# Triakis tetrahedron

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.

Triakis tetrahedron | |
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(Click here for rotating model) | |

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 | T_{d}, A_{3}, [3,3], (*332) |

Rotation group | T, [3,3]^{+}, (332) |

Dihedral angle | 129°31′16″ arccos(−7/11) |

Properties | convex, face-transitive |

Truncated tetrahedron (dual polyhedron) |
Net |

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 5/3√11 and volume 25/36√2.

## Tetartoid symmetry

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

## Orthogonal projections

Centered by | Edge normal | Face normal | Face/vertex | Edge |
---|---|---|---|---|

Triakis tetrahedron |
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(Dual) Truncated 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.

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.

## Related polyhedra

The triakis tetrahedron is a part of a sequence of polyhedra and tilings, extending into the hyperbolic plane. These face-transitive figures have (**n*32) 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.∞.∞ |

Family of uniform tetrahedral polyhedra | |||||||
---|---|---|---|---|---|---|---|

Symmetry: [3,3], (*332) | [3,3]^{+}, (332) | ||||||

{3,3} | t{3,3} | r{3,3} | t{3,3} | {3,3} | rr{3,3} | tr{3,3} | sr{3,3} |

Duals to uniform polyhedra | |||||||

V3.3.3 | V3.6.6 | V3.3.3.3 | V3.6.6 | V3.3.3 | V3.4.3.4 | V4.6.6 | V3.3.3.3.3 |

## See also

## References

- Conway, Symmetries of things, p.284
- 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 (Chapter 21, Naming the Archimedean and Catalan polyhedra and tilings, page 284, Triakis tetrahedron )