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 elongated square gyrobicupola or pseudo-rhombicuboctahedron 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 isohedral, each face must be the same polygon, or that the same polygons join around each Vertex (geometry), ver ...

s (). It is not usually considered to be 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 ...

, even though its

faces The face is the front of an animal's 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 affe ...

consist of

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 p ...

s that meet in the same pattern at each of its vertices, because unlike the 13 Archimedean solids, it lacks a set of global

symmetries Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definiti ...

that map every vertex to every other vertex (though Grünbaum has suggested it should be added to the traditional list of Archimedean solids as a 14th example). It strongly resembles, but should not be mistaken for, the

rhombicuboctahedron In geometry, the rhombicuboctahedron, or small rhombicuboctahedron, is a polyhedron with eight triangular, six square, and twelve rectangular faces. There are 24 identical vertices, with one triangle, one square, and two rectangles meeting at eac ...

, which ''is'' an Archimedean solid. It is also a

canonical polyhedron In geometry, the midsphere or intersphere of a polyhedron is a sphere which is tangent to every edge of the polyhedron. That is to say, it touches any given edge at exactly one point. Not every polyhedron has a midsphere, but for every convex po ...

. This shape may have been discovered by

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

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

) and again by V. G. Ashkinuse in 1957.

Construction and relation to the rhombicuboctahedron

As the name suggests, it can be constructed by elongating a square gyrobicupola (''J''₂₉) and inserting an

octagon In geometry, an octagon (from the Greek ὀκτάγωνον ''oktágōnon'', "eight angles") is an eight-sided polygon or 8-gon. A '' regular octagon'' has Schläfli symbol and can also be constructed as a quasiregular truncated square, t, whi ...

prism Prism usually refers to: * Prism (optics), a transparent optical component with flat surfaces that refract light * Prism (geometry), a kind of polyhedron Prism may also refer to: Science and mathematics * Prism (geology), a type of sedimentary ...

between its two halves. The solid can also be seen as the result of twisting one of the

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 (''J''₄) on a

(one of the

s; a.k.a. the elongated square orthobicupola) by 45 degrees. It is therefore a gyrate rhombicuboctahedron. Its similarity to the rhombicuboctahedron gives it the alternative name pseudo-rhombicuboctahedron. It has occasionally been referred to as "the fourteenth Archimedean solid". This property does not carry over to its pentagonal-faced counterpart, the

gyrate rhombicosidodecahedron In geometry, the gyrate rhombicosidodecahedron is one of the Johnson solids (). It is also a canonical polyhedron. Related polyhedron It can be constructed as a rhombicosidodecahedron with one pentagonal cupola rotated through 36 degrees. ...

Symmetry and classification

The pseudo-rhombicuboctahedron possesses D_4d symmetry. 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 divide 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 in ...

, and consequently not usually considered to be one of the

s. With faces colored by its ''D''_4d symmetry, it can look like this: There are 8 (green) squares around its

equator The equator is a circle of latitude, about in circumference, that divides Earth into the Northern and Southern hemispheres. It is an imaginary line located at 0 degrees latitude, halfway between the North and South poles. The term can als ...

, 4 (red) triangles and 4 (yellow) squares above and below, and one (blue) square on each pole.

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 Beeswax, wax cells built by honey bees in their beehive, nests to contain their larvae and stores of honey and pollen. beekeeping, Beekee ...

with the regular

tetrahedron 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 o ...

, cube, and

cuboctahedron A cuboctahedron is a polyhedron with 8 triangular faces and 6 square faces. A cuboctahedron has 12 identical vertices, with 2 triangles and 2 squares meeting at each, and 24 identical edges, each separating a triangle from a square. As such, it ...

. It can also form another honeycomb with the tetrahedron,

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, ...

and various combinations of cubes,

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 ( ...

s, and elongated square bipyramids.
Pseudo-great rhombicuboctahedron

The pseudo great rhombicuboctahedron 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 triangles and 18 squares), 48 edges, and 24 vertices. It is represented by the Schläfli symbol rr and Coxeter-Dynkin di ...

In chemistry

The polyvanadate ion V₁₈oxygen.html"_;"title="vanadium.html"_;"title="/nowiki>vanadium">V₁₈oxygen">O₄₂.html" ;"title="vanadium">V₁₈oxygen.html" ;"title="vanadium.html" ;"title="/nowiki>vanadium">V₁₈oxygen">O₄₂">vanadium">V₁₈oxygen.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