# Compound of four triangles

(Redirected from Tetratriangle)
Compound of four triangles
Rank2
TypeRegular
Notation
Bowers style acronymTetri
Schläfli symbol{12/4}
Elements
Components4 triangles
Edges12
Vertices12
Measures (edge length 1)
Circumradius${\displaystyle {\frac {\sqrt {3}}{3}}\approx 0.57735}$
Inradius${\displaystyle {\frac {\sqrt {3}}{6}}\approx 0.28868}$
Area${\displaystyle {\sqrt {3}}\approx 1.73205}$
Angle60°
Central density4
Number of external pieces24
Level of complexity2
Related polytopes
ArmyDog, edge length ${\displaystyle {\frac {3{\sqrt {2}}-{\sqrt {6}}}{6}}}$
DualCompound of four triangles
ConjugateCompound of four triangles
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:

• ${\displaystyle \left(\pm {\frac {1}{2}},\,\pm {\frac {\sqrt {3}}{6}}\right),}$
• ${\displaystyle \left(0,\,\pm {\frac {\sqrt {3}}{3}}\right),}$
• ${\displaystyle \left(\pm {\frac {\sqrt {3}}{6}},\,\pm {\frac {1}{2}}\right),}$
• ${\displaystyle \left(\pm {\frac {\sqrt {3}}{3}},\,0\right).}$

## 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.