cuboctahedron

The cuboctahedron can be constructed in many ways:

• Its construction can be started by attaching two regular triangular cupolas base-to-base. This is similar to one of the Johnson solids, triangular orthobicupola. The difference is that the triangular orthobicupola is constructed with one of the cupolas twisted so that similar polygonal faces are adjacent, whereas the cuboctahedron is not. As a result, the cuboctahedron may also called the triangular gyrobicupola.{{multiref

|{{harvnb|Berman|1971}}

|{{harvnb|Ogievetsky|Shlosman|2021|p=[https://books.google.com/books?id=UsspEAAAQBAJ&pg=PA477 477]}}

}}

• Its construction can be started from a cube or a regular octahedron, marking the midpoints of their edges, and cutting off all the vertices at those points. This process is known as rectification, making the cuboctahedron being named the rectified cube and rectified octahedron.{{sfn|van Leeuwen|Freixa|Cano|2023|p=[https://books.google.com/books?id=8S3nEAAAQBAJ&pg=PA50 50]}}
• An alternative construction is by cutting off all vertices (truncation) of a regular tetrahedron and beveling the edges. This process is termed cantellation, lending the cuboctahedron an alternate name of cantellated tetrahedron.{{sfn|Linti|2013|p=[https://books.google.com/books?id=_4C7oid1kQQC&pg=RA7-PA41 41]}}

From all of these constructions, the cuboctahedron has 14 faces: 8 equilateral triangles and 6 squares. It also has 24 edges and 12 vertices.{{sfn|Berman|1971}}

The Cartesian coordinates for the vertices of a cuboctahedron with edge length $\backslash sqrt\{2\}$ centered at the origin are:{{sfn|Coxeter|1973|p=52|loc=§3.7 Coordinates for the vertices of the regular and quasi-regular solids}}

$$(\backslash pm\; 1,\; \backslash pm\; 1,\; 0),\; \backslash qquad\; (\backslash pm\; 1,\; 0,\; \backslash pm\; 1),\; \backslash qquad\; (0,\; \backslash pm\; 1,\; \backslash pm\; 1).$$

The surface area of a cuboctahedron $A$ can be determined by summing all the area of its polygonal faces. The volume of a cuboctahedron $V$ can be determined by slicing it off into two regular triangular cupolas, summing up their volume. Given that the edge length $a$, its surface area and volume are:{{sfn|Berman|1971}}

$$\backslash begin\{align\}$$

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}

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]]

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}}

File:Cuboctahedron.stl

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 $\backslash mathrm\{O\}\_\backslash mathrm\{h\}$, and the same symmetry as the regular tetrahedron, tetrahedral symmetry $\backslash mathrm\{T\}\_\backslash mathrm\{d\}$.{{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.

}} 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]}}

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 8-cell (tesseract).{{Sfn|Coxeter|1973|pp=292-293|loc=Table I (ii): column ₀R/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.

{{more citations needed section|date=March 2025}}

The cuboctahedron can be represented as a configuration matrix with elements grouped by symmetry transitivity classes. A configuration matrix is a matrix in which the rows and columns correspond to the elements of a polyhedron as in the vertices, edges, and faces. The diagonal of a matrix denotes the number of each element that appears in a polyhedron, whereas the non-diagonal of a matrix denotes the number of the column's elements that occur in or at the row's element.

The cuboctahedron has 1 transitivity class of 12 vertices, 1 class of 24 edges, and 2 classes of faces: 8 triangular and 6 square; each element in a matrix's diagonal.{{cite web | url=https://www.bendwavy.org/klitzing/incmats/co.htm | title=Co }} The 24 edges can be seen in 4 central hexagons.

With octahedral symmetry (orbifold 432), the squares have the 4-fold symmetry, triangles the 3-fold symmetry, and vertices the 2-fold symmetry. With tetrahedral symmetry (orbifold 332) the 24 vertices split into 2 edge classes, and the 8 triangles split into 2 face classes. The square symmetry is reduced to 2-fold.

