Rectified truncated icosahedron

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Rectified truncated icosahedron
Rectified truncated icosahedron.png
Schläfli symbol rt{3,5}
Conway notation atI[1]
Faces 92:
60 { }∨( )
12 {5}
20 {6}
Edges 180
Vertices 90
Vertex figures 3.6.3.6 (3.6)^2 vertex.svg
3.5.3.6 3.5.3.6 vertex.svg
Symmetry group Ih, [5,3], (*532) order 120
Rotation group I, [5,3]+, (532), order 60
Dual polyhedron Rhombic enneacontahedron
Properties convex
Rectified truncated icosahedron net.png
Net

The rectified truncated icosahedron is a polyhedron, constructed as a rectified truncated icosahedron. 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.

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.

Images[edit]

截角半正三十二面體.gif

Dual[edit]

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.

Related polyhedra[edit]

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 Uniform polyhedron-53-t12.svg Truncated truncated icosahedron.png Rectified truncated icosahedron.png Expanded truncated icosahedron.png Truncated rectified truncated icosahedron.png Snub rectified truncated icosahedron.png
Net Truncated icosahedron flat.png Truncated truncated icosahedron net.png Rectified truncated icosahedron net.png Expanded truncated icosahedron net.png Snub rectified truncated icosahedron net.png
Conway dtI = kD kD kdtI jtI jtI otI mtI gtI
Dual Pentakis dodecahedron.png Kissed kissed dodecahedron.png Joined truncated icosahedron.png Ortho truncated icosahedron.png Meta truncated icosahedron.png Gyro truncated icosahedron.png
Net Pentakisdodecahedron net.png Kissed kissed dodecahedron net.png Rhombic enneacontahedron flat.png Ortho truncated icosahedron net.png Gyro truncated icosahedron net.png

See also[edit]

References[edit]

  • Coxeter Regular Polytopes, Third edition, (1973), Dover edition, ISBN 0-486-61480-8 (pp. 145–154 Chapter 8: Truncation)
  • John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, ISBN 978-1-56881-220-5

External links[edit]