Helical Petrie Polygons

Paul Gailiunas
Proceedings of Bridges 2016: Mathematics, Music, Art, Architecture, Education, Culture (2016)
Pages 135–140


A Petrie polygon of a polyhedron is defined as a circuit of edges such that exactly two consecutive edges lie in the same face. In the case of an infinite polyhedron (or honeycomb or sponge) this definition leads to an infinite helical path, rather than a circuit. Since there is a one-to-one correspondence between the edges of a polyhedron and its dual it is possible to transform the Petrie polygons of one into the other, and if the transformation is continuous there can be interesting intermediate configurations. If the sponge is regular the Petrie polygons can be replaced by circular helices passing through polyhedral vertices. The helices can be transformed in an analogous way to the polygons, again with interesting intermediates. Three dimensional networks can be considered as the edges of a polyhedral space packing, although the faces of the polyhedra may not always be planar. Often a sponge can be made by removing some of the faces, and such cases are briefly considered. Arrangements of circular helices previously used by artists and researchers are recalled, and the possibility of finding further arrangements derived from 3-D networks is suggested.