Pentagonal Tilings and Undergraduate Research

The new academic year is now well underway and the OU Math Club blog is back.  Behind the curtains, the role of blogmaster for the next year has been taken over by me, Jeff Meyer.  I am excited to tell you about all sorts of fun and interesting math things.  If you know of something that everybody should hear about, email me at jmeyer at math.ou.edu and we’ll get it on the blog!

As soon as I agreed to run the blog this year, I knew exactly what I wanted my first post to be about: tiling the plane. The idea is simple, namely what are the ways one can cover the whole plane by repeating some sort of geometric pattern?

One type of tiling requires you use only a single convex polygon over and over again.  (Recall a polygon is convex if its interior angles are less than 180 degrees.)  I suspect you can find a convex quadrilateral that you can use to tile the plane.  If you stretch or shear your quadrilateral a little bit, would it still tile the plane?  I encourage you to think about this, and maybe try sketching it one a piece of paper.  Sketching tilings is a fantastic way to pass the time in a boring meeting.

So let me now ask you a question: Can you find a single convex pentagon that will tile the plane?

It turns out, this is a really hard problem.

German mathematician Karl Reinhardt in 1918 first came up with 5 ways, and since then a total of 14 had been found. That is until this past year.  Three researchers at the University of Washington, Bothell found a 15th!  They found it after a lengthy computer search.  The algorithm for the search was developed by Dr. Casey Mann and Dr. Jennifer McLoud-Mann and automated by undergraduate David Von Derau.

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All 15 known classes of pentagonal tilings, the bottom right being the one discovered by Mann, McLoud-Mann, and Von Derau. (EdPeggJr./Wikipedia)

It is amazing that there are so many open questions here:  Is this the complete list of convex pentagonal tilings? If not, are there finitely many?  Might there be infintiely many?

I think this is such a fantastic story for two reasons.  First, the problem is so easy to state and understand.  You could explain it to grade school students.  Second, this discovery was the result of a collaboration between faculty and an undergraduate.  For all you undergraduates out there, keep in mind there are lots of tangible research questions.  You just need to talk to some encouraging faculty who can help you find one.  If you do, then maybe next year there will be a post here about you!

Take a moment and check more details at the following links:

NPR:

NPR:

The Guardian:

University of Washington, Bothell:

Wikipedia:

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