Excavated expanded cuboctahedron

The excavated expanded cuboctahedron is a quasi-convex Stewart toroid. It can be obtained by excavating 12 unit-edge-length rhombic prisms from an expanded cuboctahedron (blending their rhombus faces such that the prisms lie inside the shape), then excavating a rhombic dodecahedron from the center (blending it with the remaining rhombus faces of the prisms and opening up that space). This removes all of the rhombic faces, so the toroid is regular-faced. Its convex hull is an equilateral expanded cuboctahedron, with 30 squares (divided into two sets of 6 and 24), 12 rhombi, and 8 triangles.

It can also be obtained by outer-blending six square pyramids, eight tetrahedra, and twenty-four triangular prisms together.

Vertex coordinates
An of edge length 1 has vertex coordinates given by all permutations of
 * $$\left(\pm\frac{\sqrt3}{3},\,\pm\frac{\sqrt3}{3},\,\pm\frac{\sqrt3}{3}\right),$$
 * $$\left(\pm\frac{2\sqrt3}{3},\,0,\,0\right),$$
 * $$\left(\pm\left(\frac{2\sqrt3}{3}+\frac{\sqrt2}{2}\right),\,\pm\frac{\sqrt2}{2},\,\pm\frac{\sqrt2}{2}\right),$$
 * $$\left(\pm\frac{\sqrt3}{3},\,\pm\left(\frac{\sqrt3}{3}+\frac{\sqrt2}{2}\right),\,\pm\left(\frac{\sqrt3}{3}+\frac{\sqrt2}{2}\right)\right).$$

Related polyhedra
If the square pyramids and tetrahedra are partially-expanded into their respective cupolae (square and triangular), an excavated truncated rhombicuboctahedron will be formed.

If the triangular prisms are removed (and the pyramids brought together), a dissection of the cuboctahedron will be formed. It is a part of the tetrahedral-octahedral honeycomb.