Triangular-antitegmatic icosachoron

Triangular-antitegmatic icosachoron
(OFF file)
Rank	4
Type	Uniform dual
Space	Spherical
Notation
Coxeter diagram	m3o3o3m ()
Elements
Cells	20 triangular antitegums
Faces	60 rhombi
Edges	30+40
Vertices	10+20
Vertex figure	20 triangular bipyramids, 10 tetrahedra
Measures (edge length 1)
Inradius	$\frac{\sqrt{10}}{4} \approx 0.79057$
Dichoral angle	120°
Central density	1
Related polytopes
Dual	Small prismatodecachoron
Abstract & topological properties
Flag count	960
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	A4×2, order 240
Convex	Yes
Nature	Tame

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 Minkowski sum of 5 line segments from the center to vertices of a pentachoron. This makes it a zonochoron.

Each face of this polyhedron is a rhombus with acute angle $\arccos\left(\frac14\right) \approx 75.52249^\circ$ and obtuse angle $\arccos\left(-\frac14\right) \approx 104.47751^\circ$.

Isogonal derivatives

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

• Triangular antitegum (20): Small prismatodecachoron
• Rhombus (60): Rectified small prismatodecachoron
• Edge (30): Decachoron
• Edge (40): Bitruncatodecachoron
• Vertex (10): Bidecachoron
• Vertex (20): Biambodecachoron

External links

• Bowers, Jonathan. "Four Dimensional Dice Up To Twenty Sides".

• Klitzing, Richard. "m3o3o3m".