Gyroelongated pentagonal cupola |
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Rank | 3 |
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Type | CRF |
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Notation |
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Bowers style acronym | Gyepcu |
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Coxeter diagram | oxo10sox&#xt |
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Elements |
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Faces | 5+5+5+10 triangles, 5 squares, 1 pentagon, 1 decagon |
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Edges | 5+5+5+10+10+10+10 |
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Vertices | 5+5+5+10 |
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Vertex figures | 5 isosceles trapezoids, edge lengths 1, √2, (1+√5)/2, √2 |
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| 10 irregular pentagons, edge lengths 1, 1, 1, 1, √2 |
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| 10 isosceles trapezoids, edge lengths 1, 1, 1, √(5+√5)/2 |
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Measures (edge length 1) |
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Volume | |
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Dihedral angles | 3–3 antiprismatic: |
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| 3–4 cupolaic: |
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| 4–5: |
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| 3–3 join: |
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| 3–4 join: |
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| 3–10: |
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Central density | 1 |
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Number of external pieces | 32 |
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Level of complexity | 22 |
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Related polytopes |
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Army | Gyepcu |
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Regiment | Gyepcu |
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Dual | Pentadeltadecapentagonal hemitrapezohedron |
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Conjugate | Gyroelongated retrograde pentagrammic cupola |
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Abstract & topological properties |
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Flag count | 220 |
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Euler characteristic | 2 |
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Surface | Sphere |
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Orientable | Yes |
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Genus | 0 |
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Properties |
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Symmetry | H2×I, order 10 |
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Convex | Yes |
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Nature | Tame |
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The gyroelongated pentagonal cupola is one of the 92 Johnson solids (J24). It consists of 5+5+5+10 triangles, 5 squares, 1 pentagon, and 1 decagon. It can be constructed by attaching a decagonal antiprism to the decagonal base of the pentagonal cupola.
If a second cupola is attached to the other decagonal base of the antiprism, the result is the gyroelongated pentagonal bicupola.
A gyroelongated pentagonal cupola of edge length 1 has the following vertices:
where H = is the distance between the decagonal antiprism's center and the center of one of its bases.