Snub square antiprism

In geometry, the snub square antiprism is one of the Johnson solids (J₈₅). A Johnson solid is one of 92 strictly convex polyhedra that is composed of regular polygon faces but are not uniform polyhedra (that is, they are not Platonic solids, Archimedean solids, prisms, or antiprisms). They were named by Norman Johnson, who first listed these polyhedra in 1966.[1]

Snub square antiprism
Type	Johnson J84 - J85 - J86
Faces	8+16 triangles 2 squares
Edges	40
Vertices	16
Vertex configuration	8(3^5) 8(3^4.4)
Symmetry group	D4d
Dual polyhedron	-
Properties	convex
Net

3D model of a snub square antiprism

It is one of the elementary Johnson solids that do not arise from "cut and paste" manipulations of the Platonic and Archimedean solids, although it is a relative of the icosahedron that has fourfold symmetry instead of threefold.

Construction

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.

Cartesian coordinates

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

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

Furthermore, let h ≈ 1.35374 be defined by

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

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

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

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 of edge length a as

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

and its volume as

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

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

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

Snub antiprisms

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)

References

1. Johnson, Norman W. (1966), "Convex polyhedra with regular faces", Canadian Journal of Mathematics, 18: 169–200, doi:10.4153/cjm-1966-021-8, MR 0185507, Zbl 0132.14603.
2. Snub Anti-Prisms
3. https://levskaya.github.io/polyhedronisme/?recipe=C100sY4
4. Timofeenko, A. V. (2009). "The non-Platonic and non-Archimedean noncomposite polyhedra". Journal of Mathematical Science. 162 (5): 725.
5. Wolfram Research, Inc. (2020). "Wolfram|Alpha Knowledgebase". Champaign, IL. PolyhedronData[{"Johnson", 85}, "SurfaceArea"] Cite journal requires |journal= (help)
6. Wolfram Research, Inc. (2020). "Wolfram|Alpha Knowledgebase". Champaign, IL. MinimalPolynomial[PolyhedronData[{"Johnson", 85}, "Volume"], x] Cite journal requires |journal= (help)

External links

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