Rhombicuboctahedron

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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 symbols rr{4,3}
t0,2{4,3}
Wythoff symbol 3 4 | 2
Coxeter diagram CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.png
Symmetry group Oh, 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
Small rhombicuboctahedron.png
Colored faces
Rhombicuboctahedron
3.4.4.4
(Vertex figure)
Deltoidalicositetrahedron.jpg
Deltoidal icositetrahedron
(dual polyhedron)
Rhombicuboctahedron flat.png
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. In the book "De divina proportione", this shape was given the Latin name "Vigintisexbasium Planum Vacuum" meaning regular solid with twenty-six faces.[1]

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

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.

Small rhombicuboctahedron.png
Rhombicuboctahedron
Pseudorhombicuboctahedron.png
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 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.

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 Cube t02 v.png Cube t02 e34.png Cube t02 e44.png Cube t02 f4b.png 3-cube t02 B2.svg 3-cube t02.svg
Projective
symmetry
[2] [2] [2] [2] [4] [6]

Pyritohedral symmetry[edit]

A half symmetry form of the rhombicuboctahedron, CDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node 1.png, exists with pyritohedral symmetry, [4,3+], (3*2) as Coxeter diagram CDel node h.pngCDel 3.pngCDel node h.pngCDel 4.pngCDel node 1.png, Schläfli symbol s2{3,4}, and can be called a cantic snub octahedron. This form can be visualized by alternatingly coloring the edges of the 6 squares. These squares can then be distorted into rectangles, while the 8 triangles remain equilateral. The 12 diagonal square faces will become isosceles trapezoids. In the limit, the rectangles can be reduced to edges, and the trapezoids become triangles, and a icosahedron is formed, by a snub octahedron construction, CDel node h.pngCDel 3.pngCDel node h.pngCDel 4.pngCDel node.png, s{3,4}.

Rhombicuboctahedron uniform edge coloring.png
Uniform geometry
CDel node h.pngCDel 3.pngCDel node h.pngCDel 4.pngCDel node 1.png
Cantic snub octahedron.png
Nonuniform geometry
Rhombicuboctahedron pyritohedral.png
Nonuniform geometry
Rhombicuboctahedron pyritohedral2.png
In the limit, an icosahedron
Snub octahedron, CDel node h.pngCDel 3.pngCDel node h.pngCDel 4.pngCDel node.png
Rhombicuboctahedron pyritohedral compound.png
Compound of two icosahedra
from alternate positions

Algebraic properties[edit]

Cartesian coordinates[edit]

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

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

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

In the arts[edit]

Pacioli.jpg
Rhombicuboctahedron in top left of 1495 Portrait of Luca Pacioli.[2]
Leonardo polyhedra.png
Leonardo da Vinci's rhombicuboctahedron from Divina Proportione, 1509.

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 1509 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.[3]

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).

