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 square gyrobicupola is one of 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 ...

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

square cupola In geometry, the square cupola (sometimes called lesser dome) is a cupola with an octagonal In geometry, an octagon () is an eight-sided polygon or 8-gon. A '' regular octagon'' has Schläfli symbol and can also be constructed as a quasireg ...

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 is a polyhedron constructed by two square cupolas attaching onto the bases of octagonal prism, with one of them rotated. It is a Midsphere#Canonical polyhedron, canonical polyhedron. It is not consid ...

. This polyhedron is created when an octagonal prism is inserted between the two halves of the square gyrobicupola.

Formulae

The following

formula In science, a formula is a concise way of expressing information symbolically, as in a mathematical formula or a ''chemical formula''. The informal use of the term ''formula'' in science refers to the general construct of a relationship betwe ...

e for

volume Volume is a measure of regions in 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) ...

and

surface area The surface area (symbol ''A'') 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 d ...

can be used if all

faces The face is the front of the head that features the eyes, nose and mouth, and through which animals express many of their emotions. The face is crucial for human identity, and damage such as scarring or developmental deformities may affect the ...

are

regular Regular may refer to: Arts, entertainment, and media Music * "Regular" (Badfinger song) * Regular tunings of stringed instruments, tunings with equal intervals between the paired notes of successive open strings Other uses * Regular character, ...

, 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 Triangular prismatic honeycomb#Hexagonal prismatic honeycomb, hexagonal prismatic cells built from beeswax by honey bees in their beehive, nests to contain their brood (eggs, larvae, and pupae) and stores of honey and pol ...

s with

tetrahedra In geometry, a tetrahedron (: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular Face (geometry), faces, six straight Edge (geometry), edges, and four vertex (geometry), vertices. The tet ...

cube A cube or regular hexahedron is a three-dimensional space, three-dimensional solid object in geometry, which is bounded by six congruent square (geometry), square faces, a type of polyhedron. It has twelve congruent edges and eight vertices. It i ...

s and cuboctahedra; and with tetrahedra,

square pyramid In geometry, a square pyramid is a Pyramid (geometry), pyramid with a square base and four triangles, having a total of five faces. If the Apex (geometry), apex of the pyramid is directly above the center of the square, it is a ''right square p ...

s, and

elongated square bipyramid In geometry, the elongated square bipyramid (or elongated octahedron) is the polyhedron constructed by attaching two Equilateral square pyramid, equilateral square pyramids onto a cube (geometry), cube's faces that are opposite each other. It can ...

s. (The latter unit can be decomposed into

elongated square pyramid In geometry, the elongated square pyramid is a convex polyhedron constructed from a cube by attaching an equilateral square pyramid onto one of its faces. It is an example of Johnson solid. Construction The elongated square pyramid is a comp ...

s, cubes, and/or square pyramids).

References

External links

* {{Polyhedron-stub Johnson solids