# Tetratriangle

Tetratriangle
Rank2
TypeRegular
SpaceSpherical
Notation
Bowers style acronymTetri
Schläfli symbol{12/4}
Elements
Components4 triangles
Edges12
Vertices12
Measures (edge length 1)
Circumradius${\displaystyle \frac{\sqrt3}{3} ≈ 0.57735}$
Inradius${\displaystyle \frac{\sqrt3}{6} ≈ 0.28868}$
Area${\displaystyle \sqrt3 ≈ 1.73205}$
Angle60°
Central density4
Number of external pieces24
Level of complexity2
Related polytopes
ArmyDog, edge length ${\displaystyle \frac{3\sqrt2-\sqrt6}{6}}$
DualTetratriangle
ConjugateTetratriangle
Convex coreDodecagon
Abstract & topological properties
Flag count24
Euler characteristic0
OrientableYes
Properties
SymmetryI2(12), order 24
ConvexNo
NatureTame

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

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:

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

## 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.