The Mathematics of Tiling: Let’s Start Tiling!

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! Here’s the first article (link), if you haven’t read it yet.

Someone asked a simple question: can we tile a floor with just tiles of regular polygon? Or in other words, is a monohedral tiling possible (only tiles of one shape, we can’t break it, but turn or flip it), with regular polygons?

Answer is yes and no! Let’s start with the simplest case, equilateral triangle. We can intuitively see that we can pack those tiles easily.

Packing equilateral triangles (from Wikipedia)

Going a step higher, squares, we need not even mention – just look down! Most probably they have used square tiles to floor the room you’re sitting in right now! A square is a shape with each internal angle is 90°). However, here’s a figure:

Square tiling (from Wikipedia)

Note that at each corner where four squares meet, the sum of angles is 90° + 90° + 90° + 90° = 360°. That happens with equilateral traingles, too. This is an interesting observation we will use in the next step.

Obvious next question is if we can tile using a regular pentagonal tile. Before we visually explore it, let’s make a note. A minor subnote I skipped so far is that when we say tiling, we mean tiling without cutting the tile/overlapping. We know that internal angle of a pentagon is 108°. No matter how we try to arrange it, there will be a small gap left, like you see in the figure.

Just regular pentagonal problems. (not from Wikipedia)

Interesting! So, there are some regular polygons which cannot be used for monohedral tiling! Pentagonal tiling is of our interest, and we will look into it in detail in the next article.

Six equilateral triangles make a regular hexagon. Look at the figure for equilateral triangles tiling again, can you see a hexagonal packing? In fact, the bees beat us to finding hexagonal packing of tiles as you might have already heard!

Beethematicians (from Wikipedia)

Just for fun, let’s make an observation: One interior angle of a regular hexagon is 120°; and point where 3 regular hexagons meet, 120° + 120° + 120° = 360°.

What is a regular polygon with 7 sides called? Let’s count, un, di, tri, quad, penta, hexa, hepta, octa… Oh yes, heptagon!

What about regular heptagons? Turns out, him, and his elder brothers and sisters, regular octagon, (wait! What’s nine-sided -gons called?), regular decagons…. join our poor little regular pentagon! We cannot do a monohedral tiling using regular polygons of those types.

Can you guess why?

Hint: Using any magic formula you know, find one interior angle. See if you can make any observations like you did for poor regular pentagons!

To conclude, the only regular polygons with which you can achieve a monohedral tiling is equilateral triangles, squares or regular hexagons. Out of infinitely many candidates, we pinned on these three with just… highschool mathematics!

Another interesting question is, is there any way to tile using the same regular tri-, tetra-, hexa- gon differently from what we have already done?

Questions to Ponder

Are regular polygons we used for tiling here convex?

Can all convex regular polygons be used to tile a floor (without breaking/cutting)?

Photo by La Miko on

Extra Reading

  1. Regular Polygons (Wikipedia)
  2. Triangular tiling (Wikipedia)
  3. Square tiling (Wikipedia)
  4. Pentagonal tiling (Wikipedia)
  5. Hexagonal tiling (Wikipedia)

for similar posts in future!


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