Rhombicuboctahedron

# Rhombicuboctahedron

Overview
In geometry
Geometry
Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....

, the rhombicuboctahedron, or small rhombicuboctahedron, is an Archimedean solid
Archimedean solid
In geometry an Archimedean solid is a highly symmetric, semi-regular convex polyhedron composed of two or more types of regular polygons meeting in identical vertices...

with eight triangular
Triangle
A triangle is one of the basic shapes of geometry: a polygon with three corners or vertices and three sides or edges which are line segments. A triangle with vertices A, B, and C is denoted ....

and eighteen square
Square (geometry)
In geometry, a square is a regular quadrilateral. This means that it has four equal sides and four equal angles...

faces. There are 24 identical vertices, with one triangle and three squares meeting at each. Note that six of the squares only share vertices with the triangles while the other twelve share an edge. The polyhedron
Polyhedron
In elementary geometry a polyhedron is a geometric solid in three dimensions with flat faces and straight edges...

has octahedral symmetry
Octahedral symmetry
150px|thumb|right|The [[cube]] is the most common shape with octahedral symmetryA regular octahedron has 24 rotational symmetries, and a symmetry order of 48 including transformations that combine a reflection and a rotation...

, like the cube and octahedron
Octahedron
In geometry, an octahedron is a polyhedron with eight faces. A regular octahedron is a Platonic solid composed of eight equilateral triangles, four of which meet at each vertex....

. Its dual
Dual polyhedron
In geometry, polyhedra are associated into pairs called duals, where the vertices of one correspond to the faces of the other. The dual of the dual is the original polyhedron. The dual of a polyhedron with equivalent vertices is one with equivalent faces, and of one with equivalent edges is another...

is called the deltoidal icositetrahedron
Deltoidal icositetrahedron
In geometry, a deltoidal icositetrahedron is a Catalan solid which looks a bit like an overinflated cube. Its dual polyhedron is the rhombicuboctahedron....

or trapezoidal icositetrahedron, although its faces are not really true trapezoid
Trapezoid
In Euclidean geometry, a convex quadrilateral with one pair of parallel sides is referred to as a trapezoid in American English and as a trapezium in English outside North America. A trapezoid with vertices ABCD is denoted...

s.
Discussion

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Encyclopedia
In geometry
Geometry
Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....

, the rhombicuboctahedron, or small rhombicuboctahedron, is an Archimedean solid
Archimedean solid
In geometry an Archimedean solid is a highly symmetric, semi-regular convex polyhedron composed of two or more types of regular polygons meeting in identical vertices...

with eight triangular
Triangle
A triangle is one of the basic shapes of geometry: a polygon with three corners or vertices and three sides or edges which are line segments. A triangle with vertices A, B, and C is denoted ....

and eighteen square
Square (geometry)
In geometry, a square is a regular quadrilateral. This means that it has four equal sides and four equal angles...

faces. There are 24 identical vertices, with one triangle and three squares meeting at each. Note that six of the squares only share vertices with the triangles while the other twelve share an edge. The polyhedron
Polyhedron
In elementary geometry a polyhedron is a geometric solid in three dimensions with flat faces and straight edges...

has octahedral symmetry
Octahedral symmetry
150px|thumb|right|The [[cube]] is the most common shape with octahedral symmetryA regular octahedron has 24 rotational symmetries, and a symmetry order of 48 including transformations that combine a reflection and a rotation...

, like the cube and octahedron
Octahedron
In geometry, an octahedron is a polyhedron with eight faces. A regular octahedron is a Platonic solid composed of eight equilateral triangles, four of which meet at each vertex....

. Its dual
Dual polyhedron
In geometry, polyhedra are associated into pairs called duals, where the vertices of one correspond to the faces of the other. The dual of the dual is the original polyhedron. The dual of a polyhedron with equivalent vertices is one with equivalent faces, and of one with equivalent edges is another...

is called the deltoidal icositetrahedron
Deltoidal icositetrahedron
In geometry, a deltoidal icositetrahedron is a Catalan solid which looks a bit like an overinflated cube. Its dual polyhedron is the rhombicuboctahedron....

or trapezoidal icositetrahedron, although its faces are not really true trapezoid
Trapezoid
In Euclidean geometry, a convex quadrilateral with one pair of parallel sides is referred to as a trapezoid in American English and as a trapezium in English outside North America. A trapezoid with vertices ABCD is denoted...

s.

The name rhombicuboctahedron refers to the fact that 12 of the square faces lie in the same planes as the 12 faces of the rhombic dodecahedron
Rhombic dodecahedron
In geometry, the rhombic dodecahedron is a convex polyhedron with 12 rhombic faces. It is an Archimedean dual solid, or a Catalan solid. Its dual is the cuboctahedron.-Properties:...

which is dual to the cuboctahedron
Cuboctahedron
In geometry, a cuboctahedron is a polyhedron with eight triangular faces and six square faces. A cuboctahedron has 12 identical vertices, with two triangles and two squares meeting at each, and 24 identical edges, each separating a triangle from a square. As such it is a quasiregular polyhedron,...

