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Rhombicuboctahedron
Rhombicuboctahedron
Rhombicuboctahedron
Type Archimedean solid
Elements F = 26, E = 48, V = 24 (χ = 2)
Faces by sides 8{3}+(6+12){4}
Schläfli symbol r\begin{Bmatrix} 3 \\ 4 \end{Bmatrix}
Wythoff symbol 3 4 | 2
Coxeter-Dynkin CDW ring.pngCDW 4.pngCDW dot.pngCDW 3.pngCDW ring.png
Symmetry Oh
or (*432)
References U10, C22, W13
Properties Semiregular convex
Rhombicuboctahedron color
Colored faces Rhombicuboctahedron
3.4.4.4
(Vertex figure)
Deltoidalicositetrahedron.jpg
Deltoidal icositetrahedron
(dual polyhedron) Rhombicuboctahedron Net
Net

In geometry, the rhombicuboctahedron, or small rhombicuboctahedron, is an Archimedean solid with eight triangular and eighteen square 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 has octahedral symmetry, like the cube and octahedron. Its dual is called the deltoidal icositetrahedron or trapezoidal icositetrahedron, although its faces are not really true trapezoids.

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 which is dual to the cuboctahedron. Great rhombicuboctahedron is an alternative name for a truncated cuboctahedron, whose faces are parallel to those of the (small) rhombicuboctahedron.

It can also be called a cantellated cube or a cantellated octahedron from truncation operations of the uniform polyhedron.

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

\frac{2}{7}\sqrt{10-\sqrt{2}} and \sqrt{4-2\sqrt{2}}.\

Contents
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* 1 Area and volume
* 2 Cartesian coordinates
* 3 Geometric relations
* 4 In the arts
* 5 Games and toys
* 7 Notes
* 8 References

Area and volume

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

A = (18+2\sqrt{3})a^2 \approx 21.4641016a^2
V = \frac{1}{3} (12+10\sqrt{2})a^3 \approx 8.71404521a^3.

Cartesian coordinates
Orthographic projections 3-cube t02.svg 3-cube t02 B2.svg

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

(\pm1, \pm1, \pm(1+\sqrt{2})).\

Geometric relations
Rhombicuboctahedron dissected into two square cupolae and a central octagonal prism. A rotation of one cupola creates the pseudorhombicuboctahedron. Both of these polyhedra have the same vertex figure: 3.4.4.4

There are three pairs of parallel planes that each intersect the rhombicuboctahedron through eight edges in the form of 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, which count among the Johnson solids. These can be reassembled to give a new solid called the pseudorhombicuboctahedron (or elongated square gyrobicupola) 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.
Small rhombicuboctahedron.png
Rhombicuboctahedron
Pseudorhombicuboctahedron.png
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 or the tetrahedral distortions of the cuboctahedron. 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 but different reflections.

The lines along which a Rubik's Cube can be turned are, projected onto a sphere, similar, topologically 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: the cantellated cubic honeycomb, the runcitruncated cubic honeycomb, and the runcinated alternated cubic honeycomb.

It shares its vertex arrangement with three nonconvex uniform polyhedra: the stellated truncated hexahedron, the small rhombihexahedron (having the triangular faces and 6 square faces in common), and the small cubicuboctahedron (having 12 square faces in common).
Small rhombicuboctahedron.png
Rhombicuboctahedron Small cubicuboctahedron.png
Small cubicuboctahedron Small rhombihexahedron.png
Small rhombihexahedron Stellated truncated hexahedron.png
Stellated truncated hexahedron
In the arts
The first ever printed version of the rhombicuboctahedron, by Leonardo da Vinci as appeared in the Divina Proportione

The polyhedron in the portrait of Luca Pacioli 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.[1]
National Library of Belarus in Minsk
Games and toys
Snake in a ball solution: nonuniform concave rhombicuboctahedron

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

A level in Super Mario Galaxy has a planet in the shape of a rhombicuboctahedron.

During the Rubik's Cube 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).

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

* Compound of five small rhombicuboctahedra
* Cube
* Cuboctahedron
* Elongated square gyrobicupola
* Octahedron
* Rhombicosidodecahedron
* Rubik's Snake - puzzle that can form a Rhombicuboctahedron "ball"
* The National Library of Belarus - its architectural main component has the shape of a rhombicuboctahedron.
* Truncated cuboctahedron (great rhombicuboctahedron)

Notes

1. ^ Philosphere

References

* Williams, Robert (1979). The Geometrical Foundation of Natural Structure: A Source Book of Design. Dover Publications, Inc. ISBN 0-486-23729-X. (Section 3-9)
* Coxeter, H. S. M.; Longuet-Higgins, M. S.; Miller, J. C. P. (May 13, 1954). "Uniform Polyhedra". Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences 246, (916): 401–450. doi:10.1098/rsta.1954.0003.

* Eric W. Weisstein, Rhombicuboctahedron (Archimedean solid) at MathWorld.
* The Uniform Polyhedra
* Virtual Reality Polyhedra The Encyclopedia of Polyhedra
* Rhombicuboctahedron Star by Sándor Kabai, Wolfram Demonstrations Project.
* Rhombicuboctahedron: paper strips for plaiting

[show]Archimedean solids
Truncated tetrahedron.png
Truncated tetrahedron Truncated hexahedron.png
Truncated
cube Truncated octahedron.png
Truncated octahedron Truncated dodecahedron.png
Truncated dodecahedron Truncated icosahedron.png
Truncated icosahedron
Cuboctahedron.png
Cuboctahedron Icosidodecahedron.png
Icosidodecahedron
Snub hexahedron.png
Snub
cube Small rhombicuboctahedron.png
Rhombi-
cuboctahedron Great rhombicuboctahedron.png
Truncated cuboctahedron Great rhombicosidodecahedron.png
Truncated icosidodecahedron Small rhombicosidodecahedron.png
Rhomb-
icosidodecahedron Snub dodecahedron ccw.png
Snub
dodecahedron
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Categories: Uniform polyhedra | Archimedean solids
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