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] = [3,3]
(*332)
[3+,4]
(3*2)
{4,3} t{4,3} r{4,3}
r{31,1}
t{3,4}
t{31,1}
{3,4}
{31,1}
rr{4,3}
s2{3,4}
tr{4,3} sr{4,3} h{4,3}
{3,3}
h2{4,3}
t{3,3}
s{3,4}
s{31,1}
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.png CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.png CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.png CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.png CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.png CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.png CDel node h.pngCDel 4.pngCDel node h.pngCDel 3.pngCDel node h.png CDel node h.pngCDel 3.pngCDel node h.pngCDel 4.pngCDel node.png
CDel node h0.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.png
= CDel nodes 11.pngCDel split2.pngCDel node.png
CDel node h0.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.png
= CDel nodes 11.pngCDel split2.pngCDel node 1.png
CDel node h0.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.png
= CDel nodes.pngCDel split2.pngCDel node 1.png
CDel node 1.pngCDel 4.pngCDel node h.pngCDel 3.pngCDel node h.png CDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png =
CDel nodes 10ru.pngCDel split2.pngCDel node.png or CDel nodes 01rd.pngCDel split2.pngCDel node.png
CDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.png =
CDel nodes 10ru.pngCDel split2.pngCDel node 1.png or CDel nodes 01rd.pngCDel split2.pngCDel node 1.png
CDel node h.pngCDel 3.pngCDel node h.pngCDel 4.pngCDel node h0.png =
CDel node h.pngCDel split1.pngCDel nodes hh.png
Uniform polyhedron-43-t0.svg Uniform polyhedron-43-t01.svg Uniform polyhedron-43-t1.svg
Uniform polyhedron-33-t02.png
Uniform polyhedron-43-t12.svg
Uniform polyhedron-33-t012.png
Uniform polyhedron-43-t2.svg
Uniform polyhedron-33-t1.png
Uniform polyhedron-43-t02.png
Rhombicuboctahedron uniform edge coloring.png
Uniform polyhedron-43-t012.png Uniform polyhedron-43-s012.png Uniform polyhedron-33-t0.pngUniform polyhedron-33-t2.png Uniform polyhedron-33-t01.pngUniform polyhedron-33-t12.png Uniform polyhedron-43-h01.svg
Uniform polyhedron-33-s012.png
Duals to uniform polyhedra
V43 V3.82 V(3.4)2 V4.62 V34 V3.43 V4.6.8 V34.4 V33 V3.62 V35
CDel node f1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png CDel node f1.pngCDel 4.pngCDel node f1.pngCDel 3.pngCDel node.png CDel node.pngCDel 4.pngCDel node f1.pngCDel 3.pngCDel node.png CDel node.pngCDel 4.pngCDel node f1.pngCDel 3.pngCDel node f1.png CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node f1.png CDel node f1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node f1.png CDel node f1.pngCDel 4.pngCDel node f1.pngCDel 3.pngCDel node f1.png CDel node fh.pngCDel 4.pngCDel node fh.pngCDel 3.pngCDel node fh.png CDel node fh.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png CDel node fh.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node f1.png CDel node fh.pngCDel 3.pngCDel node fh.pngCDel 4.pngCDel node.png
CDel node f1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node f1.png CDel node f1.pngCDel 3.pngCDel node f1.pngCDel 3.pngCDel node f1.png CDel node.pngCDel 3.pngCDel node f1.pngCDel 3.pngCDel node.png CDel node f1.pngCDel 4.pngCDel node fh.pngCDel 3.pngCDel node fh.png CDel node f1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png CDel node.pngCDel 3.pngCDel node f1.pngCDel 3.pngCDel node f1.png CDel node fh.pngCDel 3.pngCDel node fh.pngCDel 3.pngCDel node fh.png
Octahedron.svg Triakisoctahedron.jpg Rhombicdodecahedron.jpg Tetrakishexahedron.jpg Hexahedron.svg Deltoidalicositetrahedron.jpg Disdyakisdodecahedron.jpg Pentagonalicositetrahedronccw.jpg Tetrahedron.svg Triakistetrahedron.jpg POV-Ray-Dodecahedron.svg

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 Euclidean Compact hyperbolic Paracompact
*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]...
 
*∞32
[∞,3]
 
