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Table of polyhedron dihedral angles
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The dihedral angles for the edge-transitive polyhedra are:
| Picture | Name | Schläfli | ||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| symbol | Vertex/Face | |||||||||||||
| configuration | exact dihedral angle | |||||||||||||
| (radians) | dihedral angle | |||||||||||||
| – exact in bold, | ||||||||||||||
| else approximate | ||||||||||||||
| (degrees) | Platonic solids (regular convex) | Kepler–Poinsot polyhedra (regular nonconvex) | Quasiregular polyhedra (Rectified regular) | Ditrigonal polyhedra | Hemipolyhedra | Quasiregular dual solids | Duals of the ditrigonal polyhedra | Duals of the hemipolyhedra | ||||||
| [[File:Tetrahedron.png | 30px]] | Tetrahedron | {3,3} | (3.3.3) | \arccos(\frac{1}{3}) | 70.529° | ||||||||
| [[File:Hexahedron.png | 30px]] | Hexahedron or Cube | {4,3} | (4.4.4) | \arccos(0)=\frac{\pi}{2} | 90° | ||||||||
| [[File:Octahedron.png | 30px]] | Octahedron | {3,4} | (3.3.3.3) | \arccos(-\frac{1}{3}) | 109.471° | ||||||||
| [[File:Dodecahedron.png | 30px]] | Dodecahedron | {5,3} | (5.5.5) | \arccos(-\frac{\sqrt{5}}{5}) | 116.565° | ||||||||
| [[File:Icosahedron.png | 30px]] | Icosahedron | {3,5} | (3.3.3.3.3) | \arccos(-\frac{\sqrt{5}}{3}) | 138.190° | ||||||||
| [[File:Small stellated dodecahedron.png | 30px]] | Small stellated dodecahedron | (....) | \arccos(-\frac{\sqrt{5}}{5}) | 116.565° | |||||||||
| [[File:Great dodecahedron.png | 30px]] | Great dodecahedron | \arccos(\frac{\sqrt{5}}{5}) | 63.435° | ||||||||||
| [[File:Great stellated dodecahedron.png | 30px]] | Great stellated dodecahedron | ,3} | (..) | \arccos(\frac{\sqrt{5}}{5}) | 63.435° | ||||||||
| [[File:Great icosahedron.png | 30px]] | Great icosahedron | \arccos(\frac{\sqrt{5}}{3}) | 41.810° | ||||||||||
| [[File:Uniform polyhedron-33-t1.svg | 30px]] | Tetratetrahedron | r{3,3} | (3.3.3.3) | \arccos(-\frac{1}{3}) | 109.471° | ||||||||
| [[File:Cuboctahedron.png | 30px]] | Cuboctahedron | r{3,4} | (3.4.3.4) | \arccos(-\frac{\sqrt{3}}{3}) | 125.264° | ||||||||
| [[File:Icosidodecahedron.png | 30px]] | Icosidodecahedron | r{3,5} | (3.5.3.5) | \arccos{(-\frac{1}{15}\sqrt{75+30\sqrt{5}})} | 142.623° | ||||||||
| [[File:Dodecadodecahedron.png | 30px]] | Dodecadodecahedron | r,5} | (5..5.) | \arccos(-\frac{\sqrt{5}}{5}) | 116.565° | ||||||||
| [[File:Great icosidodecahedron.png | 30px]] | Great icosidodecahedron | r,3} | (3..3.) | \arccos{(\frac{1}{15}\sqrt{75+30\sqrt{5}})} | 37.377° | ||||||||
| [[File:Small ditrigonal icosidodecahedron.png | 30px]] | Small ditrigonal icosidodecahedron | a{5,3} | (3..3..3.) | \arccos{(-\frac{1}{15}\sqrt{75+30\sqrt{5}})} | 142.623° | ||||||||
| [[File:Ditrigonal dodecadodecahedron.png | 30px]] | Ditrigonal dodecadodecahedron | b | (5..5..5.) | \arccos(\frac{\sqrt{5}}{5}) | 63.435° | ||||||||
| [[File:Great ditrigonal icosidodecahedron.png | 30px]] | Great ditrigonal icosidodecahedron | c | \arccos{(\frac{1}{15}\sqrt{75-30\sqrt{5}})} | 79.188° | |||||||||
| [[File:Tetrahemihexahedron.png | 30px]] | Tetrahemihexahedron | o{3,3} | (3.4..4) | \arccos(\frac{\sqrt{3}}{3}) | 54.736° | ||||||||
| [[File:Cubohemioctahedron.png | 30px]] | Cubohemioctahedron | o{3,4} | (4.6..6) | \arccos(\frac{\sqrt{3}}{3}) | 54.736° | ||||||||
| [[File:Octahemioctahedron.png | 30px]] | Octahemioctahedron | o{4,3} | (3.6..6) | \arccos(\frac{1}{3}) | 70.529° | ||||||||
| [[File:Small dodecahemidodecahedron.png | 30px]] | Small dodecahemidodecahedron | o{3,5} | (5.10..10) | \arccos{(\frac{1}{15} \sqrt{195-6\sqrt{5}})} | 26.058° | ||||||||
| [[File:Small icosihemidodecahedron.png | 30px]] | Small icosihemidodecahedron | o{5,3} | (3.10..10) | \arccos(-\frac{\sqrt{5}}{5}) | 116.565° | ||||||||
| [[File:Great dodecahemicosahedron.png | 30px]] | Great dodecahemicosahedron | o,5} | (5.6..6) | \arccos{(\frac{1}{15}\sqrt{75+30\sqrt{5}})} | 37.377° | ||||||||
