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 c ...

, the snub square antiprism is one of the

Johnson solid In geometry, a Johnson solid is a strictly convex polyhedron each face of which is a regular polygon. There is no requirement that each face must be the same polygon, or that the same polygons join around each vertex. An example of a Johns ...

s (). 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 and . The plural can be either "icosahedra" () or "icosahedrons". There are infinitely many non- similar shapes of icosahedra, some of them being more symmetric ...

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 se ...

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

. It can be constructed in Conway polyhedron notation as sY4 (''snub square pyramid''). It can also be constructed as a square gyrobianticupolae, connecting two

anticupola In geometry, a cupola is a solid formed by joining two polygons, one (the base) with twice as many edges as the other, by an alternating band of isosceles triangles and rectangles. If the triangles are equilateral and the rectangles are squares ...

e 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 A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured i ...

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 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 The surface area of a solid object is a measure of the total area that the surface of the object occupies. The mathematical definition of surface area in the presence of curved surfaces is considerably more involved than the definition of ...

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 e ...

), and result as a regular

. 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 ve ...

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 In geometry, a disphenoid () is a tetrahedron whose four faces are congruent acute-angled triangles. It can also be described as a tetrahedron in which every two edges that are opposite each other have equal lengths. Other names for the same sha ...

References

External links

* Johnson solids {{polyhedron-stub