The Snub Square Antiprism
The snub square antiprism is the 85th Johnson solid (J85). Its surface consists of 24 equilateral triangles and 2 squares.
The snub square antiprism is one of the special Johnson solids at the end of Norman Johnson's list that are not directly derived from the uniform polyhedra by cut-and-paste operations. It can be indirectly constructed from a square antiprism by separating it into two halves along a zigzag cut between its triangular faces, suitably deforming the two halves, and joining them together with a belt of 16 triangles.
An analogous process can be applied to the tetrahedron, treated as a digonal antiprism, to produce the snub disphenoid (J84). Applied to the octahedron, treated as a triangular antiprism, this process produces the icosahedron. It can also be applied to the pentagonal antiprism to produce a snub pentagonal antiprism; however, the result is no longer convex, and so is not among the Johnson solids.
Here are some views of the snub square antiprism from various angles:
Oblique side view at 22.5° angle.
The Cartesian coordinates of the snub square antiprism, centered on the origin with edge length 2, are:
- (±1, ±1, C)
- (±√2A, 0, B)
- (0, ±√2A, B)
- (±A, ±A, −B)
- (0, ±√2, −C)
- (±√2, 0, −C)
where A is the root of the following polynomial between 1.7 and 1.8:
A6 − 2A5 − 13A4 + 8A3 + 32A2 − 8A − 4 = 0
|B||=||√(1 − (1 − 1/√2)A2)|
|C||=||√(2(1 + √2A − A2)) + B|
Their numerical values are approximately:
- A = 1.715731736910394
- B = 0.371214042564360
- C = 1.353737018062712