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Table of polyhedron dihedral angles

From Wikipedia, the free encyclopedia

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) ππ/2 90°
Hexahemioctacron
(Dual of cubohemioctahedron)
V(4.6.4/3.6) ππ/3 120°
Octahemioctacron
(Dual of octahemioctahedron)
V(3.6.3/2.6) ππ/3 120°
Small dodecahemidodecacron
(Dual of small dodecahemidodecacron)
V(5.10.5/4.10) ππ/5 144°
Small icosihemidodecacron
(Dual of small icosihemidodecacron)
V(3.10.3/2.10) ππ/5 144°
Great dodecahemicosacron
(Dual of great dodecahemicosahedron)
V(5.6.5/4.6) ππ/3 120°
Small dodecahemicosacron
(Dual of small dodecahemicosahedron)
V(5/2.6.5/3.6) ππ/3 120°
Great icosihemidodecacron
(Dual of great icosihemidodecacron)
V(3.10/3.3/2.10/3) π2π/5 72°
Great dodecahemidodecacron
(Dual of great dodecahemidodecacron)
V(5/2.10/3.5/3.10/3) π2π/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.