Überoctoplex

The Überoctoplex is a certain polyhedron with the property that with a light placed inside every vertex, it is not possible to illuminate the entire interior of the polyhedron. It is the smallest known non-self-intersecting polyhedron with this property by face count and by vertex count.

The context is in a three-dimensional version of the, which asks how many guards need to be placed in a (simple) $n$-gonal art gallery so that every point in the polygon's interior is visible to at least one guard. Placing a guard at every vertex always ensures the entire gallery is visible, which upper bounds the solution by n (an elegant graph-theoretical argument reduces this upper bound to $$\lfloor n / 3 \rfloor$$). The existence of polyhedra such as the Überoctoplex demonstrates that this argument does not immediately work for polyhedra.

While not defined by symmetry, the figure can be realized as an isogonal polyhedron with pyritohedral symmetry, with its convex hull being the pyritosnub cube.

It is abstractly identical to the small rhombicuboctahedron. If the rectangles are collapsed into edges, Jessen's icosahedron is formed if the proportions are appropriate.