Cubitruncated cuboctahedron

In geometry, the cubitruncated cuboctahedron or cuboctatruncated cuboctahedron is a nonconvex uniform polyhedron, indexed as U16. It has 20 faces (8 hexagons, 6 octagons, and 6 octagrams), 72 edges, and 48 vertices, and has a shäfli symbol of tr{4,3/2}

Convex hull
Its convex hull is a nonuniform truncated cuboctahedron.

Cartesian coordinates
Cartesian coordinates for the vertices of a cubitruncated cuboctahedron are all the permutations of


 * (±($\sqrt{2}$−1), ±1, ±($\sqrt{2}$+1))

Tetradyakis hexahedron
The tetradyakis hexahedron (or great disdyakis dodecahedron) is a nonconvex isohedral polyhedron. It has 48 intersecting scalene triangle faces, 72 edges, and 20 vertices.

Proportions
The triangles have one angle of $$\arccos(\frac{3}{4})\approx 41.409\,622\,109\,27^{\circ}$$, one of $$\arccos(\frac{1}{6}+\frac{7}{12}\sqrt{2})\approx 7.420\,694\,647\,42^{\circ}$$ and one of $$\arccos(\frac{1}{6}-\frac{7}{12}\sqrt{2})\approx 131.169\,683\,243\,31^{\circ}$$. The dihedral angle equals $$\arccos(-\frac{5}{7})\approx 135.584\,691\,402\,81^{\circ}$$. Part of each triangle lies within the solid, hence is invisible in solid models.

It is the dual of the uniform cubitruncated cuboctahedron.