Pentagonal-square prismantiprismoid

Pentagonal-square prismantiprismoid
(OFF file)
Rank	4
Type	Isogonal
Notation
Bowers style acronym	Pispap
Coxeter diagram	x4s2s10o ()
Elements
Cells	20 wedges, 10 rectangular trapezoprisms, 4 pentagonal prisms, 4 pentagonal antiprisms
Faces	40 isosceles triangles, 40 isosceles trapezoids, 10+20 rectangles, 8 pentagons
Edges	20+20+40+40
Vertices	40
Vertex figure	Monoaugmented isosceles trapezoidal pyramid
Measures (as derived from unit-edge octagonal-decagonal duoprism)
Edge lengths	Short edges of rectangles (20): 1
	Side edges (40): ${\sqrt {2}}\approx 1.41421$
	Edges of pentagons (40): ${\sqrt {\frac {5+{\sqrt {2}}}{2}}}\approx 1.90211$
	Long edges of rectangles (20): $1+{\sqrt {2}}\approx 2.41421$
Circumradius	${\sqrt {\frac {5+{\sqrt {2}}+{\sqrt {5}}}{2}}}\approx 2.07970$
Central density	1
Related polytopes
Army	Pispap
Regiment	Pispap
Dual	Pentagonal-square tegmantitegmoid
Abstract & topological properties
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	(B2×I2(10))/2, order 80
Convex	Yes
Nature	Tame

The pentagonal-square prismantiprismoid or pispap, also known as the edge-snub pentagonal-square duoprism or 5-4 prismantiprismoid, is a convex isogonal polychoron that consists of 4 pentagonal antiprisms, 4 pentagonal prisms, 10 rectangular trapezoprisms, and 20 wedges. 1 pentagonal antiprism, 1 pentagonal prism, 2 rectangular trapezoprisms, and 3 wedges join at each vertex. It can be obtained through the process of alternating one class of edges of the octagonal-decagonal duoprism so that the octagons become rectangles. However, it cannot be made uniform, as it generally has 4 edge lengths, which can be minimized to no fewer than 2 different sizes.

Using the ratio method, the lowest possible ratio between the longest and shortest edges is 1:${\frac {10+{\sqrt {5}}+{\sqrt {295+39{\sqrt {5}}}}}{19}}$ ≈ 1:1.67296.

Vertex coordinates

The vertices of a pentagonal-square prismantiprismoid based on an octagonal-decagonal duoprism of edge length 1, centered at the origin, rae given by:

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