Cuboctahedron

Cuboctahedron
Type	Archimedean solid
Faces	14
Edges	24
Vertices	12
Schläfli symbol	${\displaystyle {\begin{Bmatrix}4\\3\end{Bmatrix}}}$
Symmetry group	Octahedral symmetry ${\displaystyle \mathrm {O} _{\mathrm {h} }}$ Tetrahedral symmetry ${\displaystyle \mathrm {T} _{\mathrm {d} }}$
Rupert property
Vertex figure
Net

A cuboctahedron is a

• 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.^[2]
• Its construction can be started from 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.^[3]
• An alternative construction is by cutting of all of the vertices, known as
  regular tetrahedron, cutting off the vertices and beveling the edges. This process known as cantellation, making the cuboctahedron being named cantellated tetrahedron.^[4]

From all of these constructions, the cuboctahedron has 14 faces: 8 equilateral triangles and 6 squares. It also has 24 edges and 12 vertices.[5]

The

${\displaystyle {\sqrt {2}}}$

$${\displaystyle (\pm 1,\pm 1,0),\qquad (\pm 1,0,\pm 1),\qquad (0,\pm 1,\pm 1).}$$

Properties

Measurement and other metric properties

The surface area of a cuboctahedron ${\displaystyle A}$ can be determined by summing all the area of its polygonal faces. The volume of a cuboctahedron ${\displaystyle V}$ can be determined by slicing it off into two regular triangular cupolas, summing up their volume. Given that the edge length ${\displaystyle a}$, its surface area and volume are:^[5]

$${\displaystyle {\begin{aligned}A&=\left(6+2{\sqrt {3}}\right)a^{2}&&\approx 9.464a^{2}\\V&={\frac {5{\sqrt {2}}}{3}}a^{3}&&\approx 2.357a^{3}.\end{aligned}}}$$

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°.^[7]

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.^[11] 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 ${\displaystyle \mathrm {O} _{\mathrm {h} }}$, and the same symmetry as the regular tetrahedron, tetrahedral symmetry ${\displaystyle \mathrm {T} _{\mathrm {d} }}$.^[12] The polygonal faces that meet for every vertex are two equilateral triangles and two squares, and the vertex figure of a cuboctahedron is ${\displaystyle (3\cdot 4)^{2}=3^{2}\cdot 4^{2}}$. The dual of a cuboctahedron is rhombic dodecahedron.^[13]

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. 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

The cuboctahedron, cubohemioctahedron, and octahemioctahedron.

The cuboctahedron shares its

The cuboctahedron 2-covers the tetrahemihexahedron, which accordingly has the same abstract vertex figure (two triangles and two squares: ${\displaystyle 3\cdot 4\cdot 3\cdot 4}$) and half the vertices, edges, and faces. (The actual vertex figure of the tetrahemihexahedron is ${\textstyle 3\cdot 4\cdot {\frac {3}{2}}\cdot 4}$, with the ${\textstyle {\frac {a}{2}}}$ factor due to the cross.)^[15]

The cuboctahedron can be dissected into 6

The skeleton of a cuboctahedron may be represented as the graph, one of the Archimedean graph. It has 12 vertices and 24 edges. It has 12 vertices and 24 edges. It is quartic graph, which is four vertices connecting each vertex.^[17]

The graph of a cuboctahedron may be constructed as the line graph of the cube, making it becomes the locally linear graph.^[18]

Appearance

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.^[19]

References

Footnotes

Works cited