Pentagrammic double antiprismoid

Pentagrammic double antiprismoid
	(OFF file)
Rank	4
Type	Uniform
Notation
Bowers style acronym	Padiap
Coxeter diagram	xovo5/3oxov2ovxo5/3voox&#zx
Elements
Cells	100+200 tetrahedra, 20 pentagrammic retroprisms
Faces	100+200+400 triangles, 20 pentagrams
Edges	100+200+200
Vertices	100
Vertex figure	Retrosphenocorona, edge lengths 1 and (√5–1)/2
Measures (edge length 1)
Circumradius
Hypervolume
Dichoral angles	Starp–5/2–starp: 72°
	Tet–3–tet:
	Starp–3–tet:
Central density	91
Number of external pieces	30640
Level of complexity	3542
Related polytopes
Army	Gap
Regiment	Padiap
Conjugate	Grand antiprism
Abstract & topological properties
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	I2(10)+≀S2×2, order 256
Convex	No
Nature	Tame

The pentagrammatic double antiprismoid, or padiap, is a nonconvex uniform polychoron that consists of 300 tetrahedra and 20 pentagrammic retroprisms. 12 tetrahedra and 2 pentagrammic retroprisms join at each vertex.

This polychoron can be formed as a subsymmetrical faceting of the grand hexacosichoron in a similar way as its conjugate, the convex grand antiprism, can be form from the hexacosichoron. The pentagrammic retroprisms are facetings of the great icosahedra which form the grand hexacosichoron's vertex figures.

Cross-sections[edit | edit source]

Vertex coordinates[edit | edit source]

The vertices of a pentagrammatic double antiprismoid of edge length 1 are given by:

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