Expanded
figure
Spherical triangular prism.png
3.4.2.4
Uniform tiling 332-t02.png
3.4.3.4
Uniform tiling 432-t02.png
3.4.4.4
Uniform tiling 532-t02.png
3.4.5.4
Uniform polyhedron-63-t02.png
3.4.6.4
Uniform tiling 73-t02.png
3.4.7.4
Uniform tiling 83-t02.png
3.4.8.4
H2 tiling 23i-5.png
3.4.∞.4
Coxeter
Schläfli
CDel node 1.pngCDel 2.pngCDel node.pngCDel 3.pngCDel node 1.png
rr{2,3}
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
rr{3,3}
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.png
rr{4,3}
CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node 1.png
rr{5,3}
CDel node 1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node 1.png
rr{6,3}
CDel node 1.pngCDel 7.pngCDel node.pngCDel 3.pngCDel node 1.png
rr{7,3}
CDel node 1.pngCDel 8.pngCDel node.pngCDel 3.pngCDel node 1.png
rr{8,3}
CDel node 1.pngCDel infin.pngCDel node.pngCDel 3.pngCDel node 1.png
rr{∞,3}
Deltoidal figure Triangular dipyramid.png
V3.4.2.4
Rhombicdodecahedron.jpg
V3.4.3.4
Deltoidalicositetrahedron.jpg
V3.4.4.4
Deltoidalhexecontahedron.jpg
V3.4.5.4
Tiling Dual Semiregular V3-4-6-4 Deltoidal Trihexagonal.svg
V3.4.6.4
Deltoidal triheptagonal til.png
V3.4.7.4
Deltoidal trioctagonal til.png
V3.4.8.4
Deltoidal triapeirogonal til.png
V3.4.∞.4
Coxeter CDel node f1.pngCDel 2.pngCDel node.pngCDel 3.pngCDel node f1.png CDel node f1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node f1.png CDel node f1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node f1.png CDel node f1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node f1.png CDel node f1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node f1.png CDel node f1.pngCDel 7.pngCDel node.pngCDel 3.pngCDel node f1.png CDel node f1.pngCDel 8.pngCDel node.pngCDel 3.pngCDel node f1.png CDel node f1.pngCDel infin.pngCDel node.pngCDel 3.pngCDel node f1.png
Dimensional family of expanded polyhedra and tilings: n.4.4.4
Symmetry
[n,4], (*n42)
Spherical Euclidean Compact hyperbolic Paracompact
*342
[3,4]
*442
[4,4]
*542
[5,4]
*642
[6,4]
*742
[7,4]
*842
[8,4]
*∞42
[∞,4]
Expanded
figures
Uniform polyhedron-43-t02.png Uniform tiling 44-t02.png Uniform tiling 54-t02.png Uniform tiling 64-t02.png Uniform tiling 74-t02.png Uniform tiling 84-t02.png H2 tiling 24i-5.png
Coxeter
Schläfli
CDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node 1.png
rr{3,4}
CDel node 1.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node 1.png
rr{4,4}
CDel node 1.pngCDel 5.pngCDel node.pngCDel 4.pngCDel node 1.png
rr{5,4}
CDel node 1.pngCDel 6.pngCDel node.pngCDel 4.pngCDel node 1.png
rr{6,4}
CDel node 1.pngCDel 7.pngCDel node.pngCDel 4.pngCDel node 1.png
rr{7,4}
CDel node 1.pngCDel 8.pngCDel node.pngCDel 4.pngCDel node 1.png
rr{8,4}
CDel node 1.pngCDel infin.pngCDel node.pngCDel 4.pngCDel node 1.png
rr{∞,4}
Dual
(rhombic)
figures
configuration
Deltoidalicositetrahedron.jpg
V3.4.4.4
Uniform tiling 44-t1.png
V4.4.4.4
Deltoidal tetrapentagonal tiling.png
5.4.4.4
Deltoidal tetrahexagonal til.png
V6.4.4.4
Deltoidal tetraheptagonal til.png
V7.4.4.4
Deltoidal tetraoctagonal til.png
V8.4.4.4
Deltoidal tetraapeirogonal tiling.png
V∞.4.4.4
Coxeter CDel node f1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node f1.png CDel node f1.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node f1.png CDel node f1.pngCDel 5.pngCDel node.pngCDel 4.pngCDel node f1.png CDel node f1.pngCDel 6.pngCDel node.pngCDel 4.pngCDel node f1.png CDel node f1.pngCDel 7.pngCDel node.pngCDel 4.pngCDel node f1.png CDel node f1.pngCDel 8.pngCDel node.pngCDel 4.pngCDel node f1.png CDel node f1.pngCDel infin.pngCDel node.pngCDel 4.pngCDel node f1.png

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).

Small rhombicuboctahedron.png
Rhombicuboctahedron
Small cubicuboctahedron.png
Small cubicuboctahedron
Small rhombihexahedron.png
Small rhombihexahedron
Stellated truncated hexahedron.png
Stellated truncated hexahedron

See also[edit]

Notes[edit]

  1. ^ Da divina proportione, page XXXVI
  2. ^ RitrattoPacioli.it
  3. ^ Philosphere

References[edit]

External links[edit]