| [[File:Small dodecahemicosahedron.png | 30px]] | Small dodecahemicosahedron | o | (.6..6) | \arccos{(\frac{1}{15}\sqrt{75-30\sqrt{5}})} | 79.188° | ||||||||
| [[File:Great icosihemidodecahedron.png | 30px]] | Great icosihemidodecahedron | o | (3...) | \arccos{(\frac{1}{15}\sqrt{75+30\sqrt{5}})} | 37.377° | ||||||||
| [[File:Great dodecahemidodecahedron.png | 30px]] | Great dodecahemidodecahedron | o | (...) | \arccos(\frac{\sqrt{5}}{5}) | 63.435° | ||||||||
| [[File:Hexahedron.png | 30px]] | Rhombic hexahedron | ||||||||||||
| (Dual of tetratetrahedron) | — | V(3.3.3.3) | \arccos(0)=\frac{\pi}{2} | 90° | ||||||||||
| [[File:Rhombic dodecahedron.png | 30px]] | Rhombic dodecahedron | ||||||||||||
| (Dual of cuboctahedron) | — | V(3.4.3.4) | \arccos(-\frac{1}{2})=\frac{2\pi}{3} | 120° | ||||||||||
| [[File:Rhombic triacontahedron.png | 30px]] | Rhombic triacontahedron | ||||||||||||
| (Dual of icosidodecahedron) | — | V(3.5.3.5) | \arccos(-\frac{\sqrt{5}+1}{4})=\frac{4\pi}{5} | 144° | ||||||||||
| [[File:DU36 medial rhombic triacontahedron.png | 30px]] | Medial rhombic triacontahedron | ||||||||||||
| (Dual of dodecadodecahedron) | — | V(5..5.) | \arccos(-\frac{1}{2})=\frac{2\pi}{3} | 120° | ||||||||||
| [[File:DU54 great rhombic triacontahedron.png | 30px]] | Great rhombic triacontahedron | ||||||||||||
| (Dual of great icosidodecahedron) | — | V(3..3.) | \arccos(\frac{\sqrt{5}-1}{4})=\frac{2\pi}{5} | 72° | ||||||||||
| [[File:DU30 small triambic icosahedron.png | 30px]] | Small triambic icosahedron | ||||||||||||
| (Dual of small ditrigonal icosidodecahedron) | — | V(3..3..3.) | \arccos(-\frac{1}{3}) | 109.471° | ||||||||||
| [[File:DU41 medial triambic icosahedron.png | 30px]] | Medial triambic icosahedron | ||||||||||||
| (Dual of ditrigonal dodecadodecahedron) | — | V(5..5..5.) | \arccos(-\frac{1}{3}) | 109.471° | ||||||||||
| [[File:DU47 great triambic icosahedron.png | 30px]] | Great triambic icosahedron | ||||||||||||
| (Dual of great ditrigonal icosidodecahedron) | — | V | \arccos(-\frac{1}{3}) | 109.471° | ||||||||||
| [[File:Tetrahemihexacron.png | 30px]] | Tetrahemihexacron | ||||||||||||
| (Dual of tetrahemihexahedron) | — | V(3.4..4) | \pi-\frac{\pi}{2} | 90° | ||||||||||
| [[File:Hexahemioctacron.png | 30px]] | Hexahemioctacron | ||||||||||||
| (Dual of cubohemioctahedron) | — | V(4.6..6) | \pi-\frac{\pi}{3} | 120° | ||||||||||
| [[File:Hexahemioctacron.png | 30px]] | Octahemioctacron | ||||||||||||
| (Dual of octahemioctahedron) | — | V(3.6..6) | \pi-\frac{\pi}{3} | 120° | ||||||||||
| [[File:Small dodecahemidodecacron.png | 30px]] | Small dodecahemidodecacron | ||||||||||||
| (Dual of small dodecahemidodecacron) | — | V(5.10..10) | \pi-\frac{\pi}{5} | 144° | ||||||||||
| [[File:Small dodecahemidodecacron.png | 30px]] | Small icosihemidodecacron | ||||||||||||
| (Dual of small icosihemidodecacron) | — | V(3.10..10) | \pi-\frac{\pi}{5} | 144° | ||||||||||
| [[File:Small dodecahemicosacron.png | 30px]] | Great dodecahemicosacron | ||||||||||||
| (Dual of great dodecahemicosahedron) | — | V(5.6..6) | \pi-\frac{\pi}{3} | 120° | ||||||||||
| [[File:Small dodecahemicosacron.png | 30px]] | Small dodecahemicosacron | ||||||||||||
| (Dual of small dodecahemicosahedron) | — | V(.6..6) | \pi-\frac{\pi}{3} | 120° | ||||||||||
| [[File:Great dodecahemidodecacron.png | 30px]] | Great icosihemidodecacron | ||||||||||||
| (Dual of great icosihemidodecacron) | — | V(3...) | \pi-\frac{2\pi}{5} | 72° | ||||||||||
| [[File:Great dodecahemidodecacron.png | 30px]] | Great dodecahemidodecacron | ||||||||||||
| (Dual of great dodecahemidodecacron) | — | V(...) | \pi-\frac{2\pi}{5} | 72° |
References
- Coxeter, Regular Polytopes (1963), Macmillan Company
- Regular Polytopes, (3rd edition, 1973), Dover edition, (Table I: Regular Polytopes, (i) The nine regular polyhedra {p,q} in ordinary space)
- (Section 3-7 to 3-9)
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