The truncated cuboctahedron is the convex hull of a rhombicuboctahedron with cubes above its 12 squares on 2-fold symmetry axes.
The rest of its space can be dissected into 6 square cupolas below the octagons and 8 triangular cupolas below the hexagons.

A dissected truncated cuboctahedron can create a genus 5, 7 or 11 Stewart toroid by removing the central rhombicuboctahedron and either the square cupolas, the triangular cupolas or the 12 cubes respectively.
Many other lower symmetry toroids can also be constructed by removing a subset of these dissected components.
For example, removing half of the triangular cupolas creates a genus 3 torus, which (if they are chosen appropriately) has tetrahedral symmetry.
The truncated cuboctahedron is the convex hull of a rhombicuboctahedron with cubes above its 12 squares on 2-fold symmetry axes. The rest of its space can be dissected into 6 square cupolas below the octagons and 8 triangular cupolas below the hexagons.
A dissected truncated cuboctahedron can create a genus 5, 7 or 11 Stewart toroid by removing the central rhombicuboctahedron and either the square cupolas, the triangular cupolas or the 12 cubes respectively. Many other lower symmetry toroids can also be constructed by removing a subset of these dissected components. For example, removing half of the triangular cupolas creates a genus 3 torus, which (if they are chosen appropriately) has tetrahedral symmetry.
Truncated cuboctahedron