Editing Cuboctahedron (section)

== Properties ==
=== Measurement and other metric properties ===
The surface area of a cuboctahedron <math> A </math> can be determined by summing all the area of its polygonal faces. The volume of a cuboctahedron <math> V </math> can be determined by slicing it off into two regular triangular cupolas, summing up their volume. Given that the edge length <math> a </math>, its surface area and volume are:{{sfn|Berman|1971}}
<math display="block"> \begin{align}
A &= \left(6+2\sqrt{3}\right)a^2 &&\approx 9.464a^2 \\
V &= \frac{5 \sqrt{2}}{3} a^3 &&\approx 2.357a^3.
\end{align}</math>

The dihedral angle of a cuboctahedron can be calculated with the angle of triangular cupolas. The dihedral angle of a triangular cupola between square-to-triangle is approximately 125°, that between square-to-hexagon is 54.7°, and that between triangle-to-hexagon is 70.5°. Therefore, the dihedral angle of a cuboctahedron between square-to-triangle, on the edge where the base of two triangular cupolas are attached is 54.7° + 70.5° approximately 125°. Therefore, the dihedral angle of a cuboctahedron between square-to-triangle is approximately 125°.{{sfn|Johnson|1966}}

[[File:A3-P5-P3.gif|thumb|The process of ''[[jitterbug transformation]]'']]
[[Buckminster Fuller]] found that the cuboctahedron is the only polyhedron in which the distance between its center to the vertex is the same as the distance between its edges. In other words, it has the same length vectors in three-dimensional space, known as ''vector equilibrium''.{{sfn|Cockram|2020|p=[https://books.google.com/books?id=jrITEAAAQBAJ&pg=PA53 53]}} The rigid struts and the flexible vertices of a cuboctahedron may also be transformed progressively into a [[regular icosahedron]], regular octahedron, regular tetrahedron. Fuller named this the ''[[jitterbug transformation]]''.{{sfn|Verheyen|1989}}

A cuboctahedron has the [[Rupert property]], meaning there is a polyhedron of the same or larger size that can pass through its hole.{{sfn|Chai|Yuan|Zamfirescu|2018}}

=== Symmetry and classification ===
[[File:Cuboctahedron.stl|thumb|3D model of a cuboctahedron]]
The cuboctahedron 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.{{sfn|Diudea|2018|p=[https://books.google.com/books?id=p_06DwAAQBAJ&pg=PA39 39]}} The cuboctahedron has two symmetries, resulting from the constructions as has mentioned above: the same symmetry as the regular octahedron or cube, the [[octahedral symmetry]] <math> \mathrm{O}_\mathrm{h} </math>, and the same symmetry as the regular tetrahedron, [[tetrahedral symmetry]] <math> \mathrm{T}_\mathrm{d} </math>.<ref>{{multiref
|{{harvp|Koca|Koca|2013|p=[https://books.google.com/books?id=ILnBkuSxXGEC&pg=PA48 48]}}
|{{harvp|Cromwell|1997}}. For octahedral symmetry, see [https://archive.org/details/polyhedra0000crom/page/378/mode/1up p. 378], Figure 10.13. For tetrahedral symmetry, see [https://archive.org/details/polyhedra0000crom/page/380/mode/1up p. 380], Figure 10.15.
}}</ref> The polygonal faces that meet for every vertex are two equilateral triangles and two squares, and the [[vertex figure]] of a cuboctahedron is 3.4.3.4. The dual of a cuboctahedron is [[rhombic dodecahedron]].{{sfn|Williams|1979|p=[https://archive.org/details/geometricalfound00will/page/74/mode/1up?view=theater 74]}}

=== Radial equilateral symmetry ===
In a cuboctahedron, the long radius (center to vertex) is the same as the edge length; thus its long diameter (vertex to opposite vertex) is 2 edge lengths.{{Sfn|Coxeter|1973|p=69|loc=§4.7 Other honeycombs}} Its center is like the apical vertex of a canonical pyramid: one edge length away from ''all'' the other vertices. (In the case of the cuboctahedron, the center is in fact the apex of 6 square and 8 triangular pyramids). This radial equilateral symmetry is a property of only a few uniform [[polytopes]], including the two-dimensional [[hexagon]], the three-dimensional cuboctahedron, and the four-dimensional [[24-cell]] and [[tesseract|8-cell (tesseract)]].{{Sfn|Coxeter|1973|pp=292-293|loc=Table I (ii): column ''<sub>0</sub>R/l''}} ''Radially equilateral'' polytopes are those that can be constructed, with their long radii, from equilateral triangles which meet at the center of the polytope, each contributing two radii and an edge. Therefore, all the interior elements which meet at the center of these polytopes have equilateral triangle inward faces, as in the dissection of the cuboctahedron into 6 square pyramids and 8 tetrahedra. 

Each of these radially equilateral polytopes also occurs as cells of a characteristic space-filling [[tessellation]]: the tiling of regular hexagons, the [[rectified cubic honeycomb]] (of alternating cuboctahedra and octahedra), the [[24-cell honeycomb]] and the [[tesseractic honeycomb]], respectively.{{Sfn|Coxeter|1973|p=296|loc=Table II: Regular Honeycombs}} Each tessellation has a [[dual tessellation]]; the cell centers in a tessellation are cell vertices in its dual tessellation. The densest known regular [[sphere-packing]] in two, three and four dimensions uses the cell centers of one of these tessellations as sphere centers.

Because it is radially equilateral, the cuboctahedron's center is one edge length distant from the 12 vertices.