Triangular tegum

Triangular tegum
(3D model) (OFF file)
Rank	3
Type	CRF
Space	Spherical
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	$\frac{\sqrt6}{9} ≈ 0.27217$
Volume	$\frac{\sqrt2}{6} ≈ 0.23570$
Dihedral angles	3–3 equatorial: $\arccos\left(-\frac79\right) ≈ 141.05756°$
	3–3 pyramidal: $\arccos\left(\frac13\right) ≈ 70.52878°$
Height	$\frac{2\sqrt6}{3} ≈ 1.63299$
Central density	1
Related polytopes
Army	Trit
Regiment	Trit
Dual	Semi-uniform Triangular prism
Conjugate	None
Abstract properties
Euler characteristic	2
Topological properties
Surface	Sphere
Orientable	Yes
Genus	0
Properties
Symmetry	A2×A1, order 12
Convex	Yes
Nature	Tame

The triangular tegum, also called a triangular bipyramid or triangular dipyramid, is one of the 92 Johnson solids (J₁₂). 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.

Vertex coordinates

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

• $\left(±\frac{1}{2},\,-\frac{\sqrt3}{6},\,0\right),$
• $\left(0,\,\frac{\sqrt3}{3},\,0\right),$
• $\left(0,\,0,\,±\frac{\sqrt6}{3}\right).$

Representations

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

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 $\frac23$ times the length of the edges of the base triangle, and all the dihedral angles are $\arccos\left(-\frac17\right) ≈ 98.21321°$. Each face has apex angle $\arccos\left(-\frac18\right) \approx 97.18076°$ and base angles $\arccos\left(\frac34\right) \approx 41.40962°$. If the base triangle has edge length 1, its height is $\frac23 ≈ 0.66667$.

A triangular tegum with base edges of length b and side edges of length l has volume given by $\frac{\sqrt3b^2}{6}\sqrt{l^2-\frac{b^2}{3}}$.

Other triangular bipyramids

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

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

• Klitzing, Richard. "tridpy".

• Quickfur. "The Triangular Bipyramid".

• Wikipedia Contributors. "Triangular bipyramid".
• McCooey, David. "Triangular Dipyramid J12"