Kaleido
Uniform solution for uniform polyhedra. A uniform polyhedron in 3-dimensional space has faces which are planar regular polygons (not necessarily convex), and finitely many vertices which are equivalent under its symmetry group. H. S. M. Coxeter, M. S. Longuet-Higgins and J. C. P. Miller [Phil. Trans. Roy. Soc. London Ser. A 246, 401-409 (1953; Zbl 0055.142)] (hereafter cited as [CLM]) provided a complete enumeration of the uniform polyhedra, together with a limited amount of metrical information about them. In this paper, the author shows how this and further such information can be obtained in a systematic way, that is, without the need to consider each polyhedron on individually. The basic idea is to work with the Schwarz triangle which can usually be associated with the polyhedron, and obtain simultaneously the edge-length (for a given circumradius) and dihedral angles. Only one polyhedron does not fit into this or a related scheme, and that can be dealt with directly. Similar techniques treat the dual polyhedra, which were not dealt with in [CLM]. The author includes tables of Schwarz triangles, and tables and computer generated drawings of the edge-graphs of the polyhedra and their duals (with a handful of prisms and anti-prisms); in [CLM] the drawings show the other intersections of faces, and there are also some photographs of wire models.
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References in zbMATH (referenced in 8 articles , 1 standard article )
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Sorted by year (- Schulte, Egon; Williams, Abigail: Wythoffian skeletal polyhedra in ordinary space. I (2016)
- Gévay, Gábor: Constructions for large spatial point-line $( n _k)$ configurations (2014)
- Grünbaum, Branko: Polygons: Meister was right and Poinsot was wrong but prevailed (2012)
- Grünbaum, Branko: An enduring error (2009)
- Temme, F.P.: Geodesic and re-coupling-induced limits to $S_2n$ group invariants of uniform $(k_1\dots k_2n)$ dual tensorial sets in spin physics, or NMR: SR dynamical structure on $\Hv$ via polyhedral re-coupling and time-reversal invariance$ (2005)
- Grünbaum, Branko: Are your polyhedra the same as my polyhedra? (2003)
- Messer, Peter W.: Closed-form expressions for uniform polyhedra and their duals (2002)
- Har’el, Zvi: Uniform solution for uniform polyhedra (1993)