Ромбикубооктаэдр: 26 қырлы геометриялық дене

Archimedean solid with 26 faces

In geometry, the rhombicuboctahedron, or small rhombicuboctahedron, is a polyhedron with eight triangular, six square, and twelve rectangular faces. There are 24 identical vertices, with one triangle, one square, and two rectangles meeting at each one. If all the rectangles are themselves square (equivalently, all the edges are the same length, ensuring the triangles are equilateral), it is an Archimedean solid. 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.

Johannes Kepler in his 1618 Harmonices Mundi named this polyhedron a rhombicuboctahedron, being short for truncated cuboctahedral rhombus, with cuboctahedral rhombus being his name for a rhombic dodecahedron. There are different truncations of a rhombic dodecahedron into a topological rhombicuboctahedron: Prominently its rectification (left), the one that creates the uniform solid (center), and the rectification of the dual cuboctahedron (right), which is the core of the dual compound. It can also be called an expanded or cantellated cube or octahedron, from truncation operations on either uniform polyhedron.

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.

A half symmetry form of the rhombicuboctahedron, , exists with pyritohedral symmetry, [4,3+], (3*2) as Coxeter diagram , 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 an icosahedron is formed, by a snub octahedron construction, , s{3,4}. (The compound of two icosahedra is constructed from both alternated positions.) Pyritohedral symmetry variationsUniform geometryNonuniform geometryNonuniform geometryIn the limit, an icosahedronsnub octahedron, ,from one of two positions. Compound of two icosahedrafrom both alternated positions.

The 1495 Portrait of Luca Pacioli, traditionally attributed to Jacopo de' Barbari, includes a glass rhombicuboctahedron half filled with water, which may have been painted by Leonardo da Vinci. The first printed version of the rhombicuboctahedron was by Leonardo and appeared in Pacioli's Divina proportione (1509). 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.

During the Rubik's Cube craze of the 1980s, at least two twisty puzzles sold had the form of a rhombicuboctahedron (the mechanism was similar to that of a Rubik's Cube).