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 does not belong to the class of 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, A)
- (±B, 0, C)
- (0, ±B, C)
- (±D, ±D, −C)
- (0, ±√2, −A)
- (±√2, 0, −A)
where A, B, C, and D are the roots of the following polynomials within the indicated ranges:
|2A12 + 24A10 + 3A8 − 132A6 − 32A4 + 172A2 − 1 = 0|
|1.2 ≤ A ≤ 1.4|
|B6 + 4B5 − 22B4 − 48B3 + 96B2 + 128B + 32 = 0|
|2 ≤ B ≤ 3|
|2C12 + 72C10 + 355C8 − 748C6 − 352C4 − 28C2 − 1 = 0|
|0.3 ≤ C ≤ 0.4|
|D6 − 2D5 − 13D4 + 8D3 + 32D2 − 8D − 4 = 0|
|1.7 ≤ D ≤ 1.8|
Their numerical values are approximately:
- A = 1.353737018062712
- B = 2.426411091732627
- C = 0.371214042564360
- D = 1.715731736910394