Regular tessellations. Examples in Alhambra.

Exercise 1. Cover some space on your notebook by drawing polygons. Do it your own way. The only rule is to not leave gaps between the polygons.

Each piece of this drawing is a tile.

For practical reasons, for example, when you have to cover the ground with paving stone, it is easier and cheaper to get all the tiles having the same shape.

There is an easy way to do that: by using rectangles.

When you cover a surface with a pattern of tiles with no gaps or overlaps, this is called a tessellation.

What about if you use regular polygons? You can make tessellations by using squares, equilateral triangles and hexangons.

Is it possible to cover a surface with other regular polygons?

To answer that question you have to measure the interior angle of a regular polygon. There is no need to use any measuring tool to get that. Just use maths and the following rule: "the interior angles of a triangle add up to 180°". Besides, you can divide a polygon into triangles so, for example, to work out the interior angles of a square...

Exercise 2. What is the interior angle of the first five regular polygons?

So what happen if we get together regular polygons to make a tessellation? The addition of the interior angles of the polygons that share one vertex must be 360º.

Then it is impossible to use any other regular polygons but squares, equilateral triangles and hexagons because the only ones whose interior angles are divisors of 360º are equilateral triangles (60º · 6 = 360º), squares (90º · 4 = 360º) and hexagons (120º · 3 = 360)

First tessellation: hoja or pétalo

The ancient artisans knew that restriction and tried to make prettier tessellations by using different colours or making some distortion to the original tile and repeating it alongside the whole surface.

Starting with a tessellation of squares, you can cut one side of the square and repeat it by the whole surface by translation.

Then do the same to the other side to get

Exercise 3. Draw in your notebook the previous tessellation. Colour it to your own liking.

And this is how this tessellation appears in Alhambra (with slight modifications):

Second tessellation: hueso.

For the second tessellation they used the simmetry of a square by turning it 90º.

Take a look at this video to find out how it is constructed.

And this is how it looks in Alhambra:

Exercise 4. Draw the bone using this pattern:

Third tessellation: Pajarita.

For the third tessellation we use a property of an equilateral triangle: it has rotational symmetry by 60º, that is, you can turn an equilateral triangle by 60º and get the same shape.

If you translate that property to our triangle tessellation:

Any transformation you do to one side of a triangle can be replicated by rotating it 60º and there'll be no gaps.

By rotating once, twice and so on and then applying some other ornaments, we can get this tessellation:

Notice that there are other methods to get this beautiful tessellation: one example, another example.

And this is how it looks in Alhambra.

Exercise 5. Draw on your notebook the "pajarita" tessellation by following these steps:

Step one: draw a grid of equilateral triangles. The following one is, approximately, equilateral.

Step two: mark the middle point of each side of the triangles.
Draw this kind of wave...

Exercise 6. Try to find if you can mix some different polygons to make a tessellation.

Regular tessellations. Examples in Alhambra.

First tessellation: hoja or pétalo

Second tessellation: hueso.

Third tessellation: Pajarita.

Links