Perplexing Pentagons (Exploration Idea #499)
Sunday 13 September 2015
Recently, a Maths SL student asked for my thoughts about her writing an Exploration (IA) on tilings of the plane (Euclidean tessellations). I must admit that my initial thought was that it would be easy for a student to write about tessellations and only use fairly elementary mathematics. Criterion E: Use of Mathematics contributes the largest portion of the total marks (6 out of 20) for a student Exploration – and the IB’s document Additional Notes and Guidance on the Exploration informs us that, “If the level of mathematics is not commensurate with the level of the course, a maximum of two marks can be awarded for this criterion.” The resulting loss of 4 marks is 20% of the total possible marks – ouch. However, with some effort it is possible to get into some interesting mathematics with tessellations that would be at least commensurate with the level of mathematics in Maths SL.
I wanted to point her in the right direction so that she would encounter some mathematics beyond just simple plane geometry. Tiling the plane by a single convex polygon turns out to be a very intriguing area of mathematics – and one where new discoveries are being made and where there are still unresolved questions. Which convex polygons – on its own – will tessellate (tile) the plane? Obviously any triangle will the tessellate the plane; and any quadrilateral will also. A regular hexagon certainly tessellates the plane; but, not any convex hexagon tessellates. It turns out that there are three different types of regular hexagons that can tessellate the plane. These are illustrated in the image below. Even better is this interactive Geogebra applet that illustrates the three types of hexagonal tilings.
It gets quite a bit more interesting with the pentagon. Although a regular pentagon has its own inherent beauty and the golden ratio is embedded in its shape, it does not tessellate the plane. But there are some convex pentagons that do tessellate. What about polygons beyond the hexagon? It has been mathematically proven that it is not possible to tile the plane with a convex polygon that has more than 6 sides.
During the 20th century, mathematicians – including a self-taught amateur mathematician (interesting story of housewife Marjorie Rice) determined that there were 14 different types of convex pentagons that tile the plane. In her 1978 award-winning article Tiling the Plane with Congruent Pentagons, American mathematician Doris Schattschneider, gives a thorough mathematical treatment of the 13 different convex pentagonal tilings that were known at that time (in 1985, a German mathematics graduate student found a 14th type). Schattschneider is an American mathematician and retired professor of mathematics who is known for her work on tessellations and being the co-director of a project in the 1980s that lead to the development of the powerful mathematics software Geometer’s Sketchpad. I borrowed the title for this blog post, Perplexing Pentagons, from a short article written by Schattschneider in 1996 which she concludes by writing “Are all the types of convex pentagons that tessellate now known? The tessellating pentagon problem remains unsolved.”
So, in my conversation with the Maths SL student I told her that tessellations involving convex pentagons is an active area of mathematics and recently featured in the news. It had been 30 years since the German student Rolf Stein found the 14th type of convex pentagon that tiles the plane but two months ago a 15th convex pentagonal tiling was found. Two mathematics professors and a student at the University of Washington Bothell found the new convex pentagon that tiles the plane with the aid of a specially designed computer algorithm. (this new tiling is shown in the first image above)
Attack on the pentagon results in discovery of new mathematical tile (Guardian, 11 August 2015)
Mathematicians find new 'perfect shape' that solves one of their most complex problems (Daily Mail, 17 August 2015)
A great way to see all 15 of the pentagon tilings is with this Pentagon Tilings - Wolfram Demonstration. If you download Wolfram's free CDF Player, you can interact with the demonstration - choosing the particular tiling and specific characteristics of the chosen tiling.
These articles may very well stimulate some interest in a student motivating them to look deeper into the topic. The challenge is narrowing the topic enough where it's manageable for a 6-12 page report and also focusing on mathematics which is sufficiently advanced but not too advanced. The mathematical (and computer) work that contributed to the recent discovery of the 15th convex pentagonal tiling is far too advanced but a focused look on the geometric construction of a particular tessellating convex polygon might be suitable for a student Exploration.
Schattschneider's aforementioned 16-page article Tiling the Plane with Convex Polygons is a good resource - and is what I suggested that the student read. It has perhaps too much detail but near the end of the article (pages 12-14), Schattschneider provides a table with geometric details on how to construct the different pentagonal tilings (the 13 that were known at that time).
Another direction to take is to focus on just one of the pentagonal tilings which has particularly interesting properties or aesthetic appeal. One of the pentagonal tilings that has received a lot of attention is the Cairo pentagonal tiling (shown left). It was given the name because of the claim (though not substantiated until fairly recently) that some streets and squares in Cairo have paving with this design. A Cairo-type of tiling can be constructed on isometric paper but it will have two different pentagons so it's using more than one type of pentagon - hence, not a true pentagonal tiling (like the 15 that are known). It can be interesting to mathematically determine the angle measures of the single pentagon that does create a Cairo pentagonal tiling as shown left and below. What are the exact measures of the angles?
And, lastly, there is a short 5-page article, Tiling Problem of Convex Pentagon, written by two Japanese mathematicians in 2000 on convex pentagon tilings and it references Euler's formula in graph theory on the number of vertices, edges and faces. It could be a good resource for students considering a student Exploration involving pentagonal tessellations.