Rectified truncated icosahedron

Rectified truncated icosahedron
Rank	3
Type	Near-miss
Space	Spherical
Notation
Conway notation	atI
Elements
Faces	60 isosceles triangles, 12 pentagons, 20 triambi
Edges	60+120
Vertices	30+60
Vertex figures	60 rectangles
	30 isosceles trapezoids
Measures (based on regular hexagons of edge length 1)
Edge length ratio	$\frac{3+4\sqrt2-\sqrt5}{6} ≈ 1.07013$
Central density	1
Number of external pieces	28
Related polytopes
Dual	Rhombic enneacontahedron
Conjugate	None
Abstract & topological properties
Euler characteristic	2
Surface	Sphere
Orientable	Yes
Genus	0
Properties
Symmetry	H3, order 120
Convex	Yes
Nature	Tame

The rectified truncated icosahedron is a near-miss Johnson solid. Topologically, its faces are 60 isosceles triangles, 12 pentagons and 20 triambi. It can be constructed by applying a rectification process to the truncated icosahedron.

Variations

There are three notable variants of the rectified truncated icosahedron. The first variant is obtained by making the hexagons regular, in which the ratio between the two edges is equivalent to 1:$\frac{3+4\sqrt2-\sqrt5}{6}$ ≈ 1:1.07013. The second variant is obtained by making all the edges have the same length, but the hexagons only have triangular symmetry, with the internal angles being approximately 115.28055° and 124.71945°. The third variant has the unique property of being canonical. If the canonical variant has a midradius of 1, then the edge lengths are approximately 0.34561 and 0.35956 and the hexagons' internal angles are approximately 118.07524° and 121.92476°.

Net:

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External links

• McCooey, David. "Rectified Truncated Icosahedron"

• Kaplan, Craig S. Near Misses