Editing Rhombicuboctahedron (section)

== Properties ==
=== Measurement and metric properties ===
The surface area of a rhombicuboctahedron <math> A </math> can be determined by adding the area of all faces: 8 equilateral triangles and 18 squares. The volume of a rhombicuboctahedron <math> V </math> can be determined by slicing it into two square cupolas and one octagonal prism. Given that the edge length <math> a </math>, its surface area and volume is:{{sfnp|Berman|1971|p=336|loc=See table IV, the Properties of regular-faced convex polyhedra, line 13.}}
<math display="block"> \begin{align}
 A &= \left(18+2\sqrt{3}\right)a^2 &\approx 21.464a^2,\\
 V &= \frac{12+10\sqrt{2}}{3}a^3 &\approx 8.714a^3.
\end{align} </math>

The optimal [[Packing density|packing fraction]] of rhombicuboctahedra is given by
<math display="block"> \eta = \frac{4}{3} \left( 4\sqrt{2} - 5 \right). </math>
It was noticed that this optimal value is obtained in a [[Bravais lattice]] by {{harvtxt|de Graaf|van Roij|Dijkstra|2011}}.{{sfnp|de Graaf|van Roij|Dijkstra|2011}} 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.{{cn|date=May 2024}}

The [[dihedral angle]] of a rhombicuboctahedron can be determined by adding the dihedral angle of a square cupola and an octagonal prism:{{sfnp|Johnson|1966}}
* the dihedral angle of a rhombicuboctahedron between two adjacent squares on both the top and bottom is that of a square cupola 135°. The dihedral angle of an octagonal prism between two adjacent squares is the internal angle of a [[regular octagon]] 135°. The dihedral angle between two adjacent squares on the edge where a square cupola is attached to an octagonal prism is the sum of the dihedral angle of a square cupola square-to-octagon and the dihedral angle of an octagonal prism square-to-octagon 45° + 90° = 135°. Therefore, the dihedral angle of a rhombicuboctahedron for every two adjacent squares is 135°.
* the dihedral angle of a rhombicuboctahedron square-to-triangle is that of a square cupola between those 144.7°. The dihedral angle between square-to-triangle, on the edge where a square cupola is attached to an octagonal prism is the sum of the dihedral angle of a square cupola triangle-to-octagon and the dihedral angle of an octagonal prism square-to-octagon 54.7° + 90° = 144.7°. Therefore, the dihedral angle of a rhombicuboctahedron for every square-to-triangle is 144.7°.

A rhombicuboctahedron has the [[Rupert property]], meaning there is a polyhedron with the same or larger size that can pass through its hole.<ref>{{multiref
|{{harvp|Hoffmann|2019}}
|{{harvp|Chai|Yuan|Zamfirescu|2018}}
}}</ref>

=== 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]}}
}}</ref>

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}}
}}</ref>