Small cubicuboctahedron

The small suffix serves to distinguish it from the great cubicuboctahedron, which also has faces in the aforementioned directions.

In fact, every automorphism of the abstract genus 3 surface with this tiling is realized by an isometry of Euclidean space.

Higher genus surfaces (genus 2 or greater) admit a metric of negative constant curvature (by the uniformization theorem), and the universal cover of the resulting Riemann surface is the hyperbolic plane.

If the surface is given the appropriate metric of curvature = −1, the covering map is a local isometry and thus the abstract vertex figure is the same.

[3] This regular tiling is significant as it is a tiling of the Klein quartic, the genus 3 surface with the most symmetric metric (automorphisms of this tiling equal isometries of the surface), and the orientation-preseserving automorphism group of this surface is isomorphic to the projective special linear group PSL(2,7), equivalently GL(3,2) (the order 168 group of all orientation-preserving isometries).

3D model of a small cubicuboctahedron
The 3 4 | 4 tiling is the tiling on the universal cover of the small cubicuboctahedron.
(Yellow and red reversed in this tiling, compared to polyhedron.)
The small cubicuboctahedron can also be interpreted as a polyhedral immersion (a coloring of) the Klein quartic , [ 3 ] which is a quotient of the order-7 triangular tiling .