Tetratriangle

The tetratriangle, or tetri, is a polygon compound composed of 4 triangles. As such it has 12 edges and 12 vertices.

Tetratriangle
(OFF file)
Rank	2
Type	Regular
Space	Spherical
Notation
Bowers style acronym	Tetri
Schläfli symbol	{12/4}
Elements
Components	4 triangles
Edges	12
Vertices	12
Vertex figure	Dyad, length 1
Measures (edge length 1)
Circumradius	$\frac{\sqrt3}{3} ≈ 0.57735$
Inradius	$\frac{\sqrt3}{6} ≈ 0.28868$
Area	$\sqrt3 ≈ 1.73205$
Angle	60°
Central density	4
Number of external pieces	24
Level of complexity	2
Related polytopes
Army	Dog, edge length $\frac{3\sqrt2-\sqrt6}{6}$
Dual	Tetratriangle
Conjugate	Tetratriangle
Convex core	Dodecagon
Abstract & topological properties
Flag count	24
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	I2(12), order 24
Convex	No
Nature	Tame

It is the third stellation of the dodecagon.

Its quotient prismatic equivalent is the triangular tetrahedroorthowedge, which is five-dimensional.

Vertex coordinates

Coordinates for the vertices of a tetratriangle of edge length 1 centered at the origin are given by:

• $\left(±\frac12,\,±\frac{\sqrt3}{6}\right),$
• $\left(0,\,±\frac{\sqrt3}{3}\right),$
• $\left(±\frac{\sqrt3}{6},\,±\frac12\right),$
• $\left(±\frac{\sqrt3}{3},\,0\right).$

Variations

The tetratriangle can be varied by seeing it as a compound of 2 hexagrams and changing the angle between the two component hexagrams from the usual 30°. These 4-triangle compounds generally have a dihexagon as their convex hull and remain uniform, but not regular, with hexagonal symmetry only.

External links

• Bowers, Jonathan. "Regular Polygons and Other Two Dimensional Shapes".