Frieze Patterns from Dissections

Finite frieze patterns of positive integers were shown by Conway and Coxeter to be in bijection with triangulated polygons. Baur, Parsons, and Tschabold generalized this result, showing that infinite frieze patterns of positive integers are in bijection with triangulated annuli and once-punctured discs. More recently, Holm and Jorgensen investigated frieze patterns arising from dissected polygons. The frieze patterns of Holm and Jorgensen involve algebraic integers of the form 2cos(pi/p) for an integer p. We combine these generalizations and present results on frieze patterns from dissected annuli, using these same algebraic integers. We additionally introduce combinatorial interpretations of the entries of these frieze patterns, which naturally extend interpretations in the positive integer/triangulation case. This work is joint with Lena (Jiuqi) Chen.