Following the lecture on 28/9/15 on creative mathematics, I had some idea of what tessellation meant, but had never considered it in any great detail. I decided to look it up and see what I could find out about it. I decided to investigate what shapes would and would not tessellate…

I started by choosing some random shapes to see if I could tessellate them myself. I chose an isosceles triangle (orange) [photo 1], a pentagon [photo 2], an equilateral triangle (purple) [photo 3], a hexagon [photo 4], a star [photo 5], and a right-angle triangle (pink) [photo 6]. I made copies of the shapes, cut them out and attempted to tessellate them. Now my cutting and sticking skills are not up to scratch, but think it’s still clear that some definitely tessellate while others do not.

Following what was said about the angles at the meeting points on the tessellation, I decided to check that this was true. I found that that I could measure that point and it did add up to 360°. The equilateral triangle [photo 8] and the hexagon [photo 9] had angles which were exactly the same, whereas the isosceles triangle [photo 7] and right angle triangle [photo 10] had two different angles which helped to add up to the full 360°

The star and the pentagon did not tesselate at all, so I decided to look into this a little more, and came up with this website. {https://plus.maths.org/content/five-fits}. It explains that we get convex and non-convex pentagons. Non-convex pentagons have points which are bent into the shape, see example [photo 11], and these are not included. In the example, it can be seen that the bottom ends in to the shape, which is why it is non-convex. They set out to find an irregular convex pentagon which would be able to tesselate. It appears that there are 15 pentagons which will tessellate, the most recent of which has been published this year by Casey Mann, Jennifer McLoud and David Von Derau at the University of Washington Bothell, which looks like this [I will make it clear at this point, that this is not my own work, simply a summary of research conducted by Marianne Freiberger and those she has cited]:

My conclusion from this research is that clearly mathematics is not as rigid as I previously believed. If this new ‘tessellateable’ pentagon was only discovered this year, then clearly mathematics is still  continually evolving and updating, meaning that mathematics really is all around us and there will always be new discoveries to be made. From a teaching perspective, I think that this is an important message to impart to children, that mathematics should not be done to them, but they should be doing it and trying things out and perhaps one day they will be discovering new shapes or theories, and they should be able to use the basic skills learned at school to do so.

Freiberger, M. (2015) A five that fits. Available at https://plus.maths.org/content/five-fits (Accessed 29 September 2015).

Math-Salamanders (2015) Shapes Clip Art Triangles & Quadrilaterals. Available at:http://www.math-salamanders.com/shapes-clip-art.html (Accessed 29 September 2015)

Wikimedia Commons (2006) File:Isosceles-right-triangle.jpg. Available at: https://commons.wikimedia.org/wiki/File:Isosceles-right-triangle.jpg (Accessed 29 September 2015)

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