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The skeleton of a cuboctahedron may be represented as the graph, one of the Archimedean graph. It has 12 vertices and 24 edges. It is quartic graph, which is four vertices connecting each vertex.{{sfn|Read|Wilson|1998|p=269}}

The graph of a cuboctahedron may be constructed as the line graph of the cubical graph, making it becomes the locally linear graph.{{sfn|Fan|1996}}

The 24 edges can be partitioned into 2 sets isomorphic to tetrahedral symmetry. The edges can also be partitioned into 4 hexagonal cycles, representing centrosymmetry, with only opposite vertices and edges in the same transitivity class.{{citation needed|date=March 2025}}

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{{multiple image

| image1 = cuboctahedron.png

| image2 = cubohemioctahedron.png

| image3 = octahemioctahedron.png

| footer = The cuboctahedron, cubohemioctahedron, and octahemioctahedron.

| total_width = 400

}}

The cuboctahedron shares its skeleton with the two nonconvex uniform polyhedra, the cubohemioctahedron and octahemioctahedron. These polyhedrons are constructed from the skeleton of a cuboctahedron in which the four hexagonal planes bisect its diagonal, intersecting its interior. Adding six squares or eight equilateral triangles results in the cubohemicotahedron or octahemioctahedron, respectively.{{multiref

|{{harvnb|Pisanski|Servatius|2013|p=[https://books.google.com/books?id=3vnEcMCx0HkC&pg=PA108 108]}}

|{{harvnb|Barnes|2012|p=[https://books.google.com/books?id=BQhEAAAAQBAJ&pg=PA53 53]}}

}}

The cuboctahedron 2-covers the tetrahemihexahedron, which accordingly has the same abstract vertex figure (two triangles and two squares: $3\; \backslash cdot\; 4\; \backslash cdot\; 3\; \backslash cdot\; 4$) and half the vertices, edges, and faces. (The actual vertex figure of the tetrahemihexahedron is $3\; \backslash cdot\; 4\; \backslash cdot\; \backslash frac\{3\}\{2\}\; \backslash cdot\; 4$, with the $\backslash frac\{a\}\{2\}$ factor due to the cross.){{sfn|Grünbaum|2003|p=[https://books.google.com/books?id=WoaxgpHu19gC&pg=PA338 338]}}

File:TetraOctaHoneycomb-VertexConfig.svg

The cuboctahedron can be dissected into 6 square pyramids and 8 tetrahedra meeting at a central point. This dissection is expressed in the tetrahedral-octahedral honeycomb where pairs of square pyramids are combined into octahedra.{{sfn|Posamentier|Thaller|Dorner|Geretschläger|2022|p=[https://books.google.com/books?id=DGxYEAAAQBAJ&pg=PA234 233–235]}}

The cuboctahedron was probably known to Plato: Heron's Definitiones quotes Archimedes as saying that Plato knew of a solid made of 8 triangles and 6 squares.{{sfn|Turnball|1931}}

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{{refend}}

• [http://www.mathconsult.ch/showroom/unipoly/ The Uniform Polyhedra]
• [http://www.georgehart.com/virtual-polyhedra/vp.html Virtual Reality Polyhedra] The Encyclopedia of Polyhedra
• {{mathworld2 |urlname=Cuboctahedron |title=Cuboctahedron |urlname2=ArchimedeanSolid |title2=Archimedean solid}}
• [https://hexnet.org/content/cuboctahedron The Cuboctahedron] on [https://hexnet.org Hexnet] a website devoted to hexagon mathematics.
• {{KlitzingPolytopes|polyhedra.htm|3D convex uniform polyhedra|o3x4o - co}}
• [http://www.dr-mikes-math-games-for-kids.com/polyhedral-nets.html?net=dYI7PStK037OedkFHPiwHK2fbnxjxylGZCjWfNh0UwMgy82zEaWFzVL3PBfYB9SDz8RMvhNkpb8sS9R&name=Cuboctahedron#applet Editable printable net of a Cuboctahedron with interactive 3D view]

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