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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.

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

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, 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
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

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.

Orthogonal projections
Centered by Vertex Edge
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
[2] [2] [2] [2] [4] [6]

Pyritohedral symmetry

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}.

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