Pentagonal antiprismatic pyramid

Pentagonal antiprismatic pyramid
Rank	4
Type	Segmentotope
Notation
Bowers style acronym	Pappy
Coxeter diagram	oxo5oox&#x
Elements
Cells	10 tetrahedra, 2 pentagonal pyramids, 1 pentagonal antiprism
Faces	10+10+10 triangles, 2 pentagons
Edges	10+10+10
Vertices	1+10
Vertex figures	1 pentagonal antiprism, edge length 1
	10 isosceles trapezoidal pyramids, base edge lengths 1, 1, 1, (1+√5)/2, side edge length 1
Measures (edge length 1)
Circumradius	${\frac {1+{\sqrt {5}}}{2}}\approx 1.61083$
Hypervolume	${\frac {5+3{\sqrt {5}}}{96}}\approx 0.12196$
Dichoral angles	Tet–3–tet: $\arccos \left(-{\frac {1+3{\sqrt {5}}}{8}}\right)\approx 164.47751^{\circ }$
	Pap–5–peppy: 36º
	Tet–3–peppy: $\arccos \left({\frac {\sqrt {7+3{\sqrt {5}}}}{4}}\right)\approx 22.23876^{\circ }$
	Pap–3–tet: $\arccos \left({\frac {\sqrt {7+3{\sqrt {5}}}}{4}}\right)\approx 22.23876^{\circ }$
Heights	Peg atop gyro peppy: ${\frac {1}{2}}=0.5$
	point atop pap: ${\frac {{\sqrt {5}}-1}{4}}\approx 0.30902$
Central density	1
Related polytopes
Army	Pappy
Regiment	Pappy
Dual	Pentagonal antitegmatic pyramid
Conjugate	Pentagrammic retroprismatic pyramid
Abstract & topological properties
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	(I2(10)×A1)/2×I, order 20
Convex	Yes
Nature	Tame

The pentagonal antiprismatic pyramid, or pappy, is a CRF segmentochoron (designated K-4.80 on Richard Klitzing's list). It consists of 10 regular tetrahedra, 2 pentagonal pyramids, and 1 pentagonal antiprism. As the name suggests, it is a pyramid based on the pentagonal antiprism.

It can be obtained as the middle piece of an icosahedral pyramid, with the remainder formed by augmenting two pentagonal scalenes onto the pentagonal pyramids. It also occurs as a part of icosahedron atop dodecahedron, as the vertex pyramid of the icosahedral vertices.

Vertex coordinates

The vertices of a pentagonal antiprismatic pyramid of edge length 1 are given by:

• $\pm \left(\pm {\frac {1}{2}},\,-{\sqrt {\frac {5+2{\sqrt {5}}}{20}}},\,{\sqrt {\frac {5+{\sqrt {5}}}{40}}},\,0\right),$
• $\pm \left(\pm {\frac {1+{\sqrt {5}}}{4}},\,{\sqrt {f{\frac {5-{\sqrt {5}}}{40}}}},\,{\sqrt {\frac {5+{\sqrt {5}}}{40}}},\,0\right),$
• $\pm \left(0,\,{\sqrt {\frac {5+{\sqrt {5}}}{10}}},\,{\sqrt {\frac {5+{\sqrt {5}}}{40}}},\,0\right),$
• $\left(0,\,0,\,0,\,{\frac {{\sqrt {5}}-1}{4}}\right).$

External links

• Klitzing, Richard. "pappy".