geometry Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...

, the snub square antiprism is the

Johnson solid In geometry, a Johnson solid, sometimes also known as a Johnson–Zalgaller solid, is a convex polyhedron whose faces are regular polygons. They are sometimes defined to exclude the uniform polyhedrons. There are ninety-two Solid geometry, s ...

that can be constructed by snubbing the

square antiprism In geometry, the square antiprism is the second in an infinite family of antiprisms formed by an even number, even-numbered sequence of triangle sides closed by two polygon caps. It is also known as an ''anticube''. If all its faces are regular ...

. 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 In geometry, an icosahedron ( or ) is a polyhedron with 20 faces. The name comes . The plural can be either "icosahedra" () or "icosahedrons". There are infinitely many non- similar shapes of icosahedra, some of them being more symmetrical tha ...

that has fourfold symmetry instead of threefold.

Construction and properties

The snub is the process of constructing polyhedra by cutting loose the edge's faces, twisting them, and then attaching

equilateral triangles An equilateral triangle is a triangle in which all three sides have the same length, and all three angles are equal. Because of these properties, the equilateral triangle is a regular polygon, occasionally known as the regular triangle. It is the ...

to their edges. As the name suggested, the snub square antiprism is constructed by snubbing the

, and this construction results in 24 equilateral triangles and 2 squares as its faces. The

s are the convex polyhedra whose faces are regular, and the snub square antiprism is one of them, enumerated as

J_

, the 85th Johnson solid. Let

k \approx 0.82354

be the positive root of the

cubic polynomial In mathematics, a cubic function is a function (mathematics), function of the form f(x)=ax^3+bx^2+cx+d, that is, a polynomial function of degree three. In many texts, the ''coefficients'' , , , and are supposed to be real numbers, and the func ...

9x^3+3\sqrt\left(5-\sqrt\right)x^2-3\left(5-2\sqrt\right)x-17\sqrt+7\sqrt.

Furthermore, let

h \approx 1.35374

be defined by

h = \frac.

Then,

Cartesian coordinates In geometry, a Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of real numbers called ''coordinates'', which are the signed distances to the point from two fixed perpendicular o ...

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

(1,1,h),\,\left(1+\sqrtk,0,h-\sqrt\right)

under the action of the

group A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic iden ...

generated by a rotation around the axis by 90° and by a rotation by 180° around a straight line perpendicular to the axis and making an angle of 22.5° with the axis. It has the three-dimensional symmetry of

dihedral group In mathematics, a dihedral group is the group (mathematics), group of symmetry, symmetries of a regular polygon, which includes rotational symmetry, rotations and reflection symmetry, reflections. Dihedral groups are among the simplest example ...

D_

of order 16. The surface area and volume of a snub square antiprism with edge length

a

can be calculated as:

\begin
 A = \left(2+6\sqrt\right)a^2 &\approx 12.392a^2, \\
 V &\approx 3.602 a^3.
\end

References

External links

* Johnson solids {{Johnson solids