Truncated icosidodecahedron
Truncated icosidodecahedron | |
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(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 |
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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
|
Colored faces |
4.6.10 (Vertex figure) |
Disdyakis triacontahedron (dual polyhedron) |
Net |
In geometry, a truncated icosidodecahedron, rhombitruncated icosidodecahedron,[1] great rhombicosidodecahedron,[2][3] omnitruncated dodecahedron or omnitruncated icosahedron[4] is an Archimedean solid, one of thirteen convex, isogonal, non-prismatic solids constructed by two or more types of regular polygon faces.
It has 62 faces: 30
Names
The name truncated icosidodecahedron, given originally by topologically equivalent to the Archimedean solid.
Alternate interchangeable names are:
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The name great rhombicosidodecahedron refers to the relationship with the (small) rhombicosidodecahedron (compare section Dissection).
There is a
Area and volume
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
- (±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
The truncated icosidodecahedron is the convex hull of a rhombicosidodecahedron with cuboids above its 30 squares, whose height to base ratio is φ. The rest of its space can be dissected into nonuniform cupolas, namely 12 between inner pentagons and outer decagons and 20 between inner triangles and outer hexagons.
An alternative dissection also has a rhombicosidodecahedral core. It has 12 pentagonal rotundae between inner pentagons and outer decagons. The remaining part is a toroidal polyhedron.
dissection images |
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Orthogonal projections
The truncated icosidodecahedron has seven special
Centered by | Vertex | Edge 4-6 |
Edge 4-10 |
Edge 6-10 |
Face square |
Face hexagon |
Face decagon |
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Solid | |||||||
Wireframe | |||||||
Projective symmetry |
[2]+ | [2] | [2] | [2] | [2] | [6] | [10] |
Dual image |
Spherical tilings and Schlegel diagrams
The truncated icosidodecahedron can also be represented as a
Orthographic projection | Stereographic projections | ||
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Decagon-centered | Hexagon-centered | Square-centered | |
Geometric 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.
Truncated icosidodecahedral graph
Truncated icosidodecahedral graph | |
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Hamiltonian, regular, zero-symmetric | |
Table of graphs and parameters |
In the
3-fold symmetry |
2-fold symmetry |
Related polyhedra and tilings
Bowtie icosahedron and dodecahedron contain two trapezoidal faces in place of the square.[7] |
Family of uniform icosahedral polyhedra | |||||||
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Symmetry: [5,3], (*532) | [5,3]+, (532) | ||||||
{5,3} | t{5,3} | r{5,3} | t{3,5} | {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
*n32 symmetry mutation of omnitruncated tilings: 4.6.2n | ||||||||||||
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Sym. *n32 [n,3] |
Spherical | Euclid. | Compact hyperb. | Paraco. | Noncompact hyperbolic | |||||||
*232 [2,3] |
*332 [3,3] |
*432 [4,3] |
*532 [5,3] |
*632 [6,3] |
*732 [7,3] |
*832 [8,3] |
*∞32 [∞,3] |
[12i,3] |
[9i,3] |
[6i,3] |
[3i,3] | |
Figures | ||||||||||||
Config. | 4.6.4 | 4.6.6 | 4.6.8 | 4.6.10 | 4.6.12 | 4.6.14 | 4.6.16 | 4.6.∞ | 4.6.24i | 4.6.18i | 4.6.12i | 4.6.6i |
Duals | ||||||||||||
Config.
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V4.6.4 | V4.6.6 | V4.6.8 | V4.6.10 | V4.6.12
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V4.6.14
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V4.6.16
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V4.6.∞ | V4.6.24i | V4.6.18i | V4.6.12i | V4.6.6i |
Notes
- ^ a b Wenninger Model Number 16
- ^ a b Williams (Section 3-9, p. 94)
- ^ a b Cromwell (p. 82)
- ^ a b Norman Woodason Johnson, "The Theory of Uniform Polytopes and Honeycombs", 1966
- ^ Weisstein, Eric W. "Icosahedral group". MathWorld.
- ^ Read, R. C.; Wilson, R. J. (1998), An Atlas of Graphs, Oxford University Press, p. 269
- ^ Symmetrohedra: Polyhedra from Symmetric Placement of Regular Polygons Craig S. Kaplan
References
- MR 0467493
- Cromwell, P. (1997). Polyhedra. United Kingdom: Cambridge. pp. 79–86 Archimedean solids. ISBN 0-521-55432-2.
- ISBN 0-486-23729-X.
- Cromwell, P.; Polyhedra, CUP hbk (1997), pbk. (1999).
- Weisstein, Eric W., "GreatRhombicosidodecahedron" ("Archimedean solid") at MathWorld.
- Klitzing, Richard. "3D convex uniform polyhedra x3x5x - grid".