| 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 (Template:Sfrac)
|
70.529°
|
|
Hexahedron or Cube
|
{4,3}
|
(4.4.4)
|
arccos (0) = Template:Sfrac
|
90°
|
|
Octahedron
|
{3,4}
|
(3.3.3.3)
|
arccos (-Template:Sfrac)
|
109.471°
|
|
Dodecahedron
|
{5,3}
|
(5.5.5)
|
arccos (-Template:Sfrac)
|
116.565°
|
|
Icosahedron
|
{3,5}
|
(3.3.3.3.3)
|
arccos (-Template:Sfrac)
|
138.190°
|
| Kepler–Poinsot solids (regular nonconvex)
|
|
Small stellated dodecahedron |
Template:Mset
|
(Template:Sfrac.Template:Sfrac.Template:Sfrac.Template:Sfrac.Template:Sfrac)
|
arccos (-Template:Sfrac)
|
116.565°
|
|
Great dodecahedron |
Template:Mset
|
Template:Sfrac
|
arccos (Template:Sfrac)
|
63.435°
|
|
Great stellated dodecahedron |
{Template:Sfrac,3}
|
(Template:Sfrac.Template:Sfrac.Template:Sfrac)
|
arccos (Template:Sfrac)
|
63.435°
|
|
Great icosahedron |
Template:Mset
|
Template:Sfrac
|
arccos (Template:Sfrac)
|
41.810°
|
| Quasiregular polyhedra (Rectified regular)
|
|
Tetratetrahedron
|
r{3,3}
|
(3.3.3.3)
|
arccos (-Template:Sfrac)
|
109.471°
|
|
Cuboctahedron
|
r{3,4}
|
(3.4.3.4)
|
arccos (-Template:Sfrac)
|
125.264°
|
|
Icosidodecahedron
|
r{3,5}
|
(3.5.3.5)
|
|
142.623°
|
|
Dodecadodecahedron
|
r{Template:Sfrac,5}
|
(5.Template:Sfrac.5.Template:Sfrac)
|
arccos (-Template:Sfrac)
|
116.565°
|
|
Great icosidodecahedron
|
r{Template:Sfrac,3}
|
(3.Template:Sfrac.3.Template:Sfrac)
|
|
37.377°
|
| Ditrigonal polyhedra
|
|
Small ditrigonal icosidodecahedron
|
a{5,3}
|
(3.Template:Sfrac.3.Template:Sfrac.3.Template:Sfrac)
|
|
|
|
Ditrigonal dodecadodecahedron
|
bTemplate:Mset
|
(5.Template:Sfrac.5.Template:Sfrac.5.Template:Sfrac)
|
|
|
|
Great ditrigonal icosidodecahedron
|
cTemplate:Mset
|
Template:Sfrac
|
|
|
| Hemipolyhedra
|
|
Tetrahemihexahedron
|
o{3,3}
|
(3.4.Template:Sfrac.4)
|
arccos (Template:Sfrac)
|
54.736°
|
|
Cubohemioctahedron
|
o{3,4}
|
(4.6.Template:Sfrac.6)
|
arccos (Template:Sfrac)
|
54.736°
|
|
Octahemioctahedron
|
o{4,3}
|
(3.6.Template:Sfrac.6)
|
arccos (Template:Sfrac)
|
70.529°
|
|
Small dodecahemidodecahedron
|
o{3,5}
|
(5.10.Template:Sfrac.10)
|
|
26.058°
|
|
Small icosihemidodecahedron
|
o{5,3}
|
(3.10.Template:Sfrac.10)
|
arccos (-Template:Sfrac)
|
116.56°
|
|
Great dodecahemicosahedron
|
o{Template:Sfrac,5}
|
(5.6.Template:Sfrac.6)
|
|
|
|
Small dodecahemicosahedron
|
oTemplate:Mset
|
(Template:Sfrac.6.Template:Sfrac.6)
|
|
|
|
Great icosihemidodecahedron
|
oTemplate:Mset
|
(3.Template:Sfrac.Template:Sfrac.Template:Sfrac)
|
|
|
|
