Chamfered icosahedron

Chamfered icosahedron
Rank	3
Notation
Conway notation	cI t3daD
Elements
Faces	20 triangles, 30 point-symmetric hexagons
Edges	60+60
Vertices	12+60
Vertex figures	12 pentagons
	60 isosceles triangles
Measures (edge length 1)
Dihedral angles	3-6: $\arccos \left(-{\frac {{\sqrt {3}}+{\sqrt {15}}}{6}}\right)\approx 159.09484^{\circ }$
	6-6: 144°
Central density	1
Number of external pieces	50
Level of complexity	4
Related polytopes
Army	Chamfered icosahedron
Regiment	Chamfered icosahedron
Dual	Triakis icosidodecahedron
Conjugate	None
Abstract & topological properties
Flag count	480
Euler characteristic	2
Surface	Sphere
Orientable	Yes
Genus	0
Properties
Symmetry	H3, order 120
Flag orbits	4
Convex	Yes
Nature	Tame

The chamfered icosahedron is a modification of the icosahedron that can have one edge length but has irregular faces. It has 20 triangles and 30 hexagons as faces, and 12 order-5 vertices that can be thought of as coming from the icosahedron as well as 60 new order-3 vertices.

The hexagonal faces have angles of $\arccos \left({\frac {\sqrt {5}}{5}}\right)\approx 63.43495^{\circ }$ on one pair of opposite vertices, and angles of $\arccos \left(-{\sqrt {\frac {5+{\sqrt {5}}}{10}}}\right)\approx 148.28253^{\circ }$ on the four remaining vertices.

It can be modified such that it has a single inradius, or such that it has a single midradius or "edge radius". The latter version is called the "canonical" version.

It can also be viewed as an order-3-truncated rhombic triacontahedron, or as an icosahedrally-symmetric Goldberg polyhedron.

External links

• Klitzing, Richard. "Chamfered ike".
• Wikipedia contributors. "Chamfered icosahedron".
• McCooey, David. "Chamfered Icosahedron (all edges equal)"