# Triangular-antitegmatic icosachoron

Triangular-antitegmatic icosachoron
Rank4
TypeUniform dual
Notation
Coxeter diagramm3o3o3m ()
Elements
Cells20 triangular antitegums
Faces60 rhombi
Edges30+40
Vertices10+20
Vertex figure20 triangular bipyramids, 10 tetrahedra
Measures (edge length 1)
Inradius${\displaystyle {\frac {\sqrt {10}}{4}}\approx 0.79057}$
Dichoral angle120°
Central density1
Related polytopes
DualSmall prismatodecachoron
Abstract & topological properties
Flag count960
Euler characteristic0
OrientableYes
Properties
SymmetryA4×2, order 240
Flag orbits4
ConvexYes
NatureTame

The triangular-antitegmatic icosachoron is a convex isochoric polychoron with 20 triangular antitegums as cells. It can be obtained as the dual of the small prismatodecachoron.

It can also be constructed as the convex hull of 2 dual pentachora and 2 opposite rectified pentachora, all of the same edge length. Related to this fact is that it is the 4D vertex-first projection of the regular 5-cube, or in other words, it is the zonohedrification of a pentachoron.

Each face of this polyhedron is a rhombus with acute angle ${\displaystyle \arccos \left({\frac {1}{4}}\right)\approx 75.52249^{\circ }}$ and obtuse angle ${\displaystyle \arccos \left(-{\frac {1}{4}}\right)\approx 104.47751^{\circ }}$.

## Vertex coordinates

Vertex coordinates for a triangular-antitegmatic icosachoron can be given as:

• ${\displaystyle \pm \left({\dfrac {\sqrt {10}}{10}},\,{\dfrac {\sqrt {6}}{6}},\,{\dfrac {\sqrt {3}}{3}},\,\pm 1\right)}$,
• ${\displaystyle \pm \left({\dfrac {\sqrt {10}}{10}},\,{\dfrac {\sqrt {6}}{6}},\,-{\dfrac {2{\sqrt {3}}}{3}},\,0\right)}$,
• ${\displaystyle \pm \left({\dfrac {\sqrt {10}}{10}},\,-{\dfrac {\sqrt {6}}{2}},\,0,\,0\right)}$,
• ${\displaystyle \pm \left(\pm {\dfrac {\sqrt {10}}{5}},\,{\dfrac {\sqrt {6}}{3}},\,-{\dfrac {\sqrt {3}}{3}},\,\pm 1\right)}$,
• ${\displaystyle \pm \left(\pm {\dfrac {\sqrt {10}}{5}},\,{\dfrac {\sqrt {6}}{3}},\,{\dfrac {2{\sqrt {3}}}{3}},\,0\right)}$,
• ${\displaystyle \pm \left({\dfrac {3{\sqrt {10}}}{10}},\,-{\dfrac {\sqrt {6}}{6}},\,-{\dfrac {\sqrt {3}}{3}},\,\pm 1\right)}$,
• ${\displaystyle \pm \left({\dfrac {3{\sqrt {10}}}{10}},\,-{\dfrac {\sqrt {6}}{6}},\,{\dfrac {2{\sqrt {3}}}{3}},\,0\right)}$,
• ${\displaystyle \pm \left({\dfrac {3{\sqrt {10}}}{10}},\,{\dfrac {\sqrt {6}}{2}},\,0,\,0\right)}$,
• ${\displaystyle \pm \left({\dfrac {2{\sqrt {10}}}{5}},\,0,\,0,\,0\right)}$.

## Isogonal derivatives

Substitution by vertices of these following elements will produce these convex isogonal polychora: