| Small cubicuboctahedron | |
|---|---|
| Type | Uniform star polyhedron |
| Elements | F = 20, E = 48 V = 24 (χ = −4) |
| Faces by sides | 8{3}+6{4}+6{8} |
| Wythoff symbol | 3/2 4 | 4 |
| Symmetry group | Oh, [4,3], *432 |
| Index references | U13, C38, W69 |
| Bowers acronym | Socco |
4.8.3/2.8 (Vertex figure) |
Small hexacronic icositetrahedron (dual polyhedron) |
In geometry, the small cubicuboctahedron is a uniform star polyhedron, indexed as U13. It has 20 faces (8 triangles, 6 squares, and 6 octagons), 48 edges, and 24 vertices. Its vertex figure is a crossed quadrilateral.
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It shares the vertex arrangement with the stellated truncated hexahedron. It additionally shares its edge arrangement with the rhombicuboctahedron (having the triangular faces and 6 square faces in common), and with the small rhombihexahedron (having the octagonal faces in common).
Rhombicuboctahedron |
Small cubicuboctahedron |
Small rhombihexahedron |
Stellated truncated hexahedron |
As the Euler characteristic suggests, the small cubicuboctahedron is a toroidal polyhedron of genus 3 (topologically it is a surface of genus 3), and thus can be interpreted as a (polyhedral) immersion of a genus 3 polyhedral surface. Stated alternatively, it corresponds to a uniform tiling of this surface. 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 – every automorphism of the abstract genus 3 surface with this tiling is realized by an isometry of Euclidean space (it is a uniform tiling, and the small cubicuboctahedron is a uniform polyhedron).
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. The corresponding tiling of the hyperbolic plane has vertex figure 3.8.4.8 (triangle, octahedron, square, octahedron) – the covering map is a local isometry and thus the abstract vertex figure is the same (disregarding the factor of ½ which described not how faces are abstractly arranged about a vertex, but how they are concretely realized in Euclidean 3-space). This tiling may be denoted by the (extended) Schläfli symbol t0,1{4, 3, 3}, and is depicted at right.
Alternatively and more subtly, the small cubicuboctahedron can be interpreted as a coloring of the regular (not just uniform) tiling of the genus 3 surface by 20 equilateral triangles, meeting at 24 vertices, each with degree 7.[1] 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 automorphism group of this surface is isomorphic to the projective special linear group PSL(2,7), equivalently GL(3,2) (order 168, orientation-preserving isometries). Note that the SCCO is not a realization of this abstract polyhedron, as it only have 24 orientation-preserving symmetries (not every abstract automorphism is realized by a Euclidean isometry) – the isometries of the SCCO 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 (the universal covering) is the order-7 triangular tiling. The automorphism group of the Klein quartic can be augmented (by a symmetry which is not realized by a symmetry of the polyhedron, namely "exchanging the two endpoints of the edges that bisect the squares and octahedrons) to yield the Mathieu group M24.[2]
