# 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 \(\Delta(a,b)\) for the triangular GC operator and \(\Box(a,b)\) for the quadrilateral GC operator. This contrasts with [Deza2004], who use \(GC_{k,l}(G_0)\) for both.

## Contents¶

- Master polygons
- Goldberg-Coxeter Operation on arbitrary tilings and polyhedra
- Goldberg-Coxeter Operation on spherical polyhedra
- Appendix: Spherical (and other) geometry with vectors
- Appendix: Goldberg-Coxeter Operation on improper spherical polyhedra
- Appendix: Towards Goldberg-Coxeter operation on mixed polyhedra