Table of polyhedron dihedral angles
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The dihedral angles for the edge-uniform polyhedra are:
| Picture | Name | Schläfli symbol | Vertex/Face configuration | exact dihedral angle (radians) | approximate dihedral angle (degrees) |
|---|---|---|---|---|---|
| Platonic solids | |||||
| Tetrahedron | {3,3} | (3)3 | arccos(1/3) | 70.53° |
| Hexahedron or Cube | {4,3} | (4)3 | π/2 | 90° |
| Octahedron | {3,4} | (3)4 | π − arccos(1/3) | 109.47° |
| Dodecahedron | {5,3} | (5)3 | π − arctan(2) | 116.56° |
| Icosahedron | {3,5} | (3)5 | π − arccos(√5/3) | 138.19° |
| Kepler-Poinsot solids | |||||
| Small stellated dodecahedron | {5/2,5} | (5/2)5 | π − arctan(2) | 116.56° |
| Great dodecahedron | {5,5/2} | (5)5/2 | arctan(2) | 63.435° |
| Great stellated dodecahedron | {5/2,3} | (5/2)3 | arctan(2) | 63.435° |
| Great icosahedron | {3,5/2} | (3)5/2 | arcsin(2/3) | 41.810° |
| Quasiregular solids (Rectified regular) | |||||
| Image:Uniform polyhedron-33-t1.png | Tetratetrahedron | <math>\begin{Bmatrix} 3 \\ 3 \end{Bmatrix}</math> | (3.3.3.3) | π − arccos(1/3) | 109.47° |
| Cuboctahedron | <math>\begin{Bmatrix} 3 \\ 4 \end{Bmatrix}</math> | (3.4.3.4) | π − arccos(1/sqr(3)) | 125.264° |
| Icosidodecahedron | <math>\begin{Bmatrix} 3 \\ 5 \end{Bmatrix}</math> | (3.5.3.5) | <math> \pi - \arccos{ ( \sqrt{ \frac{ (5 + 2\sqrt 5)}{15} } ) } </math> | 142.623° |
| Dodecadodecahedron | <math>\begin{Bmatrix} 5/2 \\ 5 \end{Bmatrix}</math> | (5.5/2.5.5/2) | ||
| Great icosidodecahedron | <math>\begin{Bmatrix} 5/2 \\ 3 \end{Bmatrix}</math> | (3.5/2.3.5/2) | ||
| Quasiregular dual solids | |||||
|
| Dual of tetratetrahedron | - | V(3.3.3.3) | π − π/2 | 90° |
| Rhombic dodecahedron (Dual of cuboctahedron) | - | V(3.4.3.4) | π − π/3 | 120° |
| Rhombic triacontahedron (Dual of icosidodecahedron) | - | V(3.5.3.5) | π − π/5 | 144° |
[edit] References
- Coxeter, Regular Polytopes (1963), Macmillian Company
- Regular Polytopes, (3rd edition, 1973), Dover edition, ISBN 0-486-61480-8 (Table I: Regular Polytopes, (i) The nine regular polyhedra {p,q} in ordinary space)
- Williams, Robert (1979). The Geometrical Foundation of Natural Structure: A Source Book of Design. Dover Publications, Inc. ISBN 0-486-23729-X. (Section 3-7 to 3-9)
