Solids with icosahedral symmetry

Solids with full icosahedral symmetry

Platonic solids - regular polyhedra (all faces of the same type)


{5,3}
{3,5}

Archimedean solids - polyhedra with more than one polygon face type.


3.10.10
4.6.10
5.6.6
3.4.5.4
3.5.3.5

Catalan solids - duals of the Archimedean solids.


V3.10.10
V4.6.10
V5.6.6
V3.4.5.4
V3.5.3.5

Platonic solids

Name Picture Faces Edges Vertices Edges per face Faces meeting
at each vertex
dodecahedron Dodecahedron

(Animation)

 12302053
icosahedron Icosahedron

(Animation)

 20301235

Achiral Archimedean solids

Name picture Faces Edges Vertices Vertex configuration
icosidodecahedron
(quasi-regular: vertex- and edge-uniform)		 Icosidodecahedron
(Video)		 3220 triangles
12 pentagons		 6030 3,5,3,5
truncated dodecahedron Truncated dodecahedron
(Video)		 3220 triangles
12 decagons		90 60 3,10,10
truncated icosahedron
or commonly football (soccer ball)		 Truncated icosahedron
(Video)		 3212 pentagons
20 hexagons		90 60 5,6,6
rhombicosidodecahedron
or small rhombicosidodecahedron		 Rhombicosidodecahedron
(Video)		 6220 triangles
30 squares
12 pentagons		 120603,4,5,4
truncated icosidodecahedron
or great rhombicosidodecahedron		 Truncated icosidodecahedron
(Video)		 6230 squares
20 hexagons
12 decagons		 1801204,6,10

Achiral Catalan solids

Name picture Dual Archimedean solid Faces Edges Vertices Face Polygon
rhombic triacontahedron
(quasi-regular dual: face- and edge-uniform)		 Rhombic triacontahedron
(Video)		 icosidodecahedron 30 60 32 rhombus
triakis icosahedron Triakis icosahedron
(Video)		 truncated dodecahedron 60 90 32 isosceles triangle
pentakis dodecahedron Pentakis dodecahedron
(Video)		 truncated icosahedron 60 90 32 isosceles triangle
deltoidal hexecontahedron Deltoidal hexecontahedron
(Video)		 rhombicosidodecahedron 60 120 62 kite
disdyakis triacontahedron
or hexakis icosahedron		 Disdyakis triacontahedron
(Video)		 truncated icosidodecahedron 120 180 62 scalene triangle

Kepler-Poinsot solids

Achiral nonconvex uniform polyhedra

Chiral Archimedean and Catalan solids

Archimedean solids:

Name picture Faces Edges Vertices Vertex configuration
snub dodecahedron
or snub icosidodecahedron (2 chiral forms)		 Snub dodecahedron (Ccw)
(Video)
Snub dodecahedron (Cw)
(Video)		 9280 triangles
12 pentagons		150 60 3,3,3,3,5

Catalan solids:

Name picture Dual Archimedean solid Faces Edges Vertices Face Polygon
pentagonal hexecontahedron Pentagonal hexecontahedron (Ccw)Pentagonal hexecontahedron (Cw)
(Video)(Video)		 snub dodecahedron 60 50 92 irregular pentagon

Chiral nonconvex uniform polyhedra

Construction

Construction instructions for the following solids are available: Platonic solids,[1] Archimedean solids,[2] Achiral Catalan solids,[3] Kepler–Poinsot solids,[4] Achiral nonconvex uniform polyhedra,[5] Chiral Archimedean and Catalan solids: Snub dodecahedron[6] and Pentagonal hexecontahedron,[7] and Chiral nonconvex uniform polyhedra.[8]

See also

Citations

  1. ↑ Wenninger 1971, pp. 17–19.
  2. ↑ Wenninger 1971, pp. 23–24, 26, 28, 30.
  3. ↑ Wenninger 1983, pp. 14–35.
  4. ↑ Wenninger 1971, pp. 38–40, 63.
  5. ↑ Wenninger 1971, pp. 106–117, 123–131, 135–143, 147–158, 161–170, 200–203.
  6. ↑ Wenninger 1971, p. 32.
  7. ↑ Wenninger 1983, p. 29.
  8. ↑ Wenninger 1971, pp. 172–199.

Works cited

  • Wenninger, Magnus J. (1971). Polyhedron models. Cambridge: Cambridge University Press. ISBN 0521069173.
  • Wenninger, Magnus J. (1983). Dual models. Cambridge; New York: Cambridge University Press. ISBN 0521245249.
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