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
- 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.
🔥 Top keywords: Main PageSpecial:SearchPage 3Wikipedia:Featured picturesHouse of the DragonUEFA Euro 2024Bryson DeChambeauJuneteenthInside Out 2Eid al-AdhaCleopatraDeaths in 2024Merrily We Roll Along (musical)Jonathan GroffJude Bellingham.xxx77th Tony AwardsBridgertonGary PlauchéKylian MbappéDaniel RadcliffeUEFA European Championship2024 ICC Men's T20 World CupUnit 731The Boys (TV series)Rory McIlroyN'Golo KantéUEFA Euro 2020YouTubeRomelu LukakuOpinion polling for the 2024 United Kingdom general electionThe Boys season 4Romania national football teamNicola CoughlanStereophonic (play)Gene WilderErin DarkeAntoine GriezmannProject 2025