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:

OU Math Club: first meeting of the year.

MATH CLUB!! MATH CLUB!! MATH CLUB!!

OU Math Club

Please join Dr. Catherine Hall in the first organizational meeting of the Math Club. All are welcome, math majors, math minors, math lovers and pizza lovers. Oh yeah, there’s free pizza!

When: Wednesday, October 1st, 5:15 pm

Where: PHSC 1105 (Physical Sciences, 11th floor, Room 1105)

Why: Free Pizza! Cool math! Nice people!!

Welcome Back!

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Class have started! (Well, they’re starting whether or not we like it, so let’s go with it being Good News)

After an entirely undeserved summer break, the OU Math Club blog is back, baby!

We have many things to tell you in the coming days and months.  Here is one juicy mathematical tidbit from over the summer:

In our last post eons ago, we talked about the Twin Primes Conjecture.  Remember, this is the conjecture that there are infinitely many pairs of primes which are at most two apart.  And we talked about Dr. Yitang Zhang’s amazing breakthrough which showed that there infinitely many primes which are at most 70,000,000 apart.  Which sounds a long ways from 2, but since there was no bound whatsoever before Dr. Zhang, it’s a heck of an improvement.

Over the summer people have been busily making improvements and shaving inefficiencies off of Dr. Zhang’s calculations, and the current world record is that there are infinitely many pairs of primes which are 5,414 (and maybe even 4,680) apart!

This has been a polymath project and they are now in the stage of writing up their calculations.  You can read all about it on Dr. Tao’s blog.

200,000+ and Counting!

Below is the latest stats from the OU Math Club visitor counter.  Check out them apples!

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That’s right!  200,402 visitors and counting!  Almost amazing is the wide range of places people come from.  Of course Oklahoma is well represented, but so is pretty much every other country on the face of the Earth:

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OU Math Grad Students and You!

If you are early in the OU math program, you may have had a grad student TA or grader for one of your courses.  If you’re late in the program, you might have even had a class with some of them in a slash listed course.  Either way, you probably noticed OU grad students in the math department.

Bart! Don’t Make Fun of Grad Students!

What you might not know is that they have lots of activities going on which might interest you.  They are a cool bunch of people to hang out with and you are always welcome to join in on any which sound interesting.

First, they have a blog (here and the link on the left side of our blog) where they talk about all their activities.

Second, they have a weekly Graduate Student Seminar (aka the Pizza Seminar) on Mondays where they eat pizza and have a cool math talk.  These are talks at a higher level then the usual Math Club talks, but still meant to be accessible to all grad students regardless of background.  So if you’re an upper level undergrad, you should definitely be checking them out.  Plus you get Pizza!  The Seminar schedule is here.

An artist's rendition of an OU grad student at the Pizza Seminar.

Third, if you want even more math, and know you’re interested in, say, algebra, then the grad students also run student seminars in all the major areas of math at OU.  You can see the schedules for those listed here on the MGSA page.

Fourth, if you’re feeling a little peckish on Tuesday afternoon, you can always stop by for cookies and tea/coffee at the Tuesday Tea organized by the grad students.  All are welcome!  It’s a relaxed atmosphere where people hang out and have a snack.  You should bring a friend and check it out.

You don't have to be a Mad Hatter to go to Tuesday Tea, but it doesn't hurt!