Level Up: Pythagorean triples in model sets - part 1
Preferred learners
- Alberta, Canada
- Academic experience
Categories
Skills
Project scope
- What is the main goal for this project?
-
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].
- What is the main goal for this project?
-
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].
- What tasks will learners need to complete to achieve the project goal?
-
No deliverables exist for this project.
- How will you support learners in completing the project?
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The students will be under the supervision of Dr. Chris Ramsey.
About the company
- https://www.macewan.ca/academics/academic-departments/mathematics-statistics/
- 0 - 1 employees
- Education
The Department of Mathematics and Statistics provides a vibrant and supportive home base for students who want to study mathematics and statistics. Our growing department is home to excellent scholars and has a deep commitment to student success, whether taking math as a requirement or as a pursuit of passion. Our faculty members are dedicated to providing an education that emphasizes the knowledge and broad analytical skills that are valuable in today’s world.
Beyond the classroom, our department organizes Torus Talks, a regular presentation series where faculty and students share insights into the uses and versatility of mathematics and statistics, and a Math Help Centre, where students can access additional support in mastering their course work.
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