Triangular dissections, aperiodic tilings and Jones algebras

TL;DR

This paper explores the connection between fractal dissections of triangles, aperiodic tilings like Penrose tilings, and Jones algebras, revealing how algebraic structures encode complex geometric patterns.

Contribution

It introduces a novel link between Dynkin-Coxeter graph-based fractal dissections, Jones algebras, and aperiodic tilings, generalizing Fibonacci sequences for even n.

Findings

Triangles in dissections have angles as multiples of π/(n+1)

Jones algebra acts naturally on fractal set vector spaces

For n=4, structure leads to Penrose tilings

Abstract

The Brattelli diagram associated with a given bicolored Dynkin-Coxeter graph of type $A_n$ determines planar fractal sets obtained by infinite dissections of a given triangle. All triangles appearing in the dissection process have angles that are multiples of $\pi/ (n+1).$ There are usually several possible infinite dissections compatible with a given $n$ but a given one makes use of $n/2$ triangle types if $n$ is even. Jones algebra with index $[ 4 \ \cos^2{\pi \over n+1}]^{-1}$ (values of the discrete range) act naturally on vector spaces associated with those fractal sets. Triangles of a given type are always congruent at each step of the dissection process. In the particular case $n=4$, there are isometric and the whole structure lead, after proper inflation, to aperiodic Penrose tilings. The ``tilings'' associated with other values of the index are discussed and shown to be encoded by equivalence classes of infinite sequences (with appropriate constraints) using $n/2$ digits (if $n$ is even) and generalizing the Fibonacci numbers.