Pentagonal gyrocupolarotunda Rank 3 Type CRF Notation Bowers style acronym Pegycuro Coxeter diagram xoxo5ofxx&#xt Elements Faces 5+5+5 triangles , 5 squares , 1+1+5 pentagons Edges 5+5+5+5+10+10+10 Vertices 5+5+5+10 Vertex figures 5 isosceles trapezoids , edge lengths 1, √2 , (1+√5 }/2, √2 10 rectangles , edge lengths 1 and (1+√5 )/2 10 irregular tetragons , edge lengths 1, 1, √2 , (1+√5 )/2 Measures (edge length 1) Volume
5
11
+
5
5
12
≈
9.24181
{\displaystyle 5{\frac {11+5{\sqrt {5}}}{12}}\approx 9.24181}
Dihedral angles 3–4:
arccos
(
−
3
+
15
6
)
≈
159.09484
∘
{\displaystyle \arccos \left(-{\frac {{\sqrt {3}}+{\sqrt {15}}}{6}}\right)\approx 159.09484^{\circ }}
4–5 cupolaic:
arccos
(
−
5
+
5
10
)
≈
148.28253
∘
{\displaystyle \arccos \left(-{\sqrt {\frac {5+{\sqrt {5}}}{10}}}\right)\approx 148.28253^{\circ }}
3–5:
arccos
(
−
5
+
2
5
15
)
≈
142.62263
∘
{\displaystyle \arccos \left(-{\sqrt {\frac {5+2{\sqrt {5}}}{15}}}\right)\approx 142.62263^{\circ }}
3–3:
arccos
(
−
5
5
)
≈
116.56505
∘
{\displaystyle \arccos \left(-{\frac {\sqrt {5}}{5}}\right)\approx 116.56505^{\circ }}
4–5 join:
arccos
(
−
25
−
11
5
50
)
≈
95.15242
∘
{\displaystyle \arccos \left(-{\sqrt {\frac {25-11{\sqrt {5}}}{50}}}\right)\approx 95.15242^{\circ }}
Central density 1 Number of external pieces 27 Level of complexity 20 Related polytopes Army Pegycuro Regiment Pegycuro Dual Pentadeltodecatrapezopentadelto-pentarhombic icosipentahedron Conjugate Retrograde pentagrammic gyrocupolarotunda Abstract & topological properties Flag count200 Euler characteristic 2 Surface Sphere Orientable Yes Genus 0 Properties Symmetry H2 ×I , order 10Convex Yes Nature Tame
The pentagonal gyrocupolarotunda is one of the 92 Johnson solids (J33 ). It consists of 5+5+5 triangles , 5 squares , and 1+1+5 pentagons . It can be constructed by attaching a pentagonal cupola and a pentagonal rotunda at their decagonal bases, such that the two pentagonal bases are rotated 36° with respect to each other.
If the cupola and rotunda are joined such that the bases are in the same orientation, the result is the pentagonal orthocupolarotunda .
A pentagonal gyrocupolarotunda of edge length 1 has vertices given by the following coordinates:
(
0
,
5
+
5
10
,
5
+
2
5
5
)
{\displaystyle \left(0,\,{\sqrt {\frac {5+{\sqrt {5}}}{10}}},\,{\sqrt {\frac {5+2{\sqrt {5}}}{5}}}\right)}
(
±
1
2
,
−
5
+
2
5
20
,
5
+
2
5
5
)
{\displaystyle \left(\pm {\frac {1}{2}},\,-{\sqrt {\frac {5+2{\sqrt {5}}}{20}}},\,{\sqrt {\frac {5+2{\sqrt {5}}}{5}}}\right)}
(
±
1
+
5
4
,
5
−
5
40
,
5
+
2
5
5
)
{\displaystyle \left(\pm {\frac {1+{\sqrt {5}}}{4}},\,{\sqrt {\frac {5-{\sqrt {5}}}{40}}},\,{\sqrt {\frac {5+2{\sqrt {5}}}{5}}}\right)}
(
0
,
−
5
+
2
5
5
,
5
+
5
10
)
{\displaystyle \left(0,\,-{\sqrt {\frac {5+2{\sqrt {5}}}{5}}},\,{\sqrt {\frac {5+{\sqrt {5}}}{10}}}\right)}
(
±
1
+
5
4
,
25
+
11
5
40
,
5
+
5
10
)
{\displaystyle \left(\pm {\frac {1+{\sqrt {5}}}{4}},\,{\sqrt {\frac {25+11{\sqrt {5}}}{40}}},\,{\sqrt {\frac {5+{\sqrt {5}}}{10}}}\right)}
(
±
3
+
5
4
,
−
5
+
5
40
,
5
+
5
10
)
{\displaystyle \left(\pm {\frac {3+{\sqrt {5}}}{4}},\,-{\sqrt {\frac {5+{\sqrt {5}}}{40}}},\,{\sqrt {\frac {5+{\sqrt {5}}}{10}}}\right)}
(
±
1
2
,
±
5
+
2
5
2
,
0
)
{\displaystyle \left(\pm {\frac {1}{2}},\,\pm {\frac {\sqrt {5+2{\sqrt {5}}}}{2}},\,0\right)}
(
±
3
+
5
4
,
±
5
+
5
8
,
0
)
{\displaystyle \left(\pm {\frac {3+{\sqrt {5}}}{4}},\,\pm {\sqrt {\frac {5+{\sqrt {5}}}{8}}},\,0\right)}
(
±
1
+
5
2
,
0
,
0
)
{\displaystyle \left(\pm {\frac {1+{\sqrt {5}}}{2}},\,0,\,0\right)}
(
±
1
2
,
5
+
2
5
20
,
−
5
−
5
10
)
{\displaystyle \left(\pm {\frac {1}{2}},\,{\sqrt {\frac {5+2{\sqrt {5}}}{20}}},\,-{\sqrt {\frac {5-{\sqrt {5}}}{10}}}\right)}
(
±
1
+
5
4
,
−
5
−
5
40
,
−
5
−
5
10
)
{\displaystyle \left(\pm {\frac {1+{\sqrt {5}}}{4}},\,-{\sqrt {\frac {5-{\sqrt {5}}}{40}}},\,-{\sqrt {\frac {5-{\sqrt {5}}}{10}}}\right)}
(
0
,
−
5
+
5
10
,
−
5
−
5
10
)
{\displaystyle \left(0,\,-{\sqrt {\frac {5+{\sqrt {5}}}{10}}},\,-{\sqrt {\frac {5-{\sqrt {5}}}{10}}}\right)}
A decagonal prism can be inserted between the two halves of the pentagonal gyrocupolarotunda to produce the elongated pentagonal gyrocupolarotunda .
