Elongated pentagonal orthocupolarotunda

Elongated pentagonal orthocupolarotunda
(3D model) (OFF file)
Rank	3
Type	CRF
Notation
Bowers style acronym	Epocuro
Coxeter diagram	xoxxx5ofxxo&#xt
Elements
Faces	5+5+5 triangles, 5+5+5 squares, 1+1+5 pentagons
Edges	5+5+5+5+5+5+10+10+10+10
Vertices	5+5+5+10+10
Vertex figures	5 isosceles trapezoids, edge lengths 1, √2, (1+√5}/2, √2
	10 rectangles, edge lengths 1 and (1+√5)/2
	10 trapezoids, edge lengths 1, √2, √2, √2
	10 irregular tetragons, edge lengths 1, (1+√5)/2, √2, √2
Measures (edge length 1)
Volume	$5{\frac {11+5{\sqrt {5}}+6{\sqrt {5+2{\sqrt {5}}}}}{12}}\approx 16.93602$
Dihedral angles	3–4 rotundaic join: $\arccos \left(-{\sqrt {\frac {10+2{\sqrt {5}}}{15}}}\right)\approx 169.18768^{\circ }$
	3–4 cupolaic: $\arccos \left(-{\frac {{\sqrt {3}}+{\sqrt {15}}}{6}}\right)\approx 159.09484^{\circ }$
	4–5 join: $\arccos \left(-{\frac {2{\sqrt {5}}}{5}}\right)\approx 153.43495^{\circ }$
	4–5: $\arccos \left(-{\sqrt {\frac {5+{\sqrt {5}}}{10}}}\right)\approx 148.28253^{\circ }$
	4–4 prismatic: 144°
	3–5: $\arccos \left(-{\sqrt {\frac {5+2{\sqrt {5}}}{15}}}\right)\approx 142.62263^{\circ }$
	3–4 join: $\arccos \left(-{\sqrt {\frac {10-2{\sqrt {5}}}{15}}}\right)\approx 127.37737^{\circ }$
	4–4 join: $\arccos \left(-{\sqrt {\frac {5-{\sqrt {5}}}{10}}}\right)\approx 121.71747^{\circ }$
Central density	1
Number of external pieces	37
Level of complexity	28
Related polytopes
Army	Epocuro
Regiment	Epocuro
Dual	Pentadeltodecadeltorthodecatetragopentarhombirhombic triacontapentahedron
Conjugate	Rlongated retrograde pentagrammic orthocupolarotunda
Abstract & topological properties
Flag count	280
Euler characteristic	2
Surface	Sphere
Orientable	Yes
Genus	0
Properties
Symmetry	H2×I, order 10
Convex	Yes
Nature	Tame

The elongated pentagonal orthocupolarotunda is one of the 92 Johnson solids (J₄₀). It consists of 5+5+5 triangles, 5+5+5 squares, and 1+1+5 pentagons. It can be constructed by inserting a decagonal prism between the two halves of the pentagonal orthocupolarotunda.

Vertex coordinates

An elongated pentagonal orthocupolarotunda of edge length 1 has vertices given by the following coordinates:

• $\left(\pm {\frac {1}{2}},\,\pm {\frac {\sqrt {5+2{\sqrt {5}}}}{2}},\,\pm {\frac {1}{2}}\right),$
• $\left(\pm {\frac {3+{\sqrt {5}}}{4}},\,\pm {\sqrt {\frac {5+{\sqrt {5}}}{8}}},\,\pm {\frac {1}{2}}\right),$
• $\left(\pm {\frac {1+{\sqrt {5}}}{2}},\,0,\,\pm {\frac {1}{2}}\right),$
• $\left(0,\,{\sqrt {\frac {5+{\sqrt {5}}}{10}}},\,{\frac {1+2{\sqrt {\frac {5+2{\sqrt {5}}}{5}}}}{2}}\right),$
• $\left(\pm {\frac {1}{2}},\,-{\sqrt {\frac {5+2{\sqrt {5}}}{20}}},\,{\frac {1+2{\sqrt {\frac {5+2{\sqrt {5}}}{5}}}}{2}}\right),$
• $\left(\pm {\frac {1+{\sqrt {5}}}{4}},\,{\sqrt {\frac {5-{\sqrt {5}}}{40}}},\,{\frac {1+2{\sqrt {\frac {5+2{\sqrt {5}}}{5}}}}{2}}\right),$
• $\left(0,\,-{\sqrt {\frac {5+2{\sqrt {5}}}{5}}},\,{\frac {1+2{\sqrt {\frac {5+{\sqrt {5}}}{10}}}}{2}}\right),$
• $\left(\pm {\frac {1+{\sqrt {5}}}{4}},\,{\sqrt {\frac {25+11{\sqrt {5}}}{40}}},\,{\frac {1+2{\sqrt {\frac {5+{\sqrt {5}}}{10}}}}{2}}\right),$
• $\left(\pm {\frac {3+{\sqrt {5}}}{4}},\,-{\sqrt {\frac {5+{\sqrt {5}}}{40}}},\,{\frac {1+2{\sqrt {\frac {5+{\sqrt {5}}}{10}}}}{2}}\right),$
• $\left(\pm {\frac {1}{2}},\,-{\sqrt {\frac {5+2{\sqrt {5}}}{20}}},\,-{\frac {1+2{\sqrt {\frac {5-{\sqrt {5}}}{10}}}}{2}}\right),$
• $\left(\pm {\frac {1+{\sqrt {5}}}{4}},\,{\sqrt {\frac {5+{\sqrt {5}}}{40}}},\,-{\frac {1+2{\sqrt {\frac {5-{\sqrt {5}}}{10}}}}{2}}\right),$
• $\left(0,\,{\sqrt {\frac {5+{\sqrt {5}}}{10}}},\,-{\frac {1+2{\sqrt {\frac {5-{\sqrt {5}}}{10}}}}{2}}\right),$

External links

• Klitzing, Richard. "epocuro".
• Quickfur. "The Elongated Pentagonal Orthocupolarotunda".

• Wikipedia contributors. "Elongated pentagonal orthocupolarotunda".
• McCooey, David. "Elongated Pentagonal Orthocupolarotunda"