Editing Rhombicuboctahedron (section)

== Construction ==
The rhombicuboctahedron may be constructed from a [[Cube (geometry)|cube]] by drawing a smaller one in the middle of each face, parallel to the cube's edges. After removing the edges of a cube, the squares may be joined by adding more squares adjacent between them, and the corners may be filled by the [[equilateral triangle]]s. Another way to construct the rhombicuboctahedron is by attaching two regular [[square cupola]]s into the bases of a regular [[octagonal prism]].<ref>{{multiref
|{{harvp|Hartshorne|2000|p=[https://books.google.com/books?id=EJCSL9S6la0C&pg=PA463 463]}}
|{{harvp|Berman|1971|p=336|loc=See table IV, the Properties of regular-faced convex polyhedra, line 13. Here, <math> P_8 </math> represents the octagonal prism and <math> M_5 </math> represents the square cupola.}}
}}</ref>

[[File:P2-A5-P3.gif|left|thumb|Process of expanding the rhombicuboctahedron.]]
A rhombicuboctahedron may also be known as an ''expanded octahedron'' or ''expanded cube''. 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. This construction process is known as [[Expansion (geometry)|expansion]].{{sfnp|Viana|Xavier|Aires|Campos|2019|p=1123|loc=See Fig. 6}} By using all of these methods above, the rhombicuboctahedron has 8 equilateral triangles and 16 squares as its faces.<ref>{{multiref
|{{harvp|Cockram|2020|p=[https://books.google.com/books?id=jrITEAAAQBAJ&pg=PA52 52]}}
|{{harvp|Berman|1971|p=336|loc=See table IV, the Properties of regular-faced convex polyhedra, line 13.}}
}}</ref> Relatedly, the rhombicuboctahedron may also be constructed by cutting all edges and vertices of either cube or a regular octahedron, a process known as [[Cantellation (geometry)|cantellation]].{{sfnp|Linti|2013|p=[https://books.google.com/books?id=_4C7oid1kQQC&pg=RA7-PA41 41]}}

[[Cartesian coordinate]]s of a rhombicuboctahedron with an edge length 2 are the permutations of <math> \left(\pm \left(1 + \sqrt{2}\right), \pm 1, \pm 1 \right)</math>. {{sfnp|Shepherd|1954}}