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	t_0,2{4,3}
Wythoff symbol	3 4 \| 2
Coxeter diagram
Symmetry group	O_h, BC₃, [4,3], (*432), order 48
Rotation group	O, [4,3]⁺, (432), order 24
Dihedral Angle	3-4:144°44'08" (144.74°) 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

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

Area and volume [edit]

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

$A = (18+2\sqrt{3})a^2 \approx 21.4641016a^2$

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

Close-Packing Packing Fraction [edit]

The packing-fraction of the close-packed crystal formed by rhombicuboctahedra is given by:

$\eta = \frac{4}{3} \left( 4\sqrt{2} - 5 \right)$

It was proven that this close-packed value is assumed in a Bravais-type lattice by de Graaf (2011), who also described the lattice. The proof is conditionally dependent on Hales (2005) proof of the Kepler Conjecture and the proof of the inscribed-sphere upper bound for the packing of particles by Torquato (2009). The packing and lattice were originally found by Betke and Henk (2000), who did not prove its close-packed nature.

Orthogonal projections [edit]

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 [edit]

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 [edit]

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 either an expanded cube (the blue faces) or an expanded octahedron (the red faces).

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 [edit]

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

Uniform octahedral polyhedra
Symmetry: [4,3], (*432)							[4,3]⁺, (432)	[1⁺,4,3], (*332)	[4,3⁺], (3*2)


{4,3}	t_0,1{4,3}	t₁{4,3}	t_1,2{4,3}	{3,4}	t_0,2{4,3}	t_0,1,2{4,3}	s{4,3}	h{4,3}	h_1,2{4,3}
Duals to uniform polyhedra


V4.4.4	V3.8.8	V3.4.3.4	V4.6.6	V3.3.3.3	V3.4.4.4	V4.6.8	V3.3.3.3.4	V3.3.3	V3.3.3.3.3

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.

Dimensional family of expanded polyhedra and tilings: *3.4.n.4*
Symmetry *n32 [n,3]	Spherical				Planar	Hyperbolic...
Symmetry *n32 [n,3]	*232 [2,3] D_3h	*332 [3,3] T_d	*432 [4,3] O_h	*532 [5,3] I_h	*632 [6,3] P6m	*732 [7,3]	*832 [8,3]...	*∞32 [∞,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	3.4.∞.4
Coxeter Schläfli	t_0,2{2,3}	t_0,2{3,3}	t_0,2{4,3}	t_0,2{5,3}	t_0,2{6,3}	t_0,2{7,3}	t_0,2{8,3}	t_0,2{∞,3}
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	V3.4.8.4	V3.4.∞.4
Coxeter

Dimensional family of expanded polyhedra and tilings: *n.4.4.4*
Symmetry [n,4], (*n42)	Spherical	Euclidean	Hyperbolic tiling
Symmetry [n,4], (*n42)	*342 [3,4]	*442 [4,4]	*542 [5,4]	*642 [6,4]	*742 [7,4]	*842 [8,4]	*∞42 [∞,4]
Quasiregular figures
Coxeter Schläfli	t_0,2{3,4}	t_0,2{4,4}	t_0,2{5,4}	t_0,2{6,4}	t_0,2{7,4}	t_0,2{8,4}	t_0,2{∞,4}
Dual (rhombic) figures configuration	V3.4.4.4	V4.4.4.4	5.4.4.4	V6.4.4.4	V7.4.4.4	V8.4.4.4	V∞.4.4.4
Coxeter

Vertex arrangement [edit]

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 [edit]

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]

Games and toys [edit]

Rubik's 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.

Sonic the Hedgehog 3's Icecap Zone features pillars topped with rhombicuboctahedra.

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).

Notes [edit]

References [edit]

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.
de Graaf, J.; van Roij, R.; Dijkstra, M. (2011), "Dense Regular Packings of Irregular Nonconvex Particles", Phys. Rev. Lett. 107: 155501, Bibcode:2011PhRvL.107o5501D, doi:10.1103/PhysRevLett.107.155501
Betke, U.; Henk, M. (2000), "Densest Lattice Packings of 3-Polytopes", Comput. Geom. 16: 157, doi:10.1016/S0925-7721(00)00007-9
Torquato, S.; Jiao, Y. (2009), "Dense packings of the Platonic and Archimedean solids", Nature 460: 876, doi:10.1038/nature08239, PMID 19675649
Hales, Thomas C. (2005), "A proof of the Kepler conjecture", Annals of Mathematics 162: 1065, doi:10.4007/annals.2005.162.1065

External links [edit]

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