Triangular-rectified pentachoric duoprism

Triangular-rectified pentachoric duoprism
Rank	6
Type	Uniform
Space	Spherical
Notation
Bowers style acronym	Trarap
Coxeter diagram	x3o o3x3o3o ()
Elements
Peta	5 triangular-tetrahedral duoprisms, 5 triangular-octahedral duoprisms, 3 rectified pentachoric prisms
Tera	15 tetrahedral prisms, 10+20 triangular duoprisms, 15 octahedral prisms, 3 rectified pentachora
Cells	15 tetrahedra, 30+30+60 triangular prisms, 15 octahedra
Faces	10+30+60 triangles, 90 squares
Edges	30+90
Vertices	30
Vertex figure	Triangular prismatic scalene, edge lengths 1 (base tetrahedron and top edge) and √2 (sides)
Measures (edge length 1)
Circumradius	$\frac{\sqrt{210}}{15} \approx 0.96609$
Hypervolume	$\frac{11\sqrt{15}}{384} \approx 0.11095$
Dipetal angles	Tratet–triddip–troct: $\arccos{\left(-\frac14\right)} \approx 104.47751^\circ$
	Rappip–ope–troct: 90°
	Rappip–tepe–tratet: 90°
	Troct–triddip–troct: $\arccos{\left(\frac14\right)} \approx 75.52249^\circ$
	Rappip–rap–rappip: 60°
Heights	Rap atop rappip: $\frac{\sqrt3}{2} \approx 0.86603$
	Tratet atop troctt: $\frac{\sqrt{10}}{4} \approx 0.79057$
Central density	1
Number of external pieces	13
Level of complexity	45
Related polytopes
Army	Trarap
Regiment	Trarap
Dual	Triangular-joined pentachoric duotegum
Conjugate	None
Abstract & topological properties
Flag count	32400
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	A4×A2, order 720
Convex	Yes
Nature	Tame

The triangular-rectified pentachoric duoprism or trarap is a convex uniform duoprism that consists of 3 rectified pentachoric prisms, 5 triangular-octahedral duoprisms, and 5 triangular-tetrahedral duoprisms. Each vertex joins 2 rectified pentachoric prisms, 2 triangular-tetrahedral duoprisms, and 3 triangular-octahedral duoprisms. It is a duoprism based on a triangle and a rectified pentachoron, and is thus also a convex segmentopeton, as a rectified pentachoron atop rectified pentachoric prism.

Vertex coordinates

The vertices of a triangular-rectified pentachoric duoprism of edge length 1 are given by:

• $\left(\pm\frac12,\,-\frac{\sqrt3}{6},\,-\frac{3\sqrt{10}}{20},\,-\frac{\sqrt6}{4},\,0,\,0\right)$,
• $\left(0,\,\frac{\sqrt3}{3},\,-\frac{3\sqrt{10}}{20},\,-\frac{\sqrt6}{4},\,0,\,0\right)$,
• $\left(\pm\frac12,\,-\frac{\sqrt3}{6},\,-\frac{3\sqrt{10}}{20},\,\frac{\sqrt6}{12},\,-\frac{\sqrt3}{3},\,0\right)$,
• $\left(0,\,\frac{\sqrt3}{3},\,-\frac{3\sqrt{10}}{20},\,\frac{\sqrt6}{12},\,-\frac{\sqrt3}{3},\,0\right)$,
• $\left(\pm\frac12,\,-\frac{\sqrt3}{6},\,-\frac{3\sqrt{10}}{20},\,\frac{\sqrt6}{12},\,\frac{\sqrt3}{6},\,\pm\frac12\right)$,
• $\left(0,\,\frac{\sqrt3}{3},\,-\frac{3\sqrt{10}}{20},\,\frac{\sqrt6}{12},\,\frac{\sqrt3}{6},\,\pm\frac12\right)$,
• $\left(\pm\frac12,\,-\frac{\sqrt3}{6},\,\frac{\sqrt{10}}{10},\,\frac{\sqrt6}{6},\,\frac{\sqrt3}{3},\,0\right)$,
• $\left(0,\,\frac{\sqrt3}{3},\,\frac{\sqrt{10}}{10},\,\frac{\sqrt6}{6},\,\frac{\sqrt3}{3},\,0\right)$,
• $\left(\pm\frac12,\,-\frac{\sqrt3}{6},\,\frac{\sqrt{10}}{10},\,-\frac{\sqrt6}{6},\,-\frac{\sqrt3}{3},\,0\right)$,
• $\left(0,\,\frac{\sqrt3}{3},\,\frac{\sqrt{10}}{10},\,-\frac{\sqrt6}{6},\,-\frac{\sqrt3}{3},\,0\right)$,
• $\left(\pm\frac12,\,-\frac{\sqrt3}{6},\,\frac{\sqrt{10}}{10},\,\frac{\sqrt6}{6},\,-\frac{\sqrt3}{6},\,\pm\frac12\right)$,
• $\left(0,\,\frac{\sqrt3}{3},\,\frac{\sqrt{10}}{10},\,\frac{\sqrt6}{6},\,-\frac{\sqrt3}{6},\,\pm\frac12\right)$,
• $\left(\pm\frac12,\,-\frac{\sqrt3}{6},\,\frac{\sqrt{10}}{10},\,-\frac{\sqrt6}{6},\,\frac{\sqrt3}{6},\,\pm\frac12\right)$,
• $\left(0,\,-\frac{\sqrt3}{3},\,\frac{\sqrt{10}}{10},\,-\frac{\sqrt6}{6},\,\frac{\sqrt3}{6},\,\pm\frac12\right)$.

Representations

A triangular-rectified pentachoric duoprism has the following Coxeter diagrams:

• x3o o3x3o3o () (full symmetry)
• xx3oo xo3ox3oo&#x (A₃×A₂ symmetry, triangular-tetrahedral duoprism atop triangular-octahedral duoprism)
• ox oo3xx3oo3oo&#x (A₄×A₁ symmetry, rectified pentachoron atop rectified pentachoric prism)

External links

• Klitzing, Richard. "trarap".