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) | |||||
|
Tetrahedron | {3,3} | (3.3.3) | arccos (1/3) | 70.529° |
|
Hexahedron or Cube | {4,3} | (4.4.4) | arccos (0) = π/2 | 90° |
|
Octahedron | {3,4} | (3.3.3.3) | arccos (-1/3) | 109.471° |
|
Dodecahedron | {5,3} | (5.5.5) | arccos (-√5/5) | 116.565° |
|
Icosahedron | {3,5} | (3.3.3.3.3) | arccos (-√5/3) | 138.190° |
| Kepler–Poinsot solids (regular nonconvex) | |||||
|
Small stellated dodecahedron | {5/2,5} | (5/2.5/2.5/2.5/2.5/2) | arccos (-√5/5) | 116.565° |
|
Great dodecahedron | {5,5/2} | (5.5.5.5.5)/2 | arccos (√5/5) | 63.435° |
|
Great stellated dodecahedron | {5/2,3} | (5/2.5/2.5/2) | arccos (√5/5) | 63.435° |
|
Great icosahedron | {3,5/2} | (3.3.3.3.3)/2 | arccos (√5/3) | 41.810° |
| Quasiregular polyhedra (Rectified regular) | |||||
|
Tetratetrahedron | r{3,3} | (3.3.3.3) | arccos (-1/3) | 109.471° |
|
Cuboctahedron | r{3,4} | (3.4.3.4) | arccos (-√3/3) | 125.264° |
|
Icosidodecahedron | r{3,5} | (3.5.3.5) | 142.623° | |
|
Dodecadodecahedron | r{5/2,5} | (5.5/2.5.5/2) | arccos (-√5/5) | 116.565° |
|
Great icosidodecahedron | r{5/2,3} | (3.5/2.3.5/2) | 37.377° | |
| Ditrigonal polyhedra | |||||
|
Small ditrigonal icosidodecahedron | a{5,3} | (3.5/2.3.5/2.3.5/2) | ||
|
Ditrigonal dodecadodecahedron | b{5,5/2} | (5.5/3.5.5/3.5.5/3) | ||
|
Great ditrigonal icosidodecahedron | c{3,5/2} | (3.5.3.5.3.5)/2 | ||
| Hemipolyhedra | |||||
|
Tetrahemihexahedron | o{3,3} | (3.4.3/2.4) | arccos (√3/3) | 54.736° |
|
Cubohemioctahedron | o{3,4} | (4.6.4/3.6) | arccos (√3/3) | 54.736° |
|
Octahemioctahedron | o{4,3} | (3.6.3/2.6) | arccos (1/3) | 70.529° |
|
Small dodecahemidodecahedron | o{3,5} | (5.10.5/4.10) | 26.058° | |
|
Small icosihemidodecahedron | o{5,3} | (3.10.3/2.10) | arccos (-√5/5) | 116.56° |
|
Great dodecahemicosahedron | o{5/2,5} | (5.6.5/4.6) | ||
|
Small dodecahemicosahedron | o{5,5/2} | (5/2.6.5/3.6) | ||
|
Great icosihemidodecahedron | o{5/2,3} | (3.10/3.3/2.10/3) | ||
|
Great dodecahemidodecahedron | o{3,5/2} | (5/2.10/3.5/3.10/3) | ||
| Quasiregular dual solids | |||||
|
|
Rhombic hexahedron (Dual of tetratetrahedron) |
— | V(3.3.3.3) | arccos (0) = π/2 | 90° |
|
Rhombic dodecahedron (Dual of cuboctahedron) |
— | V(3.4.3.4) | arccos (-1/2) = 2π/3 | 120° |
|
Rhombic triacontahedron (Dual of icosidodecahedron) |
— | V(3.5.3.5) | arccos (-√5+1/4) = 4π/5 | 144° |
|
Medial rhombic triacontahedron (Dual of dodecadodecahedron) |
— | V(5.5/2.5.5/2) | arccos (-1/2) = 2π/3 | 120° |
|
Great rhombic triacontahedron (Dual of great icosidodecahedron) |
— | V(3.5/2.3.5/2) | arccos (√5-1/4) = 2π/5 | 72° |
| Duals of the ditrigonal polyhedra | |||||
|
Small triambic icosahedron (Dual of small ditrigonal icosidodecahedron) |
— | V(3.5/2.3.5/2.3.5/2) | ||
|
Medial triambic icosahedron (Dual of ditrigonal dodecadodecahedron) |
— | V(5.5/3.5.5/3.5.5/3) | ||
|
Great triambic icosahedron (Dual of great ditrigonal icosidodecahedron) |
— | V(3.5.3.5.3.5)/2 | ||
| Duals of the hemipolyhedra | |||||
|
Tetrahemihexacron (Dual of tetrahemihexahedron) |
— | V(3.4.3/2.4) | π − π/2 | 90° |
|
Hexahemioctacron (Dual of cubohemioctahedron) |
— | V(4.6.4/3.6) | π − π/3 | 120° |
|
|
Octahemioctacron (Dual of octahemioctahedron) |
— | V(3.6.3/2.6) | π − π/3 | 120° |
|
Small dodecahemidodecacron (Dual of small dodecahemidodecacron) |
— | V(5.10.5/4.10) | π − π/5 | 144° |
|
|
Small icosihemidodecacron (Dual of small icosihemidodecacron) |
— | V(3.10.3/2.10) | π − π/5 | 144° |
|
Great dodecahemicosacron (Dual of great dodecahemicosahedron) |
— | V(5.6.5/4.6) | π − π/3 | 120° |
|
|
Small dodecahemicosacron (Dual of small dodecahemicosahedron) |
— | V(5/2.6.5/3.6) | π − π/3 | 120° |
|
Great icosihemidodecacron (Dual of great icosihemidodecacron) |
— | V(3.10/3.3/2.10/3) | π − 2π/5 | 72° |
|
|
Great dodecahemidodecacron (Dual of great dodecahemidodecacron) |
— | V(5/2.10/3.5/3.10/3) | π − 2π/5 | 72° |
References
edit- Coxeter, Regular Polytopes (1963), Macmillan 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)
- Weisstein, Eric W. "Uniform Polyhedron". MathWorld.