Snub square antiprism

The snub square antiprism is constructed as its name suggests, a square antiprism which is snubbed, and represented as ss{2,8}, with s{2,8} as a square antiprism.[2] It can be constructed in Conway polyhedron notation as sY4 (snub square pyramid).[3]

It can also be constructed as a square gyrobianticupolae, connecting two anticupolae with gyrated orientations.

Let k ≈ 0.82354 be the positive root of the cubic polynomial

${\displaystyle 9x^{3}+3{\sqrt {3}}\left(5-{\sqrt {2}}\right)x^{2}-3\left(5-2{\sqrt {2}}\right)x-17{\sqrt {3}}+7{\sqrt {6}}.}$

Furthermore, let h ≈ 1.35374 be defined by

${\displaystyle h={\frac {{\sqrt {2}}+8+2{\sqrt {3}}k-3\left(2+{\sqrt {2}}\right)k^{2}}{4{\sqrt {3-3k^{2}}}}}.}$

Then, Cartesian coordinates of a snub square antiprism with edge length 2 are given by the union of the orbits of the points

${\displaystyle (1,1,h),\,\left(1+{\sqrt {3}}k,0,h-{\sqrt {3-3k^{2}}}\right)}$

under the action of the group generated by a rotation around the z-axis by 90° and by a rotation by 180° around a straight line perpendicular to the z-axis and making an angle of 22.5° with the x-axis.[4]

We may then calculate the surface area of a snub square antiprism of edge length a as

${\displaystyle A=\left(2+6{\sqrt {3}}\right)a^{2}\approx 12.39230a^{2},}$[5]

and its volume as

${\displaystyle V=\xi a^{3},}$

where ξ ≈ 3.60122 is the greatest real root of the polynomial

${\displaystyle 531441x^{12}-85726026x^{8}-48347280x^{6}+11588832x^{4}+4759488x^{2}-892448.}$[6]

Similarly constructed, the ss{2,6} is a snub triangular antiprism (a lower symmetry octahedron), and result as a regular icosahedron. A snub pentagonal antiprism, ss{2,10}, or higher n-antiprisms can be similar constructed, but not as a convex polyhedron with equilateral triangles. The preceding Johnson solid, the snub disphenoid also fits constructionally as ss{2,4}, but one has to retain two degenerate digonal faces (drawn in red) in the digonal antiprism.

Snub antiprisms

Symmetry	D2d, [2+,4], (2*2)	D3d, [2+,6], (2*3)	D4d, [2+,8], (2*4)	D5d, [2+,10], (2*5)
Antiprisms	s{2,4} A2 (v:4; e:8; f:6)	s{2,6} A3 (v:6; e:12; f:8)	s{2,8} A4 (v:8; e:16; f:10)	s{2,10} A5 (v:10; e:20; f:12)
Truncated antiprisms	ts{2,4} tA2 (v:16;e:24;f:10)	ts{2,6} tA3 (v:24; e:36; f:14)	ts{2,8} tA4 (v:32; e:48; f:18)	ts{2,10} tA5 (v:40; e:60; f:22)
Symmetry	D2, [2,2]+, (222)	D3, [3,2]+, (322)	D4, [4,2]+, (422)	D5, [5,2]+, (522)
Snub antiprisms	J84	Icosahedron	J85	Concave
		sY3 = HtA3	sY4 = HtA4	sY5 = HtA5
	ss{2,4} (v:8; e:20; f:14)	ss{2,6} (v:12; e:30; f:20)	ss{2,8} (v:16; e:40; f:26)	ss{2,10} (v:20; e:50; f:32)

• Weisstein, Eric W., "Snub square antiprism" ("Johnson solid") at MathWorld.