Hexagonal antiprismatic prism

Hexagonal antiprismatic prism
(OFF file)
Rank	4
Type	Uniform
Notation
Bowers style acronym	Happip
Coxeter diagram	x2s2s12o ()
Elements
Cells	12 triangular prisms, 2 hexagonal prisms, 2 hexagonal antiprisms
Faces	24 triangles, 12+12 squares, 4 hexagons
Edges	12+24+24
Vertices	24
Vertex figure	Isosceles trapezoidal pyramid, edge lengths 1, 1, 1, √3 (base), √2 (legs)
Measures (edge length 1)
Circumradius	${\frac {\sqrt {4+{\sqrt {3}}}}{2}}\approx 1.19708$
Hypervolume	${\sqrt {2+2{\sqrt {3}}}}\approx 2.33754$
Dichoral angles	Trip–4–trip: $\arccos \left({\frac {1-2{\sqrt {3}}}{3}}\right)\approx 145.22189^{\circ }$
	Trip–4–hip: $\arccos \left({\frac {3-2{\sqrt {3}}}{3}}\right)\approx 98.89943^{\circ }$
	Hap–6–hip: 90°
	Hap–3–trip: 90°
Heights	Hap atop hap: 1
	Hip atop gyro hip: ${\sqrt {{\sqrt {3}}-1}}\approx 0.85560$
Central density	1
Number of external pieces	16
Level of complexity	16
Related polytopes
Army	Happip
Regiment	Happip
Dual	Hexagonal antitegmatic tegum
Conjugate	Hexagonal antiprismatic prism
Abstract & topological properties
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	(I2(12)×A1)+×A1, order 48
Convex	Yes
Nature	Tame

The hexagonal antiprismatic prism or happip is a prismatic uniform polychoron that consists of 2 hexagonal antiprisms, 2 hexagonal prisms, and 12 triangular prisms. Each vertex joins 1 hexagonal antiprism, 1 hexagonal prism, and 3 triangular prisms. As the name suggests, it is a prism based on a hexagonal antiprism. It is also a CRF segmentochoron designated K-4.53 on Richard Klitzing's list.

Vertex coordinates

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

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

Representations

A hexagonal antiprismatic prism has the following Coxeter diagrams:

• x2s2s12o (full symmetry)
• x2s2s6s ()
• xx xo6ox&#x (hexagonal prism atop gyrated hexagonal prism)

External links

• Bowers, Jonathan. "Category B: Antiduoprisms".

• Klitzing, Richard. "happip".
• Wikipedia contributors. "Uniform antiprismatic prism".