geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ...

, the snub square antiprism is one of the Johnson solids (). It is one of the elementary Johnson solids that do not arise from "cut and paste" manipulations of the

Platonic Plato's influence on Western culture was so profound that several different concepts are linked by being called Platonic or Platonist, for accepting some assumptions of Platonism, but which do not imply acceptance of that philosophy as a whole. It ...

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 In geometry, the square antiprism is the second in an infinite family of antiprisms formed by an 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 a sem ...

which is snubbed, and represented as ss, with s as a

. It can be constructed in

Conway polyhedron notation In geometry, Conway polyhedron notation, invented by John Horton Conway and promoted by George W. Hart, is used to describe polyhedra based on a seed polyhedron modified by various prefix operations. Conway and Hart extended the idea of using o ...

as sY4 (''snub square pyramid''). 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 In mathematics, a cubic function is a function of the form f(x)=ax^3+bx^2+cx+d where the coefficients , , , and are complex numbers, and the variable takes real values, and a\neq 0. In other words, it is both a polynomial function of degree ...

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

Furthermore, let ''h'' ≈ 1.35374 be defined by :

h=\frac.

Then, Cartesian coordinates 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 ide ...

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

A=\left(2+6\sqrt\right)a^2\approx12.39230a^2,

and its

volume Volume is a measure of occupied three-dimensional space. It is often quantified numerically using SI derived units (such as the cubic metre and litre) or by various imperial or US customary units (such as the gallon, quart, cubic inch). Th ...

as :

V=\xi a^3,

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

531441x^-85726026x^8-48347280x^6+11588832x^4+4759488x^2-892448.

Snub antiprisms

Similarly constructed, the ss is a ''snub triangular antiprism'' (a lower symmetry

octahedron In geometry, an octahedron (plural: octahedra, octahedrons) is a polyhedron with eight faces. The term is most commonly used to refer to the regular octahedron, a Platonic solid composed of eight equilateral triangles, four of which meet at ea ...

), and result as a regular icosahedron. A ''snub pentagonal antiprism'', ss, or higher ''n''-antiprisms can be similar constructed, but not as a convex polyhedron with equilateral triangles. The preceding Johnson solid, the

snub disphenoid In geometry, the snub disphenoid, Siamese dodecahedron, triangular dodecahedron, trigonal dodecahedron, or dodecadeltahedron is a convex polyhedron with twelve equilateral triangles as its faces. It is not a regular polyhedron because some vert ...

also fits constructionally as ss, but one has to retain two degenerate

digon In geometry, a digon is a polygon with two sides (edges) and two vertices. Its construction is degenerate in a Euclidean plane because either the two sides would coincide or one or both would have to be curved; however, it can be easily visu ...

al faces (drawn in red) in the digonal antiprism.

References

External links

* Johnson solids {{polyhedron-stub