For 100+ years physicists believed that only solids which are ordered in a periodic way can produce a clear diffraction pattern. This belief was shattered by Dan Shechtman's discovery of quasicrystals, a discovery for which he was awarded the 2011 Nobel Prize in Chemistry.

The clear diffraction of quasicrystals shows that they must be highly ordered but in a non-periodic way. The precise order present in them is still mysterious, and not understood. In this project we investigate mathematical aperiodic orders. Namely we will study the existence of infinitely many solutions to Pythagoras's equation in aperiodic model sets. Such solutions are known to be present in periodic crystals.

This is new mathematical research that will be carried out by two MacEwan undergraduate research assistants under the supervision of Dr. Chris Ramsey and Dr. Nicolae Strungaru.

Stage 1: Background research and Z[tau]

The research assistants search the literature and read about the Fibonacci substitution. They get familiar with the geometric realization, find the eigenvalues/eigenvectors of the substitution matrix. They read and get familiar with the ring Z[tau], where tau = (1+squareroot(5))/2 is the largest eigenvalue of the Fibonacci matrix. They also read about solving the Pythagoras's equation in integers and find particular solutions in Z[tau].

The students will be under the supervision of Dr. Chris Ramsey.