Truncated icosidodecahedron
Truncated icosidodecahedron  

(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} 
Schläfli symbols  tr{5,3} 
t_{0,1,2}{5,3}  
Wythoff symbol  2 3 5  
Coxeter diagram  
Symmetry group  I_{h}, H_{3}, [5,3], (*532), order 120 
Rotation group  I, [5,3]^{+}, (532), order 60 
Dihedral Angle  610:142.62° 410:148.28° 46:159.095° 
References  U_{28}, C_{31}, W_{16} 
Properties  Semiregular convex zonohedron 
Colored faces 
4.6.10 (Vertex figure) 
Disdyakis triacontahedron (dual polyhedron) 
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 30 square faces, 20 regular hexagonal faces, 12 regular decagonal faces, 120 vertices and 180 edges – more than any other nonprismatic uniform polyhedron. Since each of its faces has point symmetry (equivalently, 180° rotational symmetry), the truncated icosidodecahedron is a zonohedron.
Contents
Other names
Alternate interchangeable names include:
 Truncated icosidodecahedron (Johannes Kepler)
 Rhombitruncated icosidodecahedron (Magnus Wenninger^{1})
 Great rhombicosidodecahedron (Robert Williams,^{2} Peter Cromwell^{3})
 Omnitruncated dodecahedron or icosahedron (Norman Johnson)
The name truncated icosidodecahedron, originally given by Johannes Kepler, is somewhat misleading. If one truncates an icosidodecahedron by cutting the corners off, one does not get this uniform figure: instead of squares the truncation has golden rectangles. However, the resulting figure is topologically equivalent to this and can always be deformed until the faces are regular.
Icosidodecahedron 
A literal geometric truncation of the icosidodecahedron produces rectangular faces rather than squares. 
The alternative name great rhombicosidodecahedron (as well as rhombitruncated icosidodecahedron) refers to the fact that the 30 square faces lie in the same planes as the 30 faces of the rhombic triacontahedron which is dual to the icosidodecahedron. Compare to small rhombicosidodecahedron.
One unfortunate point of confusion is that there is a nonconvex uniform polyhedron of the same name. See nonconvex great rhombicosidodecahedron.
Variations
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.
Area and volume
The surface area A and the volume V of the truncated icosidodecahedron of edge length a are:
If a set of all 13 Archimedean solids were constructed with all edge lengths equal, the truncated icosidodecahedron would be the largest.
Cartesian coordinates
Cartesian coordinates for the vertices of a truncated icosidodecahedron with edge length 2τ − 2, centered at the origin, are all the even permutations of:^{4}
 (±1/τ, ±1/τ, ±(3+τ)),
 (±2/τ, ±τ, ±(1+2τ)),
 (±1/τ, ±τ^{2}, ±(−1+3τ)),
 (±(1+2τ), ±2, ±(2+τ)) and
 (±τ, ±3, ±2τ),
where τ = (1 + √5)/2 is the golden ratio.
Orthogonal projections
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 A_{2} and H_{2} Coxeter planes.
Centered by  Vertex  Edge 46 
Edge 410 
Edge 610 
Face square 
Face hexagon 
Face decagon 

Image  
Projective symmetry 
[2]^{+}  [2]  [2]  [2]  [2]  [6]  [10] 
Spherical tiling
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.
Decagoncentered 
Hexagoncentered 
squarecentered 

Spherical tiling  Stereographic projections (facecentered) 

Related polyhedra and tilings
Symmetry: [5,3], (*532)  [5,3]^{+}, (532)  

{5,3}  t{5,3}  r{5,3}  2t{5,3}=t{3,5}  2r{5,3}={3,5}  rr{5,3}  tr{5,3}  sr{5,3} 
Duals to uniform polyhedra  
V5.5.5  V3.10.10  V3.5.3.5  V5.6.6  V3.3.3.3.3  V3.4.5.4  V4.6.10  V3.3.3.3.5 
This polyhedron can be considered a member of a sequence of uniform patterns with vertex figure (4.6.2p) and CoxeterDynkin diagram . 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.
Symmetry *n32 [n,3] 
Spherical  Euclidean  Compact hyperbolic  Paracompact  

*232 [2,3] D_{3h} 
*332 [3,3] T_{d} 
*432 [4,3] O_{h} 
*532 [5,3] I_{h} 
*632 [6,3] P6m 
*732 [7,3] 
*832 [8,3]... 
*∞32 [∞,3] 

Coxeter Schläfli 
tr{2,3} 
tr{3,3} 
tr{4,3} 
tr{5,3} 
tr{6,3} 
tr{7,3} 
tr{8,3} 
tr{∞,3} 
Omnitruncated figure 

Vertex figure  4.6.4  4.6.6  4.6.8  4.6.10  4.6.12  4.6.14  4.6.16  4.6.∞ 
Dual figures  
Coxeter  
Omnitruncated duals 

Face configuration 
V4.6.4  V4.6.6  V4.6.8  V4.6.10  V4.6.12  V4.6.14  V4.6.16  V4.6.∞ 
See also
 Spinning great rhombicosidodecahedron
 Dodecahedron
 Great truncated icosidodecahedron
 Icosahedron
 Truncated cuboctahedron
Notes
 ^ Wenninger, (Model 16, p. 30)
 ^ Williamson (Section 39, p. 94)
 ^ Cromwell (p. 82)
 ^ Weisstein, Eric W., "Icosahedral group", MathWorld.
References
 Wenninger, Magnus (1974), Polyhedron Models, Cambridge University Press, ISBN 9780521098595, MR 0467493
 Cromwell, P. (1997). Polyhedra. United Kingdom: Cambridge. pp. 79–86 Archimedean solids. ISBN 0521554322.
 Williams, Robert (1979). The Geometrical Foundation of Natural Structure: A Source Book of Design. Dover Publications, Inc. ISBN 048623729X.
 Cromwell, P.; Polyhedra, CUP hbk (1997), pbk. (1999).
 Eric W. Weisstein, GreatRhombicosidodecahedron (Archimedean solid) at MathWorld
 Richard Klitzing, 3D convex uniform polyhedra, x3x5x  grid
External links
 Editable printable net of a truncated icosidodecahedron with interactive 3D view
 The Uniform Polyhedra
 Virtual Reality Polyhedra The Encyclopedia of Polyhedra

HPTS  Area Progetti  EduSoft  JavaEdu  N.Saperi  Ass.Scuola..  TS BCTV  TS VideoRes  TSODP  TRTWE  