# Triakis triangular tegum

Triakis triangular tegum
Rank3
Elements
Faces6 isosceles triangles, 12 scalene triangles
Edges3+6+6+12
Vertices2+3+6
Vertex figures2 triambi
3 rectangular-symmetric octagons
6 isosceles triangles
Measures (edge length 1)
Central density1
Related polytopes
DualTruncated triangular prism
Abstract & topological properties
Euler characteristic2
SurfaceSphere
OrientableYes
Genus0
Properties
SymmetryA2×A1, order 12
ConvexYes
NatureTame

The triakis triangular tegum is a polyhedron formed from the triangular tegum by augmenting its faces with shallow triangular pyramids. It has 6 isosceles triangles and 12 scalene triangles as faces.

The canonical variant with midradius 1 has four edge lengths: one of length ${\displaystyle {\frac {4{\sqrt {3}}}{9}}\approx 0.76980}$, one of length ${\displaystyle {\frac {10{\sqrt {3}}}{9}}\approx 1.92450}$, one of length ${\displaystyle {\frac {4{\sqrt {3}}}{3}}\approx 2.30940}$ and the other of length ${\displaystyle 2{\sqrt {3}}\approx 3.46410}$.

## Vertex coordinates

The vertices of a canonical triakis triangular tegum of midradius 1 are given by:

• ${\displaystyle \left(0,\,0,\,\pm {\frac {2{\sqrt {3}}}{3}}\right),}$
• ${\displaystyle \left(0,\,{\frac {2}{3}},\,\pm {\frac {4{\sqrt {3}}}{9}}\right),}$
• ${\displaystyle \left(\pm {\frac {\sqrt {3}}{3}},\,-{\frac {1}{3}},\,\pm {\frac {4{\sqrt {3}}}{9}}\right),}$
• ${\displaystyle \left(0,\,-2,\,0\right),}$
• ${\displaystyle \left(\pm {\sqrt {3}},\,1,\,0\right),}$

## In vertex figures

A variant of the triakis triangular tegum with (A2×A1)+ symmetry occurs as the vertex figure of the tetrafold tetraswirlchoron.