Cuboctahedron

An Archimedean Solid (also called the Dymaxion or Heptaparallelohedron) whose Dual is the Rhombic Dodecahedron. It is one of the two convex Quasiregular Polyhedra and has Schläfli Symbol $\left\{{3\atop 4}\right\}$. It is also Uniform Polyhedron $U_7$ and has Wythoff Symbol $2\,\vert\,3\,4$. Its faces are $8\{3\}+6\{4\}$. It has the $O_h$ Octahedral Group of symmetries.

The Vertices of a cuboctahedron with Edge length of $\sqrt{2}$ are $(0, \pm 1, \pm 1)$, $(\pm 1, 0, \pm 1)$, and $(\pm 1, \pm 1, 0)$. The Inradius, Midradius, and Circumradius for $a=1$ are

$$r = {\textstyle{3\over 4}} = 0.75$$

$$\rho = {\textstyle{1\over 2}}\sqrt{3} \approx 0.86602$$

$$R = 1.$$

Faceted versions include the Cubohemioctahedron and Octahemioctahedron.

The solid common to both the Cube and Octahedron (left figure) in a Cube-Octahedron Compound is a Cuboctahedron (right figure; Ball and Coxeter 1987).

See also Archimedean Solid, Cube, Cube-Octahedron Compound, Cubohemioctahedron, Octahedron, Octahemioctahedron, Quasiregular Polyhedron, Rhombic Dodecahedron, Rhombus

References

Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recreations and Essays, 13th ed. New York: Dover, p. 137, 1987.

Ghyka, M. The Geometry of Art and Life. New York: Dover, p. 54, 1977.