Small cubicuboctahedron
Small cubicuboctahedron | |
---|---|
Type | Uniform star polyhedron |
Elements | F = 20, E = 48 V = 24 (χ = −4) |
Faces by sides | 8{3}+6{4}+6{8} |
Coxeter diagram |
|
Wythoff symbol | 3/2 4 | 4 3 4/3 | 4 |
Symmetry group | Oh, [4,3], *432 |
Index references | U13, C38, W69 |
Dual polyhedron | Small hexacronic icositetrahedron |
Vertex figure | 4.8.3/2.8 |
Bowers acronym | Socco |
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.[1] 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.[2]
Related polyhedra
It shares its
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 (topologically it is a surface of genus 3), and thus can be interpreted as a (polyhedral) immersion of a genus 3 polyhedral surface, in the complement of its 24 vertices, into 3-space. (A neighborhood of any vertex is topologically a cone on a figure-8, which cannot occur in an immersion. Note that the Richter reference overlooks this fact.) The underlying polyhedron (ignoring self-intersections) 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 (genus 2 or greater) admit a metric of negative
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 (not just uniform) tiling of the genus 3 surface by 56 equilateral triangles, meeting at 24 vertices, each with degree 7.
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 octahedra) to yield the Mathieu group M24.[4]
See also
References
- ^ Maeder, Roman. "13: small cubicuboctahedron". MathConsult.
- ^ Webb, Robert. "Small Cubicuboctahedron". Stella: Polyhedron Navigator.
- ^ a b (Richter) Note each face in the polyhedron consist of multiple faces in the tiling, hence the description as a "coloring" – two triangular faces constitute a square face and so forth, as per this explanatory image.
- ^ (Richter)
- Richter, David A., How to Make the Mathieu Group M24, retrieved 2010-04-15