Retrograde pentagonal cuploid

Retrograde pentagonal cuploid
(3D model) (OFF file)
Rank	3
Type	Segmentotope
Notation
Bowers style acronym	Rapescu
Elements
Faces	5 triangles, 5 squares, 1 pentagon
Edges	5+5+10
Vertices	5+5
Vertex figures	5 isosceles trapezoids, edge lengths 1, √2, (1+√5)/2, √2
	5 butterflies, edge lengths 1, √2, 1, √2
Measures (edge length 1)
Circumradius	${\frac {\sqrt {3}}{2}}\approx 0.86603$
Dihedral angles	5–4: $\arccos \left({\sqrt {\frac {5+{\sqrt {5}}}{10}}}\right)\approx 31.71747^{\circ }$
	3–4: $\arccos \left(-{\frac {{\sqrt {3}}+{\sqrt {15}}}{6}}\right)\approx 20.90516^{\circ }$
Height	${\sqrt {\frac {5-{\sqrt {5}}}{10}}}\approx 0.52573$
Related polytopes
Army	Pentagonal frustum
Conjugate	Pentagrammic cuploid
Abstract & topological properties
Euler characteristic	1
Orientable	No
Genus	1
Properties
Symmetry	H2×I, order 10
Convex	No
Nature	Tame

The retrograde pentagonal cuploid, also called the retrograde pentagonal semicupola or rapescu, is an orbiform polyhedron. It consists of 5 triangles, 5 squares, and 1 pentagon. It is a cuploid based on the pentagon seen as {5/4}, with a pseudo {10/4} base.

Vertex coordinates

A retrograde pentagonal cuploid of edge length 1 has vertices given by the following coordinates:

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

Related polyhedra

The retrograde pentagonal cuploid can be edge-inscribed into the small ditrigonary icosidodecahedron; it uses triangles and pentagons of the great ditrigonary icosidodecahedron as well as squares of the rhombihedron, the inscribed compound of 5 cubes.

External links

• Klitzing, Richard. "rapescu".
• Wikipedia contributors. "Crossed pentagonal cuploid".
• Webb, Robert. "Crossed Pentagonal Cuploid".