Compound of four triangular prisms (prismatic symmetry)

Compound of four triangular prisms (prismatic symmetry)
Rank3
TypeUniform
Elements
Components4 triangular prisms
Faces12 squares, 8 triangles as 2 tetratriangles
Edges12+24
Vertices24
Vertex figureIsosceles triangle, edge lengths 1, 2, 2
Measures (edge length 1)
Circumradius${\displaystyle {\frac {\sqrt {21}}{6}}\approx 0.76376}$
Volume${\displaystyle {\sqrt {3}}\approx 1.73205}$
Dihedral angles4–3: 90°
4–4: 60°
Height1
Central density4
Number of external pieces26
Level of complexity6
Related polytopes
ArmySemi-uniform Twip, edge lengths ${\displaystyle {\frac {3{\sqrt {2}}-{\sqrt {6}}}{6}}}$ (base), 1 (sides)
Regiment*
DualCompound of four triangular tegums
ConjugateCompound of four triangular prisms
Abstract & topological properties
Flag count144
OrientableYes
Properties
SymmetryI2(12)×A1, order 48
ConvexNo
NatureTame

The tetratriangular prism or compound of four triangular prisms with prismatic symmetry is a prismatic uniform polyhedron compound. It consists of 2 tetratriangles and 12 squares. Each vertex joins one tetratriangle and two squares. As the name suggests, it is a prism based on a tetratriangle.

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

Vertex coordinates

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

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

Variations

This compound has variants where the bases are non-regular compounds of four triangles (seen as two-hexagram compounds). In these cases the compound has only hexagonal prismatic symmetry and the convex hull is a dihexagonal prism.