Solids with icosahedral symmetry

From Wikipedia, the free encyclopedia

Main article: Icosahedral symmetry

Contents 1 Solids with full icosahedral symmetry 1.1 Platonic solids 1.2 Achiral Archimedean solids 1.3 Achiral Catalan solids 1.4 Kepler-Poinsot solids 1.5 Achiral nonconvex uniform polyhedra 2 Chiral Archimedean and Catalan solids 3 Chiral nonconvex uniform polyhedra

[edit] Solids with full icosahedral symmetry

(For details see below.)

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

[edit] Platonic solids

Name	Picture	Faces	Edges	Vertices	Edges per face	Faces meeting at each vertex
dodecahedron	(Animation)	12	30	20	5	3
icosahedron	(Animation)	20	30	12	3	5

[edit] Achiral Archimedean solids

Name	picture	Faces		Edges	Vertices	Vertex configuration
icosidodecahedron (quasi-regular: vertex- and edge-uniform)	(Video)	32	20 triangles 12 pentagons	60	30	3,5,3,5
truncated dodecahedron	(Video)	32	20 triangles 12 decagons	90	60	3,10,10
truncated icosahedron or commonly football (soccer ball)	(Video)	32	12 pentagons 20 hexagons	90	60	5,6,6
rhombicosidodecahedron or small rhombicosidodecahedron	(Video)	62	20 triangles 30 squares 12 pentagons	120	60	3,4,5,4
truncated icosidodecahedron or great rhombicosidodecahedron	(Video)	62	30 squares 20 hexagons 12 decagons	180	120	4,6,10

[edit] Achiral Catalan solids

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

[edit] Kepler-Poinsot solids

[edit] Achiral nonconvex uniform polyhedra

[edit] Chiral Archimedean and Catalan solids

Archimedean solids:

Name	picture	Faces		Edges	Vertices	Vertex configuration
snub dodecahedron or snub icosidodecahedron (2 chiral forms)	(Video) (Video)	92	80 triangles 12 pentagons	150	60	3,3,3,3,5

Catalan solids:

Name	picture	Dual Archimedean solid	Faces	Edges	Vertices	Face Polygon
pentagonal hexecontahedron	(Video)(Video)	snub dodecahedron	60	150	92	irregular pentagon

Stellated Archimedean solids:

• the snub dodecadodecahedron
• the great inverted snub icosidodecahedron or great vertisnub icosidodecahedron
• the great retrosnub icosidodecahedron or great inverted retrosnub icosidodecahedron

[edit] Chiral nonconvex uniform polyhedra