Triakis tetrahedron
Triakis tetrahedron | |
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Type | Catalan solid |
Coxeter diagram | ![]() ![]() ![]() ![]() ![]() |
Conway notation | kT |
Face type | V3.6.6![]() isosceles triangle |
Faces | 12 |
Edges | 18 |
Vertices | 8 |
Vertices by type | 4{3}+4{6} |
Symmetry group | Td, A3, [3,3], (*332) |
Rotation group | T, [3,3]+, (332) |
Dihedral angle | 129° 31' 16"![]() |
Properties | convex, face-transitive |
![]() Truncated tetrahedron (dual polyhedron) |
![]() Net |
In geometry, a triakis tetrahedron (or kistetrahedron[1]) is an Archimedean dual solid, or a Catalan solid. Its dual is the truncated tetrahedron.
It can be seen as a tetrahedron with triangular pyramids added to each face; that is, it is the Kleetope of the tetrahedron. It is very similar to the net for the 5-cell, as the net for a tetrahedron is a triangle with other triangles added to each edge, the net for the 5-cell a tetrahedron with pyramids attached to each face. This interpretation is expressed in the name.
If the triakis tetrahedron has shorter edge lengths 1, it has area and volume
.
Orthogonal projections
Centered by | Edge normal | Face normal | Face/vertex | Edge |
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Truncated tetrahedron |
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Triakis tetrahedron |
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Projective symmetry |
[1] | [1] | [3] | [4] |
Variations
A triakis tetrahedron with equilateral triangle faces represents a net of the four-dimensional regular polytope known as the 5-cell.
Stellations
This chiral figure is one of thirteen stellations allowed by Miller's rules.
Related polyhedra
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The triakis tetrahedron is a part of a sequence of polyhedra and tilings, extending into the hyperbolic plane. These face-transitive figures have (*n32) reflectional symmetry.
Symmetry *n32 [n,3] |
Spherical | Euclid. | Compact hyperb. | Paraco. | Noncompact hyperbolic | ||||||
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*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 |
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Config. | 3.4.4 | 3.6.6 | 3.8.8 | 3.10.10 | 3.12.12 | 3.14.14 | 3.16.16 | 3.∞.∞ | 3.24i.24i | 3.18i.18i | 3.12i.12i |
Triakis figures |
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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
References
- ↑ 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 (1983), Dual Models, Cambridge University Press, ISBN 978-0-521-54325-5, MR 730208 (The thirteen semiregular convex polyhedra and their duals, Page 14, Triakistetrahedron)
- The Symmetries of Things 2008, John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, ISBN 978-1-56881-220-5 (Chapter 21, Naming the Archimedean and Catalan polyhedra and tilings, page 284, Triakis tetrahedron )
External links
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