Grand antiprismatic prism

Grand antiprismatic prism
File:Grand antiprismatic prism.png
Rank	5
Type	Uniform
Notation
Bowers style acronym	Gappip
Coxeter diagram	x xofo5oxof2ofxo5foox&#zx
Elements
Tera	100+200 tetrahedral prisms, 20 pentagonal antiprismatic prisms, 2 grand antiprisms
Cells	200+400 tetrahedra, 100+200+400 triangular prisms, 20 pentagonal prisms, 40 pentagonal antiprisms
Faces	200+400+800 triangles, 100+200+200 squares, 40 pentagons
Edges	100+200+400+400
Vertices	200
Vertex figure	Sphenocorona pyramid, edge lengths 1, (1+√5)/2 (base), √2 (legs)
Measures (edge length 1)
Circumradius	${\frac {\sqrt {7+2{\sqrt {5}}}}{2}}\approx 1.69353$
Hypervolume	${\frac {5\left(55+26{\sqrt {5}}\right)}{24}}\approx 23.57037$
Diteral angles	Tepe–trip–tepe: $\arccos \left(-{\frac {1+3{\sqrt {5}}}{8}}\right)\approx 164.47751^{\circ }$
	Pappip–pip–pappip: 144°
	Pappip–trip–tepe: $\arccos \left(-{\frac {\sqrt {10}}{4}}\right)\approx 142.23876^{\circ }$
	Gap–pap–pappip: 90°
	Gap–tet–tepe: 90°
Height	1
Central density	1
Number of external pieces	322
Level of complexity	110
Related polytopes
Army	Gappip
Regiment	Gappip
Dual	Pentagonal double antitegmoidal tegum
Conjugate	Pentagrammatic double antiprismoidal prism
Abstract & topological properties
Euler characteristic	2
Orientable	Yes
Properties
Symmetry	I2(10)≀S2+×A1, order 800
Convex	Yes
Nature	Tame

The grand antiprismatic prism or gappip is a prismatic uniform polyteron that consists of 2 grand antiprisms, 20 pentagonal antiprismatic prisms, and 100+200 tetrahedral prisms. 1 grand antiprism, 2 pentagonal antiprismatic prisms, and 12 tetrahedral prisms join at each vertex. As the name suggests, it can be obtained as a prism based on the grand antiprism, which also makes it a convex segmentoteron.

Vertex coordinates

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

• $\pm \left(\pm {\frac {3+{\sqrt {5}}}{4}},\,0,\,{\frac {1+{\sqrt {5}}}{4}},\,-{\frac {1}{2}},\,\pm {\frac {1}{2}}\right),$
• $\pm \left(\pm {\frac {1}{2}},\,0,\,{\frac {3+{\sqrt {5}}}{4}},\,-{\frac {1+{\sqrt {5}}}{4}},\,\pm {\frac {1}{2}}\right),$
• $\pm \left(0,\,\pm {\frac {3+{\sqrt {5}}}{4}},\,{\frac {1}{2}},\,{\frac {1+{\sqrt {5}}}{4}},\,\pm {\frac {1}{2}}\right),$
• $\pm \left(0,\,\pm {\frac {1}{2}},\,{\frac {1+{\sqrt {5}}}{4}},\,{\frac {3+{\sqrt {5}}}{4}},\,\pm {\frac {1}{2}}\right),$
• $\left(0,\,0,\,\pm {\frac {1+{\sqrt {5}}}{2}},\,0,\,\pm {\frac {1}{2}}\right),$
• $\left(0,\,0,\,0,\,\pm {\frac {1+{\sqrt {5}}}{2}},\,\pm {\frac {1}{2}}\right),$
• $\left(\pm {\frac {1+{\sqrt {5}}}{4}},\,\pm {\frac {1+{\sqrt {5}}}{4}},\,\pm {\frac {1+{\sqrt {5}}}{4}},\,\pm {\frac {1+{\sqrt {5}}}{4}},\,\pm {\frac {1}{2}}\right),$
• $\left(\pm {\frac {1+{\sqrt {5}}}{4}},\,0,\,\pm {\frac {1}{2}},\,\pm {\frac {3+{\sqrt {5}}}{4}},\,\pm {\frac {1}{2}}\right),$
• $\left(0,\,\pm {\frac {1+{\sqrt {5}}}{4}},\,\pm {\frac {3+{\sqrt {5}}}{4}},\,\pm {\frac {1}{2}},\,\pm {\frac {1}{2}}\right),$
• $\left(\pm {\frac {1+{\sqrt {5}}}{4}},\,\pm {\frac {1}{2}},\,\pm {\frac {3+{\sqrt {5}}}{4}},\,0,\,\pm {\frac {1}{2}}\right),$
• $\left(\pm {\frac {3+{\sqrt {5}}}{4}},\,\pm {\frac {1+{\sqrt {5}}}{4}},\,\pm {\frac {1}{2}},\,0,\,\pm {\frac {1}{2}}\right),$
• $\left(\pm {\frac {1}{2}},\,\pm {\frac {3+{\sqrt {5}}}{4}},\,\pm {\frac {1+{\sqrt {5}}}{4}},\,0,\,\pm {\frac {1}{2}}\right),$
• $\left(\pm {\frac {3+{\sqrt {5}}}{4}},\,\pm {\frac {1}{2}},\,0,\,\pm {\frac {1+{\sqrt {5}}}{4}},\,\pm {\frac {1}{2}}\right),$
• $\left(\pm {\frac {1}{2}},\,\pm {\frac {1+{\sqrt {5}}}{4}},\,0,\,\pm {\frac {3+{\sqrt {5}}}{4}},\,\pm {\frac {1}{2}}\right),$
• $\left(\pm {\frac {1+{\sqrt {5}}}{4}},\,\pm {\frac {3+{\sqrt {5}}}{4}},\,0,\,\pm {\frac {1}{2}},\,\pm {\frac {1}{2}}\right).$

External links

• Klitzing, Richard. "Gappip".