Rhombicuboctahedron

Rhombicuboctahedron
(Click here for rotating model)
Type	Archimedean solid Uniform polyhedron
Elements	F = 26, E = 48, V = 24 (χ = 2)
Faces by sides	8{3}+(6+12){4}
Schläfli symbol	t0,2{4,3}
Wythoff symbol	3 4 | 2
Coxeter–Dynkin
Symmetry	Oh BC3, [4,3], (*432)
Dihedral Angle	3-4:144°44'08" 4-4:135°
References	U10, C22, W13
Properties	Semiregular convex
Colored faces	3.4.4.4 (Vertex figure)
Deltoidal icositetrahedron (dual polyhedron)	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 twelve of the square faces lie in the same planes as the twelve 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 an expanded cube or 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

and

Area and volume

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

Orthogonal projections

The rhombicuboctahedron has six special orthogonal projections, centered, on a vertex, on two types of edges, and three types of faces: triangles, and two squares. The last two correspond to the B₂ and A₂ Coxeter planes.

Orthographic projections

Orthogonal projections

Centered by	Vertex	Edge 3-4	Edge 4-4	Face Square-1	Face Square-2	Face Triangle
Image
Projective symmetry	[2]	[2]	[2]	[2]	[4]	[6]

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

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 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, which count among the Johnson solids; it is thus an elongated square orthobicupola. These pieces can be reassembled to give a new solid called the elongated square gyrobicupola 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

The rhombicuboctahedron can be seen as an expanded cube.

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 T_h 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.

Related polyhedra

The rhombicuboctahedron is one of a family of uniform polyhedra related to the cube and regular octahedron.

Johnson name	Parent	Truncated	Rectified	Bitruncated (tr. dual)	Birectified (dual)	Cantellated	Omnitruncated (Cantitruncated)	Snub
Extended Schläfli symbol
	t0{4,3}	t0,1{4,3}	t1{4,3}	t1,2{4,3}	t2{4,3}	t0,2{4,3}	t0,1,2{4,3}	s{4,3}
Wythoff symbol 4-3-2	3 | 4 2	2 3 | 4	2 | 4 3	2 4 | 3	4 | 3 2	4 3 | 2	4 3 2 |	| 4 3 2
Coxeter-Dynkin diagram
Vertex figure	43	(3.2p.2p)	(4.3.4.3)	(4.2q.2q)	34	(4.4.3.4)	(4.2p.2q)	(3.3.4.3.3)
Octahedral 4-3-2	{4,3}	(3.8.8)	(3.4.3.4)	(4.6.6)	{3,4}	(3.4.4.4)	(4.6.8)	(3.3.3.3.4)

This polyhedron is topologically related as a part of sequence of cantellated polyhedra with vertex figure (3.4.n.4), and continues as tilings of the hyperbolic plane. These vertex-transitive figures have (*n32) reflectional symmetry.

Spherical/planar symmetry	*232 [2,3] D3h	*332 [3,3] Td	*432 [4,3] Oh	*532 [5,3] Ih	*632 [6,3] P6m	*732 [7,3]	*832 [8,3]
Symmetry order	12	24	48	120	∞
Coxeter Schläfli	t0,2{2,3}	t0,2{3,3}	t0,2{4,3}	t0,2{5,3}	t0,2{6,3}	t0,2{7,3}	t0,2{8,3}
Expanded figure	3.4.2.4	3.4.3.4	3.4.4.4	3.4.5.4	3.4.6.4	3.4.7.4	3.4.8.4
Deltoidal figure	V3.4.2.4	V3.4.3.4	V3.4.4.4	V3.4.5.4	V3.4.6.4	V3.4.7.4

Vertex arrangement

It shares its vertex arrangement with three nonconvex uniform polyhedra: the stellated truncated hexahedron, the small rhombihexahedron (having the triangular faces and six square faces in common), and the small cubicuboctahedron (having twelve square faces in common).

Rhombicuboctahedron

Small cubicuboctahedron

Small rhombihexahedron

Stellated truncated hexahedron

In the arts

Rhombicuboctahedron in top left of Portrait of Luca Pacioli.^[1]

Leonardo da Vinci's rhombicuboctahedron

The large polyhedron in the 1495 portrait of Luca Pacioli, traditionally though controversially attributed to Jacopo de' Barbari is a glass rhombicuboctahedron half-filled with water. The first printed version of the rhombicuboctahedron was by Leonardo da Vinci and appeared in his Divina Proportione.

A spherical 180×360° 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.^[2]

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 the videogame 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).

See also

Notes

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. 

External links

• Eric W. Weisstein, Rhombicuboctahedron (Archimedean solid) at MathWorld
• Richard Klitzing, 3D convex uniform polyhedra, x3o4x - sirco
• The Uniform Polyhedra
• Virtual Reality Polyhedra The Encyclopedia of Polyhedra
• Editable printable net of a rhombicuboctahedron with interactive 3D view
• Rhombicuboctahedron Star by Sándor Kabai, Wolfram Demonstrations Project.
• Rhombicuboctahedron: paper strips for plaiting

Archimedean solids
Truncated tetrahedron Truncated cube Truncated octahedron Truncated dodecahedron Truncated icosahedron Cuboctahedron Icosidodecahedron Snub cube Rhombicuboctahedron Truncated cuboctahedron Truncated icosidodecahedron Rhombicosidodecahedron Snub dodecahedron