# Pentagrammic double antiprismoid

Pentagrammic double antiprismoid
Rank4
TypeUniform
Notation
Coxeter diagramxovo5/3oxov2ovxo5/3voox&#zx
Elements
Cells100+200 tetrahedra, 20 pentagrammic retroprisms
Faces100+200+400 triangles, 20 pentagrams
Edges100+200+200
Vertices100
Vertex figureRetrosphenocorona, edge lengths 1 and (5–1)/2
Measures (edge length 1)
Circumradius${\displaystyle {\frac {{\sqrt {5}}-1}{2}}\approx 0.61803}$
Hypervolume${\displaystyle {\frac {5(26{\sqrt {5}}-55)}{4}}\approx 3.92221}$
Dichoral anglesStarp–5/2–starp: 72°
Tet–3–tet: ${\displaystyle \arccos \left({\frac {3{\sqrt {5}}-1}{8}}\right)\approx 44.47751^{\circ }}$
Starp–3–tet: ${\displaystyle \arccos \left({\frac {\sqrt {10}}{4}}\right)\approx 37.76124^{\circ }}$
Central density91
Number of external pieces30640
Level of complexity3542
Related polytopes
ArmyGap
ConjugateGrand antiprism
Abstract & topological properties
Euler characteristic0
OrientableYes
Properties
SymmetryI2(10)+≀S2×2, order 256
ConvexNo
NatureTame

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.

## Vertex coordinates

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

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