The Augmented Dodecahedron
The augmented dodecahedron (J58) is the 58th Johnson solid. It has 21 vertices, 35 edges, and 16 faces (11 pentagons and 5 triangles). It can be constructed augmenting a dodecahedron with a pentagonal pyramid.
Adding a second pentagonal pyramid on the opposite side produces the parabiaugmented dodecahedron (J59). Adding the second pyramid to a face one edge away from the first pyramid produces instead the metabiaugmented dodecahedron (J60).
Here are some views of the augmented dodecahedron from various angles:
The Cartesian coordinates of the augmented dodecahedron, centered on the origin and with edge length 2, are:
- (0, ±φ2, ±1)
- (±1, 0, ±φ2)
- (±φ2, ±1, 0)
- (±φ, ±φ, ±φ)
- (0, (7+φ)/5, (8φ+1)/5)
where φ=(1+√5)/2 is the Golden Ratio.