# Triangular tegum

Triangular tegum | |
---|---|

Rank | 3 |

Type | CRF |

Notation | |

Bowers style acronym | Trit |

Coxeter diagram | oxo3ooo&#xt |

Elements | |

Faces | 6 triangles |

Edges | 3+6 |

Vertices | 2+3 |

Vertex figures | 2 triangles, edge length 1 |

3 rhombi, edge length 1 | |

Measures (edge length 1) | |

Inradius | |

Volume | |

Dihedral angles | 3–3 equatorial: |

3–3 pyramidal: | |

Height | |

Central density | 1 |

Number of external pieces | 6 |

Level of complexity | 3 |

Related polytopes | |

Army | Trit |

Regiment | Trit |

Dual | Semi-uniform Triangular prism |

Conjugate | None |

Abstract & topological properties | |

Flag count | 36 |

Euler characteristic | 2 |

Surface | Sphere |

Orientable | Yes |

Genus | 0 |

Properties | |

Symmetry | A_{2}×A_{1}, order 12 |

Flag orbits | 3 |

Convex | Yes |

Nature | Tame |

The **triangular tegum**, also called a **triangular bipyramid** or **triangular dipyramid**, is one of the 92 Johnson solids (J_{12}). It has 6 equilateral triangles as faces, with 2 order-3 and 3 order-4 vertices. It can be constructed by joining two regular tetrahedra at one of their faces.

It is one of three regular polygonal tegums to be CRF. The others are the regular octahedron (square tegum) and the pentagonal tegum.

The simplicial bipyramids, a family of Blind polytopes, generalize the triangular tegum to any number of dimensions.

The triangular tegum (with theoretical edge length 1) is the vertex figure of the triangular duotegum, which cannot be made uniform, because the Johnson solid variant is not circumscribable.

## Vertex coordinates[edit | edit source]

A triangular tegum of edge length 1 has the following vertices:

## Representations[edit | edit source]

A triangular tegum has the following Coxeter diagrams:

- oxo3ooo&#xt (as tower)
- yo ox3oo&#zx (y = 2√6/3, as tegum product)
- oyo oox&#xt (digonal symmetry)

## Variations[edit | edit source]

The triangular tegum can have the height of its pyramids varied while maintaining its full symmetry. These variations generally have 6 isosceles triangles for faces.

One notable variations can be obtained as the dual of the uniform triangular prism, which can be represented by m2m3o. In this variant the side edges are exactly times the length of the edges of the base triangle, and all the dihedral angles are . Each face has apex angle and base angles . If the base triangle has edge length 1, its height is .

A triangular tegum with base edges of length b and side edges of length l has volume given by .

## Other triangular bipyramids[edit | edit source]

Besides this fully symmetric version, other 5-vertex polyhedra with 6 triangular faces exist:

- Apiculated triangular pyramid - the two pyramids are different heights, dual to a triangular frustum
- Notch - 2 isosceles and 4 scalene triangles, dual to a wedge
- Scalene notch - 3 pairs of scalene triangles, dual to a skewed wedge

## Related polyhedra[edit | edit source]

A triangular prism can be inserted between the halves of the triangular tegum to produce the elongated triangular bipyramid. Trying to insert a regular octahedron (as a triangular antiprism) would result in pairs of triangles becoming coplanar and turning into 60°/120° rhombi, resulting in a triangular antitegum, a variant of the cube.

Truncating only the vertices where four triangles meet results in the 3D associahedron.

## External links[edit | edit source]

- Klitzing, Richard. "tridpy".
- Quickfur. "The Triangular Bipyramid".

- Wikipedia contributors. "Triangular bipyramid".
- McCooey, David. "Triangular Dipyramid J12"