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 3: Solutions in Fibonacci

In this stage, the research assistants extend the approach to various other model sets such as the Silver Model set or the noble means model sets. For each model set they need to first find solutions to and then solve the Pythagoras equation in Z[lambda], where lambda is the largest eigenvalue of the corresponding matrix, and then by using this general solution to solve the question in that model set. If time permits, the research assistants will study various cases of Fermat's Last Theorem or other Diophantine equations in the above-mentioned model sets.

Students will be working under the supervision of Dr. Chris Ramsey