Thursday, December 17, 2009

Acronym in geometry: SCCO for small cubi-cuboctahedron

SCCO stands for small cubi-cuboctahedron (also non-hyphenated: small cubicuboctahedron). The SCCO is one of the 43 polyhedra collectively known as nonconvex uniform polyhedra (n-cUP). The mathematician John Lawrence Hudson illustrated the geometry and stellation of the SCCO in a recent article [1], pointing to the component forms (Platonic polyhedra) that are reflected in its name: six facial planes containing a square lying in the planes of a cube, eight facial planes containing a triangle lying in the planes of an octahedron, and six facial planes containing an octagon lying again in the planes of a cube. The latter cube is smaller than the first and lies therein with its facial planes parallel to corresponding planes of the first cube. The adjective “small” in SCCO distinguishes this solid from the “great” cubi-cuboctahedron.
The SCCO symmetry group: octahedral (also reflected in the name). Since the SCCO has three face types (equilateral triangle, square, and regular octagon), it has three different stellation diagrams.

Keywords: three-dimensional geometry, polytopes, solids, facial planes, 2-D facets

References and illustrations
[1] John Lawrence Hudson: Further Stellations of the Uniform Polyhedra. The Mathematical Intelligencer Fall 2009, Volume 31, No. 4, pp. 18-26.

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