A “Circle Limit III” Backbone Arc Formula

Douglas Dunham and Luns Tee
Bridges Leeuwarden: Mathematics, Music, Art, Architecture, Culture (2008)
Pages 341–346


M.C. Escher considered Circle Limit III to be the most successful of his four “Circle Limit” patterns. Two artistic or mathematical questions have been raised: (1) what angle do the white backbone lines make with the bounding circle, and (2) are other such patterns of fish possible? H.S.M. Coxeter provided an exact expression to the answer the first question, and a 3-parameter family of possible fish patterns was described in Dunham’s 2006 Bridges Conference paper. Dunham’s 2007 Bridges Conference paper provided a sequence of calculations that determine the intersection angle for any pattern of that family. In this paper, we derive a single expression for that angle, which agrees with Coxeter’s expression for the special case of Circle Limit III.