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 elongated square gyrobicupola is a polyhedron constructed by 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 ...

s attaching onto the bases of octagonal prism, with one of them rotated. It is a

canonical polyhedron In geometry, the midsphere or intersphere of a convex polyhedron is a sphere which is tangent to every edge of the polyhedron. Not every polyhedron has a midsphere, but the uniform polyhedra, including the regular, quasiregular and semiregul ...

. It is not considered to be an

Archimedean solid The Archimedean solids are a set of thirteen convex polyhedra whose faces are regular polygon and are vertex-transitive, although they aren't face-transitive. The solids were named after Archimedes, although he did not claim credit for them. They ...

because it lacks a set of global

symmetries Symmetry () in everyday life refers to a sense of harmonious and beautiful proportion and balance. In mathematics, the term has a more precise definition and is usually used to refer to an object that is invariant under some transformations ...

that map every vertex to every other vertex, unlike the 13 Archimedean solids. However, it was once mistakenly considered a

rhombicuboctahedron In geometry, the rhombicuboctahedron is an Archimedean solid with 26 faces, consisting of 8 equilateral triangles and 18 squares. It was named by Johannes Kepler in his 1618 Harmonices Mundi, being short for ''truncated cuboctahedral rhombus'', w ...

by many mathematicians. For this reason, it is also known as the pseudo-rhombicuboctahedron, Miller solid, or Miller–Askinuze solid.

Construction

The elongated square gyrobicupola can be constructed similarly to the

, by attaching two regular

s onto the bases of an octagonal prism, a process known as elongation. The difference between these two polyhedrons is that one of the two square cupolas is twisted by 45 degrees, a process known as ''gyration'', making the triangular faces staggered vertically. The resulting polyhedron has 8

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 ...

and 18

square In geometry, a square is a regular polygon, regular quadrilateral. It has four straight sides of equal length and four equal angles. Squares are special cases of rectangles, which have four equal angles, and of rhombuses, which have four equal si ...

s. A

convex Convex or convexity may refer to: Science and technology * Convex lens, in optics Mathematics * Convex set, containing the whole line segment that joins points ** Convex polygon, a polygon which encloses a convex set of points ** Convex polytop ...

polyhedron in which all of the faces are

regular polygon In Euclidean geometry, a regular polygon is a polygon that is Equiangular polygon, direct equiangular (all angles are equal in measure) and Equilateral polygon, equilateral (all sides have the same length). Regular polygons may be either ''convex ...

s is a

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 ...

, and the elongated square gyrobicupola is among them, enumerated as the 37th Johnson solid

J_

. The elongated square gyrobicupola may have been discovered by

Johannes Kepler Johannes Kepler (27 December 1571 – 15 November 1630) was a German astronomer, mathematician, astrologer, Natural philosophy, natural philosopher and writer on music. He is a key figure in the 17th-century Scientific Revolution, best know ...

in his enumeration of the Archimedean solids, but its first clear appearance in print appears to be the work of

Duncan Sommerville Duncan MacLaren Young Sommerville (1879–1934) was a Scottish mathematician and astronomer. He compiled a bibliography on non-Euclidean geometry and also wrote a leading textbook in that field. He also wrote ''Introduction to the Geometry of N ...

in 1905. It was independently rediscovered by

J. C. P. Miller Jeffrey Charles Percy Miller (31 August 1906 – 24 April 1981) was an English mathematician and computing pioneer. He worked in number theory and on geometry, particularly polyhedra, where Miller's monster is a nickname of the great dirhombic ...

in 1930 by mistake while attempting to construct a model of the

. This solid was discovered again by V. G. Ashkinuse in 1957.

Properties

An elongated square gyrobicupola with edge length

a

has a surface area:

\left(18+2\sqrt\right)a^2 \approx 21.464a^2,

by adding the area of 8 equilateral triangles and 10 squares. Its volume can be calculated by slicing it into two square cupolas and one octagonal prism:

\fraca^3 \approx 8.714a^3.

The elongated square gyrobicupola possesses three-dimensional symmetry group

D_

of order 16. It is locally vertex-regular – the arrangement of the four faces incident on any vertex is the same for all vertices; this is unique among the Johnson solids. However, the manner in which it is "twisted" gives it a distinct "equator" and two distinct "poles", which in turn divides its vertices into 8 "polar" vertices (4 per pole) and 16 "equatorial" vertices. It is therefore not

vertex-transitive In geometry, a polytope (e.g. a polygon or polyhedron) or a tiling is isogonal or vertex-transitive if all its vertices are equivalent under the symmetries of the figure. This implies that each vertex is surrounded by the same kinds of face i ...

, and consequently not usually considered to be the 14th

. The dihedral angle of an elongated square gyrobicupola can be ascertained in a similar way as the rhombicuboctahedron, by adding the dihedral angle of a square cupola and an octagonal prism: * the dihedral angle of a rhombicuboctahedron between two adjacent squares on both the top and bottom is that of a square cupola 135°. The dihedral angle of an octagonal prism between two adjacent squares is the internal angle of a

regular octagon In geometry, an octagon () is an eight-sided polygon or 8-gon. A ''regular polygon, regular octagon'' has Schläfli symbol and can also be constructed as a quasiregular Truncation (geometry), truncated square, t, which alternates two types of ...

135°. The dihedral angle between two adjacent squares on the edge where a square cupola is attached to an octagonal prism is the sum of the dihedral angle of a square cupola square-to-octagon and the dihedral angle of an octagonal prism square-to-octagon 45° + 90° = 135°. Therefore, the dihedral angle of a rhombicuboctahedron for every two adjacent squares is 135°. * the dihedral angle of a rhombicuboctahedron square-to-triangle is that of a square cupola between those, 144.7°. The dihedral angle between square-to-triangle, on the edge where a square cupola is attached to an octagonal prism is the sum of the dihedral angle of a square cupola triangle-to-octagon and the dihedral angle of an octagonal prism square-to-octagon 54.7° + 90° = 144.7°. Therefore, the dihedral angle of a rhombicuboctahedron for every square-to-triangle is 144.7°.

Related polyhedra and honeycombs

The elongated square gyrobicupola can form a 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 ...

with the regular

tetrahedron 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, and

cuboctahedron A cuboctahedron is a polyhedron with 8 triangular faces and 6 square faces. A cuboctahedron has 12 identical vertex (geometry), vertices, with 2 triangles and 2 squares meeting at each, and 24 identical edge (geometry), edges, each separating a tr ...

. It can also form another honeycomb with the tetrahedron,

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 ...

and various combinations of cubes,

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, 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 ...

The

pseudo great rhombicuboctahedron In geometry, the pseudo great rhombicuboctahedron is one of the two pseudo uniform polyhedra, the other being the convex elongated square gyrobicupola or pseudo rhombicuboctahedron. It has the same vertex figure as the nonconvex great rhombicuboc ...

is a nonconvex analog of the pseudo-rhombicuboctahedron, constructed in a similar way from the

nonconvex great rhombicuboctahedron In geometry, the nonconvex great rhombicuboctahedron is a nonconvex uniform polyhedron, indexed as U17. It has 26 faces (8 Triangle, triangles and 18 Square, squares), 48 edges, and 24 vertices. It is represented by the Schläfli symbol rr and C ...

In chemistry

The polyvanadate ion V₁₈oxygen">O₄₂">vanadium">V<_a><sub>18<_s.html" ;"title="vanadium.html" ;"title="/nowiki>vanadium">V₁₈oxygen">O₄₂sup>12− has a pseudo-rhombicuboctahedral structure, where each square face acts as the base of a VO₅ pyramid.

References

External links