  We give explicit formulae for fringe lengths of the Calegari-Walker Ziggurats
-- i.e. graphs of extremal rotation numbers associated to positive words in
free groups. These formulae reveal (partial) integral projective
self-similarity in ziggurat fringes, which are low-dimensional projections of
characteristic polyhedra on the bounded cohomology of free groups. This
explains phenomena observed experimentally by Gordenko and Calegari-Walker.
