In this paper, we define certain operators, each of which transforms one regular map into another. These operator are based on the notions of Petrie path and jth order “hole” introduced by Coxeter. Together with the usual dual operator, they are a powerful tool for the analysis and taxonomy of regular maps. We produce, as an example, 18 distinct maps from the icosahedron, including six of Brahana and Coble's eight pentagonal dodecahedra.
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