Rectified truncated icosahedron

In geometry, the rectified truncated icosahedron is a convex polyhedron. It has 92 faces: 60 isosceles triangles, 12 regular pentagons, and 20 regular hexagons. It is constructed as a rectified, truncated icosahedron, rectification truncating vertices down to mid-edges.

Rectified truncated icosahedron
TypeNear-miss Johnson solid
Faces92:
60 isosceles triangles
12 pentagons
20 hexagons
Edges180
Vertices90
Vertex configuration3.6.3.6
3.5.3.6
Schläfli symbolrt{3,5}
Conway notationatI[1]
Symmetry groupIh, [5,3], (*532) order 120
Rotation groupI, [5,3]+, (532), order 60
Dual polyhedronRhombic enneacontahedron
Propertiesconvex
Net

As a near-miss Johnson solid, under icosahedral symmetry, the pentagons are always regular, although the hexagons, while having equal edge lengths, do not have the same edge lengths with the pentagons, having slightly different but alternating angles, causing the triangles to be isosceles instead. The shape is a symmetrohedron with notation I(1,2,*,[2])

Images

Dual

By Conway polyhedron notation, the dual polyhedron can be called a joined truncated icosahedron, jtI, but it is topologically equivalent to the rhombic enneacontahedron with all rhombic faces.

The rectified truncated icosahedron can be seen in sequence of rectification and truncation operations from the truncated icosahedron. Further truncation, and alternation operations creates two more polyhedra:

Name Truncated
icosahedron		 Truncated
truncated
icosahedron		 Rectified
truncated
icosahedron		 Expanded
truncated
icosahedron		 Truncated
rectified
truncated
icosahedron		 Snub
rectified
truncated
icosahedron
Coxeter tI ttI rtI rrtI trtI srtI
Conway atI etI btI stI
Image
Net
Conway dtI = kD kD kdtI jtI jtI otI mtI gtI
Dual
Net

See also

References
  1. "PolyHédronisme".
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