# Snub square antiprism

Snub square antiprism
Rank3
TypeCRF
Notation
Bowers style acronymSnisquap
Elements
Faces8+16 triangles, 2 squares
Edges8+8+8+16
Vertices8+8
Vertex figures8 mirror-symmetric pentagons, edge length 1, 1, 1, 1, 2
8 pentagons, edge length 1
Measures (edge length 1)
Volume≈ 3.60122
Central density1
Number of external pieces26
Level of complexity10
Related polytopes
ArmySnisquap
RegimentSnisquap
ConjugateRetrosnub square antiprism
Abstract & topological properties
Flag count160
Euler characteristic2
SurfaceSphere
OrientableYes
Genus0
Properties
Symmetry(I2(8)×A1)/2, order 16
ConvexYes
NatureTame

The snub square antiprism is one of the 92 Johnson solids (J85). It consists of 8+16 triangles and 2 squares.

It can be constructed from a square antiprism by expanding the two halves outward and inserting a set of 16 triangles in between the halves.

## Coordinates

Coordinates for a snub square antiprism with unit edge length are given by

• (±1/2, ±1/2, C/2),
• (±√2A/2, 0, B/2),
• (0, ±√2A/2, B/2),
• (±A/2, ±A/2, −B/2),
• (0, ±√2/2, −C/2),
• (±√2/2, 0, −C/2),

where A is the second-to-greatest root of

${\displaystyle x^{6}-2x^{5}-13x^{4}+8x^{3}+32x^{2}-8x-4,}$

and where B and C are given by

${\displaystyle B={\sqrt {1-\left(1-{\frac {\sqrt {2}}{2}}\right)}}A,}$
${\displaystyle C={\sqrt {2+2{\sqrt {2}}A-2A^{2}}}+B.}$

From these coordinates, its volume can be calculated as ξ ≈ 3.60122, where ξ is the greatest real root of

${\displaystyle 531441x^{12}-85726026x^{8}-48347280x^{6}+11588832x^{4}+4759488x^{2}-892448.}$[1]

## Related polyhedra

The snub square antiprism can be considered to be the square case in the family of snub antiprisms. The snub triangular antiprism is the regular icosahedron, and the snub disphenoid or snub digonal antiprism is another Johnson solid. No other members of this family can be made convex and regular-faced (the snub pentagonal antiprism can be made with regular faces, but is concave).