Editing Rhombicuboctahedron (section)

=== Symmetry and its classification family ===
[[File:Rhombicuboctahedron.stl|thumb|3D model of a rhombicuboctahedron]]
The rhombicuboctahedron has the same symmetry as a cube and regular octahedron, the [[octahedral symmetry]] <math> \mathrm{O}_\mathrm{h} </math>.<ref>{{multiref
|{{harvp|Koca|Koca|2013|p=[https://books.google.com/books?id=ILnBkuSxXGEC&pg=PA48 48]}}
|{{harvp|Cromwell|1997|p=[https://archive.org/details/polyhedra0000crom/page/377/mode/1up 377]}}. See Figure 10.12.
}}</ref> 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 [[pyritohedral symmetry]] <math> \mathrm{T}_\mathrm{h} </math>, so they are invariant under the same rotations as the tetrahedron but different reflections.{{sfnp|Cromwell|1997|p=[https://archive.org/details/polyhedra0000crom/page/386/mode/1up 386]. See Table 10.21, Classes of vertex-transitive polyhedra.}} It is [[centrosymmetric]], meaning its symmetric is interchangeable by the appearance of [[inversion center]]. It is also non-[[Chirality (mathematics)|chiral]]; that is, it is congruent to its own mirror image.<ref>{{multiref
|{{harvp|O'Keeffe|Hyde|2020|p=[https://books.google.com/books?id=_MjPDwAAQBAJ&pg=PA54 54]}}
|{{harvp|Koca|Koca|2013|p=[https://books.google.com/books?id=ILnBkuSxXGEC&pg=PA48 48]}}
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The rhombicuboctahedron is an [[Archimedean solid]], meaning it is a highly symmetric and semi-regular polyhedron, and two or more different [[Regular polygon|regular polygonal]] faces meet in a vertex.{{sfnp|Diudea|2018|p=[https://books.google.com/books?id=p_06DwAAQBAJ&pg=PA39 39]}} The polygonal faces that meet for every vertex are one equilateral triangle and three squares, and the [[vertex figure]] is denoted as <math> 3 \cdot 4^3 </math>. Its dual is [[deltoidal icositetrahedron]], a [[Catalan solid]], shares the same symmetry as the rhombicuboctahedron.{{sfnp|Williams|1979|p=[https://archive.org/details/geometricalfound00will/page/80/mode/1up?view=theater 80]}}

The [[elongated square gyrobicupola]] is the only polyhedron resembling the rhombicuboctahedron. The difference is that the elongated square gyrobicupola is constructed by twisting one of its cupolae. It was once considered as the 14th Archimedean solid, until it was discovered that it is not [[vertex-transitive]], categorizing it as the [[Johnson solid]] instead.<ref>{{multiref
|{{harvp|Cromwell|1997|p=[https://archive.org/details/polyhedra0000crom/page/91/mode/1up 91]}}
|{{harvp|Grünbaum|2009}}
|{{harvp|Lando|Zvonkin|2004}}
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