Icosidodecahedron

Icosidodecahedron
Dimensionality	3
Schläfli Symbol	'"`UNIQ--postMath-00000001-QINU`"'
Coxeter–Dynkin diagram
Bowers acronym	id
Subfacets
Vertices	30 points
Edges	60 line segments
Faces	20 triangles 12 pentagons
Cells	1 icosidodecahedron
Hypervolumes
Surcell Volume	'"`UNIQ--postMath-00000002-QINU`"'

An icosidodecahedron is a uniform polyhedron and an Archimedean solid created from rectifying a dodecahedron or icosahedron. It has 12 pentagonal faces and 20 triangular faces. Each edge of a pentagonal face is shared with a triangle, and every vertex is shared by two pentagons and two triangles.

It's Bowers acronym is id.

Symbols

Dynkin symbols of the icosidodecahedron include:

• o5x3o (uniform)
• xoxfo5ofxox&#xt (pentagonal gyrobirotunda)
• oxFfofx3xfofFxo&#xt (triangular symmetry)
• VooFxf oVofFx ooVxfF&#zx (block symmetry)

Hypervolumes

• Vertex count = $30$
• Edge length = $60l$
• Surface area = $(5\sqrt{3}+3\sqrt{25+10\sqrt{5}})l^2$
• surcell volume = $(45+17\sqrt{5})l^3$

Subfacets

• 30 points (0D)
• 60 line segments (1D)
• 12 pentagons (2D)
• 20 triangle (2D)
• 1 icosidodecahedron (3)

Radii

• Vertex radius: $\frac{1+\sqrt{5}}{2}l$
• Edge radius: $\frac{\sqrt{5+2\sqrt{5}}}{2}l$
• Pentagon radius: $\frac{\sqrt{5+2\sqrt{5}}{5}}l$
• Triangle radius: $(3\sqrt{3}+\sqrt{15})l$

Angles

• Dihedral angle: $\arccos( \sqrt{\frac{5+2\sqrt{5}}{15}})$

​

Vertex coordinates

The coordinates of the icosidodecahedron are:

• (±2φ,0,0) and all permutations
• (±φ^2,±φ,±1) and all even permutations

These coordinates show that a regular octahedron can be inscribed into the vertices of an icosidodecahedron.

Related shapes

• Dual: Rhombic triacontahedron
• Vertex figure: Rectangle, sides 1 and φ
• Conjugate: Great icosidodecahedron
• Caps: Pentagonal rotunda
• Gyrations: Pentagonal orthobirotunda
• Regiment members: 3 (other uniform members: small icosihemidodecahedron, small dodecahemidodecahedron)

See Also

• Polytope Wiki. "Icosidodecahedron"

Regular $t_0 \{5,3\}$	Rectified $t_1 \{5,3\}$	Birectified $t_2 \{5,3\}$	Truncated $t_{0,1} \{5,3\}$	Bitruncated $t_{1,2} \{5,3\}$	Cantellated $t_{0,2} \{5,3\}$	Cantitruncated $t_{0,1,2} \{5,3\}$
Dodecahedron	Icosidodecahedron	Icosahedron	Truncated dodecahedron	Truncated icosahedron	Rhombicosidodecahedron	Great rhombicosidodecahedron

$t_1 \{2,3\}$	$t_1 \{3,3\}$	$t_1 \{4,3\}$	$t_1 \{5,3\}$	$t_1 \{6,3\}$	$t_1 \{7,3\}$	$t_1 \{8,3\}$	...	$t_1 \{\infty,3\}$	$t_1 \{\frac{i\pi}{\lambda},3\}$
Trigonal dihedron	Octahedron	Cuboctahedron	Icosidodecahedron	Trihexagonal tiling	Triheptagonal tiling	Trioctagonal tiling	...	Triapeirogonal tiling	Tripseudogonal tiling

$t_1 \{2,5\}$	$t_1 \{3,5\}$	$t_1 \{4,5\}$	$t_1 \{5,5\}$	$t_1 \{6,5\}$	...	$t_1 \{\aleph_0,5\}$
Pentagonal dihedron	Icosidodecahedron	Tetrapentagonal tiling	Order-4 pentagonal tiling	Pentahexagonal tiling	...	Pentapeirogonal tiling