Small Rhombicuboctahedron

An Archimedean Solid also (inappropriately) called the Truncated Icosidodecahedron. This name is inappropriate since truncation would yield rectangular instead of square faces. Its Dual Polyhedron is the Deltoidal Icositetrahedron, also called the Trapezoidal Icositetrahedron. It has Schläfli Symbol r $\left\{{3\atop 4}\right\}$. It is also Uniform Polyhedron $U_{10}$ and has Wythoff Symbol $3\,4\,\vert\,2$. Its Inradius, Midradius, and Circumradius for $a=1$ are

$$r = {\textstyle{1\over 17}}(6+\sqrt{2}\,)\sqrt{5+2\sqrt{2}}=1.22026\ldots$$

$$\rho = {\textstyle{1\over 2}}\sqrt{4+2\sqrt{2}}=1.30656\ldots$$

$$R = {\textstyle{1\over 2}}\sqrt{5+2\sqrt{2}}=1.39896\ldots.$$

A version in which the top and bottom halves are rotated with respect to each other is known as the Elongated Square Gyrobicupola.

See also Elongated Square Gyrobicupola, Great Rhombicuboctahedron (Archimedean), Great Rhombicuboctahedron (Uniform)

References

Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recreations and Essays, 13th ed. New York: Dover, pp. 137-138, 1987.