Truncated icosidodecahedron

From Wikipedia, the free encyclopedia
Jump to navigation Jump to search
Truncated icosidodecahedron
Truncatedicosidodecahedron.jpg
(Click here for rotating model)
Type Archimedean solid
Uniform polyhedron
Elements F = 62, E = 180, V = 120 (χ = 2)
Faces by sides 30{4}+20{6}+12{10}
Conway notation bD or taD
Schläfli symbols tr{5,3} or
t0,1,2{5,3}
Wythoff symbol 2 3 5 |
Coxeter diagram CDel node 1.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node 1.png
Symmetry group Ih, H3, [5,3], (*532), order 120
Rotation group I, [5,3]+, (532), order 60
Dihedral angle 6-10: 142.62°
4-10: 148.28°
4-6: 159.095°
References U28, C31, W16
Properties Semiregular convex zonohedron
Polyhedron great rhombi 12-20 max.png
Colored faces
Great rhombicosidodecahedron vertfig.png
4.6.10
(Vertex figure)
Polyhedron great rhombi 12-20 dual max.png
Disdyakis triacontahedron
(dual polyhedron)
Polyhedron great rhombi 12-20 net.svg
Net

In geometry, the truncated icosidodecahedron is an Archimedean solid, one of thirteen convex isogonal nonprismatic solids constructed by two or more types of regular polygon faces.

It has 62 faces: 30 squares, 20 regular hexagons, and 12 regular decagons. It has more vertices (120) and edges (180) than any other convex nonprismatic uniform polyhedron. Since each of its faces has point symmetry (equivalently, 180° rotational symmetry), the truncated icosidodecahedron is a zonohedron.

Names[edit]

The name truncated icosidodecahedron, given originally by Johannes Kepler, is misleading. An actual truncation of a icosidodecahedron has rectangles instead of squares. This nonuniform polyhedron is topologically equivalent to the Archimedean solid.

Alternate interchangeable names are:

Icosidodecahedron and its truncation

The name great rhombicosidodecahedron refers to the relationship with the (small) rhombicosidodecahedron (compare section Dissection).
There is a nonconvex uniform polyhedron with a similar name, the nonconvex great rhombicosidodecahedron.

Area and volume[edit]

The surface area A and the volume V of the truncated icosidodecahedron of edge length a are:[citation needed]

If a set of all 13 Archimedean solids were constructed with all edge lengths equal, the truncated icosidodecahedron would be the largest.

Cartesian coordinates[edit]

Cartesian coordinates for the vertices of a truncated icosidodecahedron with edge length 2φ − 2, centered at the origin, are all the even permutations of:[4]

1/φ, ±1/φ, ±(3 + φ)),
2/φ, ±φ, ±(1 + 2φ)),
1/φ, ±φ2, ±(−1 + 3φ)),
(±(2φ − 1), ±2, ±(2 + φ)) and
φ, ±3, ±2φ),

where φ = 1 + 5/2 is the golden ratio.

Dissection[edit]

The truncated icosidodecahedron is the convex hull of a rhombicosidodecahedron with cuboids above its 30 squares whose height to base ratio is the golden ratio. The rest of its space can be dissected into 12 nonuniform pentagonal cupolas below the decagons and 20 nonuniform triangular cupolas below the hexagons.

Orthogonal projections[edit]

The truncated icosidodecahedron has seven special orthogonal projections, centered on a vertex, on three types of edges, and three types of faces: square, hexagonal and decagonal. The last two correspond to the A2 and H2 Coxeter planes.

