The Pentagonal Orthobicupola
The pentagonal orthobicupola is the 30th Johnson solid (J30). It has 20 vertices, 40 edges, and 22 faces (10 equilateral triangles, 10 squares, and 2 pentagons).
The pentagonal orthobicupola can be constructed by joining two pentagonal cupolae to each other at their decagonal face, such that the square faces of each cupola touch the square faces of the other at the edges. The ortho in the name refers to how the orientation of the two cupolae are aligned with each other. Joining the two cupolae in gyro orientation produces the pentagonal gyrobicupola (J31) instead.
Here are some views of the pentagonal orthobicupola from various angles:
The Cartesian coordinates of the pentagonal orthobicupola with edge length 2 are:
- (0, √((10+2√5)/5), ±2√((3−φ)/5))
- (±φ, √((5−√5)/10), ±2√((3−φ)/5))
- (±1, −√((5+2√5)/5), ±2√((3−φ)/5))
- (±2φ, 0, 0)
- (±1, ±√(3+4φ), 0)
- (±φ2, ±√(2+φ), 0)
where φ = (1+√5)/2 is the Golden Ratio, approximately 1.61803.