Small cubicuboctahedron
In geometry, the small cubicuboctahedron is a uniform star polyhedron, indexed as U13. It has 20 faces, 48 edges, and 24 vertices. Its vertex figure is a crossed quadrilateral.
The small cubicuboctahedron is a faceting of the rhombicuboctahedron. Its square faces and its octagonal faces are parallel to those of a cube, while its triangular faces are parallel to those of an octahedron: hence the name cubicuboctahedron. The small suffix serves to distinguish it from the great cubicuboctahedron, which also has faces in the aforementioned directions.
Related polyhedra
It shares its vertex arrangement with the stellated truncated hexahedron. It additionally shares its edge arrangement with the rhombicuboctahedron, and with the small rhombihexahedron.Rhombicuboctahedron | Small cubicuboctahedron | Small rhombihexahedron | Stellated truncated hexahedron |
Related tilings
As the Euler characteristic suggests, the small cubicuboctahedron is a toroidal polyhedron of genus 3, and thus can be interpreted as a immersion of a genus 3 polyhedral surface, in the complement of its 24 vertices, into 3-space. The underlying polyhedron defines a uniform tiling of this surface, and so the small cubicuboctahedron is a uniform polyhedron. In the language of abstract polytopes, the small cubicuboctahedron is a faithful realization of this abstract toroidal polyhedron, meaning that it is a nondegenerate polyhedron and that they have the same symmetry group. In fact, every automorphism of the abstract genus 3 surface with this tiling is realized by an isometry of Euclidean space.Higher genus surfaces admit a metric of negative constant curvature, and the universal cover of the resulting Riemann surface is the hyperbolic plane. The corresponding tiling of the hyperbolic plane has vertex figure 3.8.4.8. 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. This tiling may be denoted by the Wythoff symbol 3 4 | 4, and is depicted on the right.
, which is a quotient of the order-7 triangular tiling.
Alternatively and more subtly, by chopping up each square face into 2 triangles and each octagonal face into 6 triangles, the small cubicuboctahedron can be interpreted as a non-regular coloring of the combinatorially regular tiling of the genus 3 surface by 56 equilateral triangles, meeting at 24 vertices, each with degree 7. This regular tiling is significant as it is a tiling of the Klein quartic, the genus 3 surface with the most symmetric metric, and the orientation-preseserving automorphism group of this surface is isomorphic to the projective special linear group PSL, equivalently GL. Note that the small cubicuboctahedron is not a realization of this abstract polyhedron, as it only has 24 orientation-preserving symmetries – the isometries of the small cubicuboctahedron preserve not only the triangular tiling, but also the coloring, and hence are a proper subgroup of the full isometry group.
The corresponding tiling of the hyperbolic plane is the order-7 triangular tiling. The automorphism group of the Klein quartic can be augmented to yield the Mathieu group M24.