6-simplicial prism

6-simplicial prism
File:6-simplicial prism.png
Rank	7
Type	Uniform
Notation
Bowers style acronym	Hopip
Coxeter diagram	x x3o3o3o3o3o ()
Tapertopic notation	151
Elements
Exa	2 heptapeta, 7 hexateric prisms
Peta	14 hexatera, 21 pentachoric prisms
Tera	42 pentachora, 35 tetrahedral prisms
Cells	70 tetrahedra, 35 triangular prisms
Faces	70 triangles, 21 squares
Edges	7+42
Vertices	14
Vertex figure	Hexateric pyramid, edge lengths 1 (base), √2 (legs)
Measures (edge length 1)
Circumradius	${\frac {\sqrt {133}}{14}}\approx 0.82375$
Hypervolume	${\frac {\sqrt {7}}{5760}}\approx 0.00045933$
Diexal angles	Hop–hix–hixip: 90°
	Hixip–penp–hixip: $\arccos \left({\frac {1}{6}}\right)\approx 80.40593^{\circ }$
Heights	Hop atop hop: 1
	Dyad atop hixip: ${\frac {\sqrt {21}}{6}}\approx 0.76376$
Central density	1
Number of external pieces	9
Level of complexity	7
Related polytopes
Army	Hopip
Regiment	Hopip
Dual	Heptapetic tegum
Conjugate	None
Abstract & topological properties
Flag count	70560
Euler characteristic	2
Orientable	Yes
Properties
Symmetry	A6×A1, order 10080
Convex	Yes
Nature	Tame

The heptapetic prism or hopip is a prismatic uniform polyexon that consists of 2 heptapeta and 7 hexateric prisms as facets. Each vertex joins 1 heptapeton and 6 hexateric prisms. As the name suggests, it is a prism based on the heptapeton, which also makes it a convex segmentoexon.

Vertex coordinates

The vertices of a heptapetic prism of edge length 1 are given by:

• $\left(\pm {\frac {1}{2}},\,-{\frac {\sqrt {3}}{6}},\,-{\frac {\sqrt {6}}{12}},\,-{\frac {\sqrt {10}}{20}},\,-{\frac {\sqrt {15}}{30}},\,-{\frac {\sqrt {21}}{42}},\,\pm {\frac {1}{2}}\right),$
• $\left(0,\,{\frac {\sqrt {3}}{3}},\,-{\frac {\sqrt {6}}{12}},\,-{\frac {\sqrt {10}}{20}},\,-{\frac {\sqrt {15}}{30}},\,-{\frac {\sqrt {21}}{42}},\,\pm {\frac {1}{2}}\right),$
• $\left(0,\,0,\,{\frac {\sqrt {6}}{4}},\,-{\frac {\sqrt {10}}{20}},\,-{\frac {\sqrt {15}}{30}},\,-{\frac {\sqrt {21}}{42}},\,\pm {\frac {1}{2}}\right),$
• $\left(0,\,0,\,0,\,{\frac {\sqrt {10}}{5}},\,-{\frac {\sqrt {15}}{30}},\,-{\frac {\sqrt {21}}{42}},\,\pm {\frac {1}{2}}\right),$
• $\left(0,\,0,\,0,\,0,\,{\frac {\sqrt {15}}{6}},\,-{\frac {\sqrt {21}}{42}},\,\pm {\frac {1}{2}}\right),$
• $\left(0,\,0,\,0,\,0,\,0,\,{\frac {\sqrt {21}}{7}},\,\pm {\frac {1}{2}}\right).$

External links

• Klitzing, Richard. "hopip".