Great dodecahemidodecahedron
|
oTemplate:Mset
|
(Template:Sfrac.Template:Sfrac.Template:Sfrac.Template:Sfrac)
|
|
|
| Quasiregular dual solids
|
|
|
Rhombic hexahedron
(Dual of tetratetrahedron)
|
—
|
V(3.3.3.3)
|
arccos (0) = Template:Sfrac
|
90°
|
|
Rhombic dodecahedron
(Dual of cuboctahedron)
|
—
|
V(3.4.3.4)
|
arccos (-Template:Sfrac) = Template:Sfrac
|
120°
|
|
Rhombic triacontahedron
(Dual of icosidodecahedron)
|
—
|
V(3.5.3.5)
|
arccos (-Template:Sfrac) = Template:Sfrac
|
144°
|
|
Medial rhombic triacontahedron
(Dual of dodecadodecahedron)
|
—
|
V(5.Template:Sfrac.5.Template:Sfrac)
|
arccos (-Template:Sfrac) = Template:Sfrac
|
120°
|
|
Great rhombic triacontahedron
(Dual of great icosidodecahedron)
|
—
|
V(3.Template:Sfrac.3.Template:Sfrac)
|
arccos (Template:Sfrac) = Template:Sfrac
|
72°
|
| Duals of the ditrigonal polyhedra
|
|
Small triambic icosahedron
(Dual of small ditrigonal icosidodecahedron)
|
—
|
V(3.Template:Sfrac.3.Template:Sfrac.3.Template:Sfrac)
|
|
|
|
Medial triambic icosahedron
(Dual of ditrigonal dodecadodecahedron)
|
—
|
V(5.Template:Sfrac.5.Template:Sfrac.5.Template:Sfrac)
|
|
|
|
Great triambic icosahedron
(Dual of great ditrigonal icosidodecahedron)
|
—
|
VTemplate:Sfrac
|
|
|
| Duals of the hemipolyhedra
|
|
Tetrahemihexacron
(Dual of tetrahemihexahedron)
|
—
|
V(3.4.Template:Sfrac.4)
|
Template:Pi − Template:Sfrac
|
90°
|
|
Hexahemioctacron
(Dual of cubohemioctahedron)
|
—
|
V(4.6.Template:Sfrac.6)
|
Template:Pi − Template:Sfrac
|
120°
|
|
|
Octahemioctacron
(Dual of octahemioctahedron)
|
—
|
V(3.6.Template:Sfrac.6)
|
Template:Pi − Template:Sfrac
|
120°
|
|
Small dodecahemidodecacron
(Dual of small dodecahemidodecacron)
|
—
|
V(5.10.Template:Sfrac.10)
|
Template:Pi − Template:Sfrac
|
144°
|
|
|
Small icosihemidodecacron
(Dual of small icosihemidodecacron)
|
—
|
V(3.10.Template:Sfrac.10)
|
Template:Pi − Template:Sfrac
|
144°
|
|
Great dodecahemicosacron
(Dual of great dodecahemicosahedron)
|
—
|
V(5.6.Template:Sfrac.6)
|
Template:Pi − Template:Sfrac
|
120°
|
|
|
Small dodecahemicosacron
(Dual of small dodecahemicosahedron)
|
—
|
V(Template:Sfrac.6.Template:Sfrac.6)
|
Template:Pi − Template:Sfrac
|
120°
|
|
Great icosihemidodecacron
(Dual of great icosihemidodecacron)
|
—
|
V(3.Template:Sfrac.Template:Sfrac.Template:Sfrac)
|
Template:Pi − Template:Sfrac
|
72°
|
|
|
Great dodecahemidodecacron
(Dual of great dodecahemidodecacron)
|
—
|
V(Template:Sfrac.Template:Sfrac.Template:Sfrac.Template:Sfrac)
|
Template:Pi − Template:Sfrac
|
72°
|
