The Gyroelongated Pentagonal Cupolarotunda
The gyroelongated pentagonal cupolarotunda is the 47th Johnson solid (J47). It has 35 vertices, 80 edges, and 47 faces (35 equilateral triangles, 5 squares, and 7 pentagons).
The gyroelongated pentagonal cupolarotunda can be constructed by attaching a pentagonal rotunda and a pentagonal cupola to a decagonal antiprism, or equivalently, adding a pentagonal rotunda to a gyroelongated pentagonal cupola (J24), or adding a pentagonal cupola to a gyroelongated pentagonal rotunda (J25).
J47 is one of the few Johnson solids that are chiral; it is distinct from its own mirror image:
Projections
Here are some views of the gyroelongated pentagonal cupolarotunda from various angles:
Projection | Description |
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Top view. |
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Front view. |
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Side view. |
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9° side view. |
Coordinates
The Cartesian coordinates of the gyroelongated pentagonal cupolarotunda with edge length 2 are:
- (0, −√((10+2√5)/5), H+√((20+8√5)/5))
- (±1, √((5+2√5)/5), H+√((20+8√5)/5))
- (±φ, −√((5−√5)/10), H+√((20+8√5)/5))
- (0, √((20+8√5)/5), H+√((10+2√5)/5))
- (±φ, −√((25+11√5)/10), H+√((10+2√5)/5))
- (±φ2, √((5+√5)/10), H+√((10+2√5)/5))
- (±1, ±√(3+4φ), H)
- (±φ2, ±√(2+φ), H)
- (±2φ, 0, H)
- (0, ±2φ, −H)
- (±√(3+4φ), ±1, −H)
- (±√(2+φ), ±φ2, −H)
- (√((10+2√5)/5), 0, −(H + 2√((3−φ)/5)))
- (√((5−√5)/10), ±φ, −(H + 2√((3−φ)/5)))
- (−√((5+2√5)/5), ±1, −(H + 2√((3−φ)/5)))
where H = √(√(11φ+7)−2φ−1), approximately 0.862397, is half the height of a decagonal antiprism of edge length 2, and φ = (1+√5)/2 is the Golden Ratio.