If you moved house at some point as a child, you can probably still remember (I most certainly still do) your mum complaining about the state of your new kitchen floor: “Hey! These tiles are so worn out and ugly! We need to replace them all.” Now, unless your mum is an architect or someone with a keen eye for interior design, you will most likely end up with regular-shaped, boring square tiles for the floor, like this:
For reasons of both aesthetics and practicality, when you cover your kitchen floor, you neither want to leave any spaces in between the tiles, nor pieces to overlap, so as to avoid wasting materials. Imagine now that your kitchen is no longer of fixed length and width, but instead it extends forever. This problem is called tessellation.
One can construct both aperiodic (non-repeating) patterns and periodic (repeating) ones. In this article we’ll be focusing on the latter.
Let’s consider first the case in which we only use regular shapes for the tiles. As we saw in the kitchen example above, we could tile the plane with regular-shaped square tiles. But could you use a different shape of tile?
If you try using triangles, we observe that you can in fact cover the plane – which is great!
How about pentagons? If we try to tile the plane, we encounter a problem quite soon, either through the overlap of 2 pieces, or by leaving gaps between the tiles.
So, the question then becomes how can we know which regular shapes are permitted?
Consider an empty plane and a regular polygon with n sides. Now, for a complete and proper tessellation, when we add similar shaped regular polygons around our “central” piece, we should observe no overlap or spaces. That is, when looking at any of the n vertices of the polygon, the space around it has to be covered by an integer number of other regular polygons, say m. Since they are regular, the arches formed at that vertex will all be equal and, hence, they have to add to exactly 360°.
Now, fortunately, there is a neat formula which we can use to calculate the size of these equal angles: 180 * (n-2) is the sum of the degrees of all the angles of a polygon with n sides.
We can see the formula works for the familiar examples of a triangle: 180 * (3-2) = 180°, and quadrilateral: 180 * (4-2) = 360°. So, to get the size of the n equal-sized angles, we simply divide by n to get 180 * (n-2) / n.
Since we have m other polygons to fit at each vertex of size 180 * (n-2) / n, we obtain the final formula: m * 180 * (n-2) / n = 360.
Simplifying this a little, we have 180 * m * (n-2) = 360 * n, or m * (n-2) = 2 * n. Expanding the brackets and adding 4 to each side gives (m-2) * (n-2) = 4. This means that each bracket must represent a whole number (or integer) divisor of 4.
Now, you may be wondering why exactly integer divisors, but we need to keep in mind that both m and n are whole numbers as they represent the number of polygons n, and the number of sides of said polygons m respectively, which means both (m-2) and (n-2) must also be integer quantities.
Moreover, n needs to be greater or equal to 3, since we need to have an actual polygon – two sides isn’t enough to create a 2D shape. Therefore, (n-2) will always be bigger than 1 and so negative divisors of 4 won’t work in our case.
We therefore have one of the three possible cases:
Case 1: (m-2) = 1 and (n-2) = 4 which leads to m = 3 and n = 6, i.e. we can tile around a point with 3 hexagons, so we can do a tessellation with hexagons of the plane.
Case 2: (m-2) = 2 and (n-2) = 2 which leads to m = 4 and n = 4, i.e. we can tile around a point with 4 squares, as we have seen before with your mum’s kitchen.
Case 3: (m-2) = 4 and (n-2) = 1 which leads to m = 6 and n = 3, i.e. we can tile around a point with 6 equilateral triangles, again as we saw earlier.
SO, what this shows is that we can only tile our plane with regular polygons if they are triangles, squares or hexagons. This result is what we call the Uniform Tiling Theorem.
This kind of regular tiling appears also in nature: take for example the hexagonal shapes of the honeycomb (image courtesy of AMSI).
One can only assume that the bees have chosen to construct it in such a manner to maximise storage space capacity for the honey while using as little material for the building as possible, since it is very expensive for them to produce.
But, back to our kitchen floor remodelling. There are of course more complicated ways in which to tile it if we drop the regular polygon rule, and, if you want to go full ‘Oxford’, choose to use the Penrose tiling (found outside the Oxford Maths Institute).
This particular tiling scheme has the peculiar property that no pattern repeats itself, no matter how much you expand your kitchen…
Or if you would like something less math-inducing but still eye-catchy, you may take inspiration from the halls of the Alhambra. The Alhambra Palace of Granada in southern Spain is renowned for its architectural beauty and it is said that the mosaics are so varied that you can observe all of the seventeen groups of wallpaper patterns. The wallpaper patterns are repetitive two-dimensional patterns, grouped by the type of symmetry present. You can see below some basic tilings following these wallpaper patterns, constructed only from regular polygons (image from Quadibloc).
Some more colourful examples are shown below, with each image labelled according to the mathematical name of the wallpaper group that it represents (image from Wolfram Mathworld).
So, if you ever find yourself moving into a new house and needing to change the kitchen floor, you will hopefully now be able to come up with something more creative than the usual square tile pattern! Sorry mum.