Triakis icosahedron
| Triakis icosahedron | |
|---|---|
(Click here for rotating model) |
|
| Type | Catalan solid |
| Coxeter diagram | |
| Conway notation | kI |
| Face type | V3.10.10
isosceles triangle |
| Faces | 60 |
| Edges | 90 |
| Vertices | 32 |
| Vertices by type | 20{3}+12{10} |
| Symmetry group | Ih, H3, [5,3], (*532) |
| Rotation group | I, [5,3]+, (532) |
| Dihedral angle | 160°36′45″ arccos(−24 + 15√5/61) |
| Properties | convex, face-transitive |
Truncated dodecahedron (dual polyhedron) |
Net |
In geometry, the triakis icosahedron (or kisicosahedron[1]) is an Archimedean dual solid, or a Catalan solid. Its dual is the truncated dodecahedron.
Contents
Orthogonal projections[edit]
The triakis icosahedron has three symmetry positions, two on vertices, and one on a midedge: The Triakis icosahedron has five special orthogonal projections, centered on a vertex, on two types of edges, and two types of faces: hexagonal and pentagonal. The last two correspond to the A2 and H2 Coxeter planes.
| Projective symmetry |
[2] | [6] | [10] |
|---|---|---|---|
| Image | |||
| Dual image |
Kleetope[edit]
It can be seen as an icosahedron with triangular pyramids augmented to each face; that is, it is the Kleetope of the icosahedron. This interpretation is expressed in the name, triakis.
Other triakis icosahedra[edit]
This interpretation can also apply to other similar nonconvex polyhedra with pyramids of different heights:
First stellation of icosahedron, or small triambic icosahedron (sometimes called a triakis icosahedron)
Great stellated dodecahedron (with very tall pyramids)
Great dodecahedron (with inverted pyramids)
Stellations[edit]
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The triakis icosahedron has numerous stellations, including this one.
Related polyhedra[edit]
| Family of uniform icosahedral polyhedra | |||||||
|---|---|---|---|---|---|---|---|
| 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 |
The triakis icosahedron is a part of a sequence of polyhedra and tilings, extending into the hyperbolic plane. These face-transitive figures have (*n32) reflectional symmetry.
| *n32 symmetry mutation of truncated tilings: t{n,3} | |||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|
| Symmetry *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] | |
| Truncated figures |
|||||||||||
| Symbol | t{2,3} | t{3,3} | t{4,3} | t{5,3} | t{6,3} | t{7,3} | t{8,3} | t{∞,3} | t{12i,3} | t{9i,3} | t{6i,3} |
| Triakis figures |
|||||||||||
| Config. | V3.4.4 | V3.6.6 | V3.8.8 | V3.10.10 | V3.12.12 | V3.14.14 | V3.16.16 | V3.∞.∞ | |||
See also[edit]
- Triakis triangular tiling for other "triakis" polyhedral forms.
- Great triakis icosahedron
References[edit]
- ^ Conway, Symmetries of things, p.284
- Williams, Robert (1979). The Geometrical Foundation of Natural Structure: A Source Book of Design. Dover Publications, Inc. ISBN 0-486-23729-X. (Section 3-9)
- Wenninger, Magnus (1974). Polyhedron Models. Cambridge University Press. ISBN 0-521-09859-9.
- Wenninger, Magnus (1983). Dual Models. Cambridge University Press. ISBN 978-0-521-54325-5. MR 730208. (The thirteen semiregular convex polyhedra and their duals, Page 19, Triakisicosahedron)
- The Symmetries of Things 2008, John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, ISBN 978-1-56881-220-5 [1] (Chapter 21, Naming the Archimedean and Catalan polyhedra and tilings, page 284, Triakis icosahedron )
External links[edit]
- Eric W. Weisstein, Triakis icosahedron (Catalan solid) at MathWorld.
- Triakis Icosahedron – Interactive Polyhedron Model
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