# Rectified truncated icosahedron

Rectified truncated icosahedron Rank3
TypeNear-miss
SpaceSpherical
Notation
Conway notationatI
Elements
Faces60 isosceles triangles, 12 pentagons, 20 triambi
Edges60+120
Vertices30+60
Vertex figures60 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 density1
Number of external pieces28
Related polytopes
DualRhombic enneacontahedron
ConjugateNone
Abstract & topological properties
Euler characteristic2
SurfaceSphere
OrientableYes
Genus0
Properties
SymmetryH3, order 120
ConvexYes
NatureTame

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°.