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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 a Catalan solid with 12 faces. Each Catalan solid is the dual of an Archimedean solid. The dual of the triakis tetrahedron is the truncated tetrahedron.

The triakis tetrahedron can be seen as a tetrahedron with a triangular pyramid 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.

The length of the shorter edges is 3/5 that of the longer edges[2]. If the triakis tetrahedron has shorter edge length 1, it has area 5/311 and volume 25/362.

Contents

Tetartoid symmetryEdit

The triakis tetrahedron can be made as a degenerate limit of a tetartoid:

Example tetartoid variations
       
       

Orthogonal projectionsEdit

Orthogonal projection
Centered by Edge normal Face normal Face/vertex Edge
Triakis
tetrahedron
       
(Dual)
Truncated
tetrahedron
       
Projective
symmetry
[1] [1] [3] [4]

VariationsEdit

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

 

If the triangles are right-angled isosceles, the faces will be coplanar and form a cubic volume. This can be seen by adding the 6 edges of tetrahedron inside of a cube.

 

StellationsEdit

 

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

Related polyhedraEdit

 
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 alsoEdit

ReferencesEdit

  1. ^ Conway, Symmetries of things, p.284
  2. ^ https://rechneronline.de/pi/triakis-tetrahedron.php
  • 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, doi:10.1017/CBO9780511569371, ISBN 978-0-521-54325-5, MR 0730208 (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 [1] (Chapter 21, Naming the Archimedean and Catalan polyhedra and tilings, page 284, Triakis tetrahedron )

External linksEdit