The Mathematics of Tiling: Some Basic Ideas

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

Tessellation/Tiling is the process of covering a surface with tiles (mostly polygons) of one or more than one shape.

Fig 1: Tessellation (from Wikipedia)

We call it a monohedral tessellation if we use tiles of only one shape.

Fig 2: Example of Monohedral Tiling (from Wikipedia)

A tile (geometric shape) is said to be convex if straight line joining any two points inside (in the interior) of the tile lies completely within the tile.

Fig 3: Here a straight line joining any 2 points will lie inside the shape.
Fig 4: … whereas, in this case, for the given two points, the straight line may not lie inside the shape.

A polygon with all sides of the same length and same internal angle, like Equilateral Triangle, Square etc. is called a regular polygon. Like any mathematicians would do, let’s use our learning so far and ask some interesting questions.

Digging Deeper: Some Simple Puzzles

Meanwhile, look around! What is the shape of the tiles used to cover the floor of your house? Is that shape convex? How are they spread? Do they overlap each other? Is it a monohedral packing?

Look at Fig. 2: Is it a monohedral tiling? Is that shape convex? What about tiling in Fig. 1?

Can we always tile a flood using the same shape (for example, can we tile a floor with circular tiles alone)? Think about it, make a note of your observations.

If those questions were too easy, try drawing some tiling patterns yourself. Try see if it’s convex and if your new pattern is a monohedral tiling. You can also look around, copy tiling pattern you see in nature/sidewalks.

That’s it, for now! Watch out for more posts in this series in coming days, including teaching materials!

Photo by Scott Webb on

Extra Reading

  1. Tessellation/Tiling (Wikipedia)
  2. Monohedral Tiling (Wikipedia)
  3. Convex Set (Wikipedia)
  4. Regular Polygons (Wikipedia)

for similar posts in future!


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