Triakis tetrahedron

From Wikipedia, the free encyclopedia
Jump to: navigation, search
Triakis tetrahedron
Triakistetrahedron.jpg
(Click here for rotating model)
Type Catalan solid
Coxeter diagram CDel node f1.pngCDel 3.pngCDel node f1.pngCDel 3.pngCDel node.png
Conway notation kT
Face type V3.6.6
DU02 facets.png

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″
arccos(−7/11)
Properties convex, face-transitive
Truncated tetrahedron.png
Truncated tetrahedron
(dual polyhedron)
Triakis tetrahedron Net
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 5/311 and volume 25/362.

Orthogonal projections[edit]

Orthogonal projection
Centered by Edge normal Face normal Face/vertex Edge
Truncated
tetrahedron
Tetrahedron t01 ae.png Tetrahedron t01 af36.png 3-simplex t01 A2.svg 3-simplex t01.svg
Triakis
tetrahedron
Dual tetrahedron t01 ae.png Dual tetrahedron t01 af36.png Dual tetrahedron t01 A2.png Dual tetrahedron t01.png
Projective
symmetry
[1] [1] [3] [4]

Variations[edit]

A triakis tetrahedron with equilateral triangle faces represents a net of the four-dimensional regular polytope known as the 5-cell.

5-cell net.png

Stellations[edit]

Stellation of triakis tetrahedron.png

This chiral figure is one of thirteen stellations allowed by Miller's rules.

Related polyhedra[edit]

Spherical triakis tetrahedron

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.

See also[edit]

References[edit]

  1. ^ Conway, Symmetries of things, p.284

External links[edit]