Decagrammic antiprism

Decagrammic antiprism
(3D model) (OFF file)
Rank	3
Type	Uniform
Notation
Bowers style acronym	Stidap
Coxeter diagram	s2s20/3o
Elements
Faces	20 triangles, 2 decagrams
Edges	20+20
Vertices	20
Vertex figure	Isosceles trapezoid, edge lengths 1, 1, 1, √(5–√5)/2
Measures (edge length 1)
Circumradius	${\sqrt {\frac {8-2{\sqrt {5}}+{\sqrt {50-22{\sqrt {5}}}}}{8}}}\approx 0.74380$
Volume	${\frac {5{\sqrt {-2+2{\sqrt {5}}+2{\sqrt {650-290{\sqrt {5}}}}}}}{6}}\approx 1.85486$
Dihedral angles	3–3: $\arccos \left({\frac {1-{\sqrt {10-2{\sqrt {5}}}}}{3}}\right)\approx 116.76809^{\circ }$
	10/3–3: $\arccos \left(-{\sqrt {\frac {11-4{\sqrt {5}}-2{\sqrt {50-22{\sqrt {5}}}}}{3}}}\right)\approx 107.10805^{\circ }$
Height	${\sqrt {\frac {-4+2{\sqrt {5}}+{\sqrt {50-22{\sqrt {5}}}}}{2}}}\approx 0.82771$
Central density	3
Number of external pieces	102
Level of complexity	24
Related polytopes
Army	Non-uniform Dap, edge lengths ${\frac {3-{\sqrt {5}}}{2}}$ (base), ${\sqrt {\frac {2-{\sqrt {130-58{\sqrt {5}}}}}{2}}}$ (sides)
Regiment	Stidap
Dual	Decagrammic antitegum
Conjugate	Decagonal antiprism
Convex core	Decagonal antibifrustum
Abstract & topological properties
Flag count	160
Euler characteristic	2
Orientable	Yes
Genus	0
Properties
Symmetry	(I2(20)×A1)/2, order 40
Convex	No
Nature	Tame

The decagrammic antiprism, or stidap, is a prismatic uniform polyhedron. It consists of 20 triangles and 2 decagrams. Each vertex joins one decagram and three triangles. As the name suggests, it is an antiprism based on a decagram.

Vertex coordinates

A decagrammic antiprism of edge length 1 has vertex coordinates given by:

• $\left(\pm {\frac {1}{2}},\,\pm {\frac {\sqrt {5-2{\sqrt {5}}}}{2}},\,H\right),$
• $\left(\pm {\frac {3-{\sqrt {5}}}{4}},\,\pm {\sqrt {\frac {5-{\sqrt {5}}}{8}}},\,H\right),$
• $\left(\pm {\frac {{\sqrt {5}}-1}{2}},\,0,\,H\right),$
• $\left(\pm {\frac {\sqrt {5-2{\sqrt {5}}}}{2}},\,\pm {\frac {1}{2}},\,-H\right),$
• $\left(\pm {\sqrt {\frac {5-{\sqrt {5}}}{8}}},\,\pm {\frac {3-{\sqrt {5}}}{4}},\,-H\right),$
• $\left(0,\,\pm {\frac {{\sqrt {5}}-1}{2}},\,-H\right),$

where H = ${\sqrt {\frac {-4+2{\sqrt {5}}+{\sqrt {50-22{\sqrt {5}}}}}{8}}}$ is the distance between the antiprism's center and the center of one of its bases.

External links

• Wikipedia contributors. "Decagrammic antiprism".