The Mathematics of Tiling: Irregular Polygons #1

This is a part of the series of posts introducing an active research field called Tessellation – The Mathematics of Tiling, discussing some basic, intuitive tiling patterns. Subscribe now (link) to get notified about more posts in coming days!


In the last article, we looked at regular polygons and we found that there are only 3 possible regular tessellations. Obvious next question is, what about irregular polygons? When it comes to triangles and quadrilaterals, it becomes too trivial – would you like to give it a shot? Two isosceles triangles (with two sides equal) can be used to form a rectangle (try it yourself) and then you know how to tile with it!

The question gets more interesting when we get to pentagons – especially since regular pentagons, as we observed already, does not form a neat tiling. If we keep away the condition that the tiles should be monohedral, we have some interesting tiling like Penrose tiling.

Then what about Pentagons? We had earlier noticed that we cannot achieve a monohedral tiling with a regular pentagon. What about it in general?

Monohedral Pentagonal tiling (from Wikipedia)

This looks like a mountain range, right? This gave me another idea, from this painting by Melvin:

Playing around with this idea, I ended up finding this pattern:

5 sides 4 corners forms a packing

Puzzles

  1. Can you take more inspiration from nature and find pentagonal tiling patterns? Upload them in Imgur (link) and leave it in comments below, along with your inspiration for the design!
  2. Can every irregular pentagon be used for ‘neat’ monohedral tiling? If not, can you find one tile, which does not play nice?
  3. What about irregular hexagon, heptagon, octagon and so on? Is it possible to form tiling with it?
  4. Are these tiles above convex (line joining any two points inside the tile lies inside the tile)? Can you try to find some monohedral convex pentagonal tiling? How many different tiles can you find?

In the next post, we will be digging into Pentagonal tiling. Do you know? It was a mathematical puzzle, solved over 100 years! Another exciting fact is that

Extra Reading

  1. Penrose Tiling (Wikipedia)
  2. Tiling with rectangles (Wikipedia)
  3. Exploring tessellation with regular and irregular polygons – Grade 8 Maths Links (WordPress)

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