In geometry, the rhombicuboctahedron is an Archimedean solid with 26 faces, consisting of 8 equilateral triangles and 18 squares.

The elongated square gyrobicupola is a polyhedron that is similar to a rhombicuboctahedron, but it is not an Archimedean solid because it is not vertex-transitive.

The rhombicuboctahedron is found in diverse cultures in architecture, toys, the arts, and elsewhere.

This is because the rhombicuboctahedron may also be constructed by separating and pushing away the faces of a cube or a regular octahedron from their centroid (in blue or red, respectively, in the animation), and filling between them with the squares and equilateral triangles.

[4] Relatedly, the rhombicuboctahedron may also be constructed by cutting all edges and vertices of either cube or a regular octahedron, a process known as rectification.

can be determined by adding the area of all faces: 8 equilateral triangles and 18 squares.

It was noticed that this optimal value is obtained in a Bravais lattice by de Graaf, van Roij & Dijkstra (2011).

[8] Since the rhombicuboctahedron is contained in a rhombic dodecahedron whose inscribed sphere is identical to its inscribed sphere, the value of the optimal packing fraction is a corollary of the Kepler conjecture: it can be achieved by putting a rhombicuboctahedron in each cell of the rhombic dodecahedral honeycomb, and it cannot be surpassed, since otherwise the optimal packing density of spheres could be surpassed by putting a sphere in each rhombicuboctahedron of the hypothetical packing which surpasses it.

[citation needed] The dihedral angle of a rhombicuboctahedron can be determined by adding the dihedral angle of a square cupola and an octagonal prism:[9] A rhombicuboctahedron has the Rupert property, meaning there is a polyhedron with the same or larger size that can pass through its hole.

[12] It is centrosymmetric, meaning its symmetric is interchangeable by the appearance of inversion center.

[13] The rhombicuboctahedron is an Archimedean solid, meaning it is a highly symmetric and semi-regular polyhedron, and two or more different regular polygonal faces meet in a vertex.

Its dual is deltoidal icositetrahedron, a Catalan solid, shares the same symmetry as the rhombicuboctahedron.

The difference is that the elongated square gyrobicupola is constructed by twisting one of its cupolae.

In other words, the edges of a graph are not crossed while being drawn, and removing any two of its vertices leaves a connected subgraph.

[17] The rhombicuboctahedron sometimes appears in architecture, with an example being the building of the National Library located at Minsk.

[18] The Wilson House by Bruce Goff is another example of a rhombicuboctahedral building, although its module was depicted as a truncated cube in which the edges are all cut off.

For example, if the lines along which a Rubik's Cube can be turned are projected onto a sphere, they are topologically identical to a rhombicuboctahedron's edges.

Variants using the Rubik's Cube mechanism have been produced, which closely resemble the rhombicuboctahedron.

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)[20][21] Another example may be found in dice from Corfe Castle, each of whose square faces have marks of pairs of letters and pips.

An example is the 1495 Portrait of Luca Pacioli, traditionally attributed to Jacopo de' Barbari, which includes a glass rhombicuboctahedron half-filled with water, which may have been painted by Leonardo da Vinci.

[23] The first printed version of the rhombicuboctahedron was by Leonardo da Vinci and appeared in Pacioli's Divina proportione (1509).