Goldberg-Coxeter Operations on Polyhedra and Tilings ==================================================== The Goldberg-Coxeter (GC) operation can be used to subdivide the faces of a polyhedron or tiling with triangular or square faces, by replacing the faces with a portion of a lattice of triangular or square faces. Geodesic subdivision is another name for the method on triangular faces; this method can be attributed to [Goldberg]_, [Coxeter]_, Fuller, or [Caspar]_ and Klug, depending on whether you're a mathematician or a scientist :) This method often produces polyhedra with nice geometric qualities, for instance, local symmetry preservation, minimal distortion, etc. Strictly, the GC operation defined in e.g. [Deza2004]_ and [Deza2015]_ is defined on a polyhedral graph, not a polyhedron. Embedding the graph into R3 produces a polyhedron. In the embedding, it is necessary to consider geometric questions like the exact placement of vertices that aren't an issue with graphs; those are addressed here. Notation -------- This text uses :math:`\Delta(a,b)` for the triangular GC operator and :math:`\Box(a,b)` for the quadrilateral GC operator. This contrasts with [Deza2004]_, who use :math:`GC_{k,l}(G_0)` for both. Contents -------- .. toctree:: gco-planar gco-arb gco-spherical gco-a-spherical gco-a-improper gco-a-mixed