Icosidodecadodecahedron

The icosidodecadodecahedron, or ided, is a uniform polyhedron. It consists of 12 pentagons, 12 pentagrams, and 20 hexagons. One pentagon, one pentagram, and two hexagons join at each vertex.

Icosidodecadodecahedron
(3D model) (OFF file)
Rank	3
Type	Uniform
Space	Spherical
Notation
Bowers style acronym	Ided
Coxeter diagram	o5/3x3x5*a ()
Elements
Faces	12 pentagons, 12 pentagrams, 20 hexagons
Edges	60+60
Vertices	60
Vertex figure	Crossed isosceles trapezoid, edge lengths (√5–1)/2, √3, (1+√5)/2, √3
Measures (edge length 1)
Circumradius	$\frac{\sqrt7}{2} \approx 1.32288$
Volume	20
Dihedral angles	5–6: $\arccos\left(\sqrt{\frac{5-2\sqrt5}{15}}\right) \approx 79.18768^\circ$
	5/2–6: $\arccos\left(\sqrt{\frac{5+2\sqrt5}{15}}\right) \approx 37.37737^\circ$
Central density	4
Number of external pieces	408
Level of complexity	25
Related polytopes
Army	Semi-uniform ti, edge lengths $\frac{\sqrt5-1}{2}$ (pentagons), $\frac{3-\sqrt5}{2}$ (between ditrigons)
Regiment	Raded
Dual	Medial icosacronic hexecontahedron
Conjugate	Icosidodecadodecahedron
Convex core	Truncated icosahedron
Abstract & topological properties
Flag count	480
Euler characteristic	–16
Orientable	Yes
Properties
Symmetry	H3, order 120
Convex	No
Nature	Tame

It is a faceting of the rhombidodecadodecahedron, using its 12 pentagrams and 12 pentagons along with 20 additional hexagons.

Vertex coordinates

Its vertices are the same as those of its regiment colonel, the rhombidodecadodecahedron.

External links

• Bowers, Jonathan. "Polyhedron Category 4: Trapeziverts" (#43).

• Klitzing, Richard. "ided".

• Wikipedia Contributors. "Icosidodecadodecahedron".
• McCooey, David. "Icosidodecadodecahedron"