle 2\,}](//wikimedia.org/api/rest_v1/media/math/render/svg/53f0585b3d3c0d207a91af7a41e4173b58f309ae) | ![{\displaystyle {\sqrt {2}}}](//wikimedia.org/api/rest_v1/media/math/render/svg/b4afc1e27d418021bf10898eb44a7f5f315735ff) | ![{\displaystyle 2/\varphi }](//wikimedia.org/api/rest_v1/media/math/render/svg/8bec1875d5fb90def68cac97468154e999079e4a) | ![{\displaystyle 2\,}](//wikimedia.org/api/rest_v1/media/math/render/svg/53f0585b3d3c0d207a91af7a41e4173b58f309ae) |
Circumradius ( ) |
![{\displaystyle {\sqrt {3}}}](//wikimedia.org/api/rest_v1/media/math/render/svg/3b19c09494138b5082459afac7f9a8d99c546fcd) | ![{\displaystyle {\sqrt {3}}}](//wikimedia.org/api/rest_v1/media/math/render/svg/3b19c09494138b5082459afac7f9a8d99c546fcd) | ![{\displaystyle 1\,}](//wikimedia.org/api/rest_v1/media/math/render/svg/bfd1e7984fe6e1b79a26404a8138a6c6ee41a476) | ![{\displaystyle {\sqrt {3}}}](//wikimedia.org/api/rest_v1/media/math/render/svg/3b19c09494138b5082459afac7f9a8d99c546fcd) | ![{\displaystyle \xi \varphi }](//wikimedia.org/api/rest_v1/media/math/render/svg/482121f25f6117ebed2838261b0ad5a850e488e3) |
Midradius ( ) |
![{\displaystyle 1\,}](//wikimedia.org/api/rest_v1/media/math/render/svg/bfd1e7984fe6e1b79a26404a8138a6c6ee41a476) | ![{\displaystyle {\sqrt {2}}}](//wikimedia.org/api/rest_v1/media/math/render/svg/b4afc1e27d418021bf10898eb44a7f5f315735ff) | ![{\displaystyle 1 \over {\sqrt {2}}}](//wikimedia.org/api/rest_v1/media/math/render/svg/290f5df76f84660af20a0e20d8c3527d4480f0e5) | ![{\displaystyle \varphi }](//wikimedia.org/api/rest_v1/media/math/render/svg/33ee699558d09cf9d653f6351f9fda0b2f4aaa3e) | ![{\displaystyle \varphi }](//wikimedia.org/api/rest_v1/media/math/render/svg/33ee699558d09cf9d653f6351f9fda0b2f4aaa3e) |
Inradius ( ) |
![{\displaystyle 1 \over {\sqrt {3}}}](//wikimedia.org/api/rest_v1/media/math/render/svg/a7d6de16b86e958b902f4205962589ff9bd2055d) | ![{\displaystyle 1\,}](//wikimedia.org/api/rest_v1/media/math/render/svg/bfd1e7984fe6e1b79a26404a8138a6c6ee41a476) | ![{\displaystyle 1 \over {\sqrt {3}}}](//wikimedia.org/api/rest_v1/media/math/render/svg/a7d6de16b86e958b902f4205962589ff9bd2055d) | ![{\displaystyle \varphi /\xi }](//wikimedia.org/api/rest_v1/media/math/render/svg/55a6581bb890a945748d10af5671df4fa8193251) | ![{\displaystyle \varphi ^{2} \over 3}](//wikimedia.org/api/rest_v1/media/math/render/svg/7d9cff4d204da721f02b39b45676f8e99c49cad6) |
Surface area |
![{\displaystyle 8{\sqrt {3}}}](//wikimedia.org/api/rest_v1/media/math/render/svg/107e43cf45d5588d3c3b6b0f80de84f5db23c44f) | ![{\displaystyle 24\,}](//wikimedia.org/api/rest_v1/media/math/render/svg/982ad40dc729a0058a68f9bf04fb0fa30a65b753) | ![{\displaystyle 4{\sqrt {3}}}](//wikimedia.org/api/rest_v1/media/math/render/svg/91aa30bbdc174c99ec494a94eff697ac0387639f) | ![{\displaystyle 60 \over {\xi \varphi }}](//wikimedia.org/api/rest_v1/media/math/render/svg/ed46d9eae4a4edace3b67ce44eec3ef6fae3db66) | ![{\displaystyle 20{\sqrt {3}}}](//wikimedia.org/api/rest_v1/media/math/render/svg/905c2c3259eee2103a66fe07c266f8ae6f671a88) |
Volume |
![{\displaystyle 8 \over 3}](//wikimedia.org/api/rest_v1/media/math/render/svg/bb613c60a56d129d6ba18e887477fccd96f2087d) | ![{\displaystyle 8\,}](//wikimedia.org/api/rest_v1/media/math/render/svg/333ca76beefe158eccf6636fe15e471a5093fcac) | ![{\displaystyle 4 \over 3}](//wikimedia.org/api/rest_v1/media/math/render/svg/81cd3d0e50de1ae56edc100b08cbfa84b68bea8d) | ![{\displaystyle 4(\varphi +2)}](//wikimedia.org/api/rest_v1/media/math/render/svg/6c7eef43090f0db4d18500175dabe44e2837407b) | ![{\displaystyle {20 \over 3}\varphi ^{2}}](//wikimedia.org/api/rest_v1/media/math/render/svg/156d7be5548eae4c84d30d58dd548b552ab74a35) |