Orthogonal projections
Centered by Vertex Edge
4-6
Edge
4-10
Edge
6-10
Face
square
Face
hexagon
Face
decagon
Solid Polyhedron great rhombi 12-20 from blue max.png Polyhedron great rhombi 12-20 from yellow max.png Polyhedron great rhombi 12-20 from red max.png
Wireframe Dodecahedron t012 v.png Dodecahedron t012 e46.png Dodecahedron t012 e4x.png Dodecahedron t012 e6x.png Dodecahedron t012 f4.png Dodecahedron t012 A2.png Dodecahedron t012 H3.png
Projective
symmetry
[2]+ [2] [2] [2] [2] [6] [10]
Dual
image
Dual dodecahedron t012 v.png Dual dodecahedron t012 e46.png Dual dodecahedron t012 e4x.png Dual dodecahedron t012 e6x.png Dual dodecahedron t012 f4.png Dual dodecahedron t012 A2.png Dual dodecahedron t012 H3.png

Spherical tilings and Schlegel diagrams[edit]

The truncated icosidodecahedron can also be represented as a spherical tiling, and projected onto the plane via a stereographic projection. This projection is conformal, preserving angles but not areas or lengths. Straight lines on the sphere are projected as circular arcs on the plane.

Schlegel diagrams are similar, with a perspective projection and straight edges.

Orthographic projection Stereographic projections
Decagon-centered Hexagon-centered Square-centered
Uniform tiling 532-t012.png Truncated icosidodecahedron stereographic projection decagon.png Truncated icosidodecahedron stereographic projection hexagon.png Truncated icosidodecahedron stereographic projection square.png

Geometric variations[edit]

Within Icosahedral symmetry there are unlimited geometric variations of the truncated icosidodecahedron with isogonal faces. The truncated dodecahedron, rhombicosidodecahedron, and truncated icosahedron as degenerate limiting cases.

Truncated dodecahedron.png Great truncated icosidodecahedron convex hull.png Nonuniform truncated icosidodecahedron.png Uniform polyhedron-53-t012.png Truncated dodecadodecahedron convex hull.png Icositruncated dodecadodecahedron convex hull.png Truncated icosahedron.png Small rhombicosidodecahedron.png
CDel node 1.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node.png CDel node 1.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node 1.png CDel node.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node 1.png CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node 1.png

Truncated icosidodecahedral graph[edit]

Truncated icosidodecahedral graph
Truncated icosidodecahedral graph.png
5-fold symmetry
Vertices120
Edges180
Radius15
Diameter15
Girth4
Automorphisms120 (A5×2)
Chromatic number2
PropertiesCubic, Hamiltonian, regular, zero-symmetric
Table of graphs and parameters

In the mathematical field of graph theory, a truncated icosidodecahedral graph (or great rhombicosidodecahedral graph) is the graph of vertices and edges of the truncated icosidodecahedron, one of the Archimedean solids. It has 120 vertices and 180 edges, and is a zero-symmetric and cubic Archimedean graph.[5]

Schlegel diagram graphs
Truncated icosidodecahedral graph-hexcenter.png
3-fold symmetry
Truncated icosidodecahedral graph-squarecenter.png
2-fold symmetry

Related polyhedra and tilings[edit]

Conway polyhedron b3I.png Conway polyhedron b3D.png
Bowtie icosahedron and dodecahedron contain two trapezoidal faces in place of the square.[6]

This polyhedron can be considered a member of a sequence of uniform patterns with vertex figure (4.6.2p) and Coxeter-Dynkin diagram CDel node 1.pngCDel p.pngCDel node 1.pngCDel 3.pngCDel node 1.png. For p < 6, the members of the sequence are omnitruncated polyhedra (zonohedrons), shown below as spherical tilings. For p > 6, they are tilings of the hyperbolic plane, starting with the truncated triheptagonal tiling.

Notes[edit]

  1. ^ Wenninger, (Model 16, p. 30)
  2. ^ Williamson (Section 3-9, p. 94)
  3. ^ Cromwell (p. 82)
  4. ^ Weisstein, Eric W. "Icosahedral group". MathWorld.
  5. ^ Read, R. C.; Wilson, R. J. (1998), An Atlas of Graphs, Oxford University Press, p. 269
  6. ^ Symmetrohedra: Polyhedra from Symmetric Placement of Regular Polygons Craig S. Kaplan

References[edit]

External links[edit]