. Great rhombicuboctahedron is an alternative name for a truncated cuboctahedron
Truncated cuboctahedron
In geometry, the truncated cuboctahedron is an Archimedean solid. It has 12 square faces, 8 regular hexagonal faces, 6 regular octagonal faces, 48 vertices and 72 edges...

, whose faces are parallel to those of the (small) rhombicuboctahedron.

It can also be called an expanded
Expansion (geometry)
In geometry, expansion is a polytope operation where facets are separated and moved radially apart, and new facets are formed at separated elements...

cube
or cantellated
Cantellation (geometry)
In geometry, a cantellation is an operation in any dimension that cuts a regular polytope at its edges and vertices, creating a new facet in place of each edge and vertex. The operation also applies to regular tilings and honeycombs...

cube or a cantellated octahedron from truncation operations of the uniform polyhedron
Uniform polyhedron
A uniform polyhedron is a polyhedron which has regular polygons as faces and is vertex-transitive...

.

If the original rhombicuboctahedron has unit edge length, its dual strombic icositetrahedron has edge lengths and

## Area and volume

The area A and the volume V of the rhombicuboctahedron of edge length a are:

## Cartesian coordinates

Cartesian coordinates for the vertices of a rhombicuboctahedron centred at the origin, with edge length 2 units, are all permutations of

## Geometric relations

There are three pairs of parallel planes that each intersect the rhombicuboctahedron in a regular octagon. The rhombicuboctahedron may be divided along any of these to obtain an octagonal prism with regular faces and two additional polyhedra called square cupolae
Cupola (geometry)
In geometry, a cupola is a solid formed by joining two polygons, one with twice as many edges as the other, by an alternating band of triangles and rectangles...

, which count among the Johnson solid
Johnson solid
In geometry, a Johnson solid is a strictly convex polyhedron, each face of which is a regular polygon, but which is not uniform, i.e., not a Platonic solid, Archimedean solid, prism or antiprism. There is no requirement that each face must be the same polygon, or that the same polygons join around...

s; it is thus an elongated square orthobicupola. These pieces can be reassembled to give a new solid called the elongated square gyrobicupola
Elongated square gyrobicupola
In geometry, the elongated square gyrobicupola or pseudorhombicuboctahedron is one of the Johnson solids . The 92 Johnson solids were named and described by Norman Johnson in 1966.- Relation to Rhombicuboctahedron :...

or pseudorhombicuboctahedron, with the symmetry of a square antiprism. In this the vertices are all locally the same as those of a rhombicuboctahedron, with one triangle and three squares meeting at each, but are not all identical with respect to the entire polyhedron, since some are closer to the symmetry axis than others.
 Rhombicuboctahedron Pseudorhombicuboctahedron

There are distortions of the rhombicuboctahedron that, while some of the faces are not regular polygons, are still vertex-uniform. Some of these can be made by taking a cube or octahedron and cutting off the edges, then trimming the corners, so the resulting polyhedron has six square and twelve rectangular faces. These have octahedral symmetry and form a continuous series between the cube and the octahedron, analogous to the distortions of the rhombicosidodecahedron
Rhombicosidodecahedron
In geometry, the rhombicosidodecahedron, or small rhombicosidodecahedron, is an Archimedean solid, one of thirteen convex isogonal nonprismatic solids constructed of two or more types of regular polygon faces....

or the tetrahedral distortions of the cuboctahedron
Cuboctahedron
In geometry, a cuboctahedron is a polyhedron with eight triangular faces and six square faces. A cuboctahedron has 12 identical vertices, with two triangles and two squares meeting at each, and 24 identical edges, each separating a triangle from a square. As such it is a quasiregular polyhedron,...

. However, the rhombicuboctahedron also has a second set of distortions with six rectangular and sixteen trapezoidal faces, which do not have octahedral symmetry but rather Th symmetry, so they are invariant under the same rotations as the tetrahedron
Tetrahedron
In geometry, a tetrahedron is a polyhedron composed of four triangular faces, three of which meet at each vertex. A regular tetrahedron is one in which the four triangles are regular, or "equilateral", and is one of the Platonic solids...

but different reflections.

The lines along which a Rubik's Cube
Rubik's Cube
Rubik's Cube is a 3-D mechanical puzzle invented in 1974 by Hungarian sculptor and professor of architecture Ernő Rubik.Originally called the "Magic Cube", the puzzle was licensed by Rubik to be sold by Ideal Toy Corp. in 1980 and won the German Game of the Year special award for Best Puzzle that...

can be turned are, projected onto a sphere, similar, topologically
Topology
Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...

identical, to a rhombicuboctahedron's edges. In fact, variants using the Rubik's Cube mechanism have been produced which closely resemble the rhombicuboctahedron.

The rhombicuboctahedron is used in three uniform space-filling tessellations
Honeycomb (geometry)
In geometry, a honeycomb is a space filling or close packing of polyhedral or higher-dimensional cells, so that there are no gaps. It is an example of the more general mathematical tiling or tessellation in any number of dimensions....

