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 square gyrobicupola 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 (). Like the square orthobicupola (), it can be obtained by joining two

square cupola In geometry, the square cupola, sometimes called lesser dome, is one of the Johnson solids (). It can be obtained as a slice of the rhombicuboctahedron. As in all cupolae, the base polygon has twice as many edges and vertices as the top; in t ...

e () along their bases. The difference is that in this solid, the two halves are rotated 45 degrees with respect to one another. The ''square gyrobicupola'' is the second in an infinite set of gyrobicupolae. Related to the square gyrobicupola is the

elongated square gyrobicupola In geometry, the elongated square gyrobicupola or pseudo-rhombicuboctahedron is one of the Johnson solids (). It is not usually considered to be an Archimedean solid, even though its faces consist of regular polygons that meet in the same pat ...

. This polyhedron is created when an

octagonal prism In geometry, the octagonal prism is the sixth in an infinite set of prisms, formed by rectangular sides and two regular octagon caps. If faces are all regular, it is a semiregular polyhedron. Symmetry Images The octagonal prism can also ...

is inserted between the two halves of the square gyrobicupola. It is argued whether or not the elongated square gyrobicupola is an

Archimedean solid In geometry, an Archimedean solid is one of the 13 solids first enumerated by Archimedes. They are the convex uniform polyhedra composed of regular polygons meeting in identical vertices, excluding the five Platonic solids (which are composed ...

because, although it meets every other standard necessary to be an Archimedean solid, it is not highly symmetric.

Formulae

The following formulae for

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 ...

and

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 ...

can be used if all faces are

regular The term regular can mean normal or in accordance with rules. It may refer to: People * Moses Regular (born 1971), America football player Arts, entertainment, and media Music * "Regular" (Badfinger song) * Regular tunings of stringed instrum ...

, with edge length ''a'': :

V=\left(2+\frac\right)a^3\approx3.88562...a^3

A=2\left(5+\sqrt\right)a^2\approx13.4641...a^2

Related polyhedra and honeycombs

The square gyrobicupola forms space-filling

honeycomb A honeycomb is a mass of hexagonal prismatic wax cells built by honey bees in their nests to contain their larvae and stores of honey and pollen. Beekeepers may remove the entire honeycomb to harvest honey. Honey bees consume about of honey t ...

s with

tetrahedra In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all the ...

cube In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. Viewed from a corner it is a hexagon and its net is usually depicted as a cross. The cube is the on ...

s and cuboctahedra; and with tetrahedra,

square pyramid In geometry, a square pyramid is a pyramid having a square base. If the apex is perpendicularly above the center of the square, it is a right square pyramid, and has symmetry. If all edge lengths are equal, it is an equilateral square pyramid ...

s, and

elongated square bipyramid In geometry, the elongated square bipyramid (or elongated octahedron) is one of the Johnson solids (). As the name suggests, it can be constructed by elongating an octahedron by inserting a cube (geometry), cube between its Square pyramid, congr ...

s. (The latter unit can be decomposed into

elongated square pyramid In geometry, the elongated square pyramid is one of the Johnson solids (). As the name suggests, it can be constructed by elongating a square pyramid () by attaching a cube to its square base. Like any elongated pyramid, it is topologically (b ...

s, cubes, and/or square pyramids).

References

External links

* {{Polyhedron-stub Johnson solids