: the cantellated cubic honeycomb
Cantellated cubic honeycomb
The cantellated cubic honeycomb is a uniform space-filling tessellation in Euclidean 3-space. It is composed of small rhombicuboctahedra, cuboctahedra, and cubes in a ratio of 1:1:3.- References :...

, the runcitruncated cubic honeycomb
Runcitruncated cubic honeycomb
The runcitruncated cubic honeycomb is a uniform space-filling tessellation in Euclidean 3-space. It is composed of small rhombicuboctahedra, truncated cubes, octagonal prisms, and cubes in a ratio of 1:1:3:3....

, and the runcinated alternated cubic honeycomb
Runcinated alternated cubic honeycomb
The runcinated alternated cubic honeycomb is a uniform space-filling tessellation in Euclidean 3-space. It is composed of small rhombicuboctahedra, cubes, and tetrahedra in a ratio of 1:1:2.- References :...

.

It shares its vertex arrangement with three nonconvex uniform polyhedra
Nonconvex uniform polyhedron
In geometry, a uniform star polyhedron is a self-intersecting uniform polyhedron. They are also sometimes called nonconvex polyhedra to imply self-intersecting...

: the stellated truncated hexahedron, the small rhombihexahedron
Small rhombihexahedron
In geometry, the small rhombihexahedron is a nonconvex uniform polyhedron, indexed as U18. It has 18 faces , 48 edges, and 24 vertices. Its vertex figure is an antiparallelogram.-Related polyhedra:...

(having the triangular faces and 6 square faces in common), and the small cubicuboctahedron
Small cubicuboctahedron
In geometry, the small cubicuboctahedron is a uniform star polyhedron, indexed as U13. It has 20 faces , 48 edges, and 24 vertices. Its vertex figure is a crossed quadrilateral.- Related polyhedra :...

(having 12 square faces in common).
 Rhombicuboctahedron Small cubicuboctahedronSmall cubicuboctahedronIn geometry, the small cubicuboctahedron is a uniform star polyhedron, indexed as U13. It has 20 faces , 48 edges, and 24 vertices. Its vertex figure is a crossed quadrilateral.- Related polyhedra :... Small rhombihexahedronSmall rhombihexahedronIn geometry, the small rhombihexahedron is a nonconvex uniform polyhedron, indexed as U18. It has 18 faces , 48 edges, and 24 vertices. Its vertex figure is an antiparallelogram.-Related polyhedra:... Stellated truncated hexahedron

## In the arts

The polyhedron in the portrait of Luca Pacioli
Luca Pacioli
Fra Luca Bartolomeo de Pacioli was an Italian mathematician, Franciscan friar, collaborator with Leonardo da Vinci, and seminal contributor to the field now known as accounting...

is a glass rhombicuboctahedron half-filled with water.

A spherical 180x360° panorama can be projected onto any polyhedron; but the rhombicuboctahedron provides a good enough approximation of a sphere while being easy to build. This type of projection, called 'Philosphere', is possible from some panorama assembly software. It consists of two images that are printed separately and cut with scissors while leaving some flaps for assembly with glue.

## Games and toys

The Freescape
Freescape
thumb|The Freescape logo.The Freescape engine was an early 3D game engine used in games such as 1987's Driller.-History:Developed in-house by Incentive Software, Freescape is considered to be one of the first proprietary 3D engines to be used in computer games, although the engine was not used...

games Driller and Dark Side both had a game map in the form of a rhombicuboctahedron.

A level in the videogame Super Mario Galaxy
Super Mario Galaxy
is a 3D platform game developed by Nintendo EAD Tokyo and published by Nintendo for the Wii. It was released in most regions in November 2007, and is the third 3D original platformer in the Mario series, after Super Mario 64 and Super Mario Sunshine. The game follows the protagonist, Mario, on a...

has a planet in the shape of a rhombicuboctahedron.

During the Rubik's Cube
Rubik's Cube
Rubik's Cube is a 3-D mechanical puzzle invented in 1974 by Hungarian sculptor and professor of architecture Ernő Rubik.Originally called the "Magic Cube", the puzzle was licensed by Rubik to be sold by Ideal Toy Corp. in 1980 and won the German Game of the Year special award for Best Puzzle that...

craze of the 1980s, one combinatorial puzzle sold had the form of a rhombicuboctahedron (the mechanism was of course that of a Rubik's Cube
Rubik's Cube
Rubik's Cube is a 3-D mechanical puzzle invented in 1974 by Hungarian sculptor and professor of architecture Ernő Rubik.Originally called the "Magic Cube", the puzzle was licensed by Rubik to be sold by Ideal Toy Corp. in 1980 and won the German Game of the Year special award for Best Puzzle that...

).

The Rubik's Snake
Rubik's Snake
A Rubik's Snake is a toy with twenty-four wedges identically shaped liked prisms, specifically right isosceles triangular prisms. The wedges are connected, by spring bolts, such that they can be twisted, but not separated...

toy was usually sold in the shape of a stretched rhombicuboctahedron (12 of the squares being replaced with 1:√2 rectangles).