# Polygons and PolyhedraTessellations

**tessellations**

Humans have copied many of these natural patterns in art, architecture and technology – from ancient Rome to the present. Here are a few examples:

Here you can create your own tessellations using regular polygons. Simply drag new shapes from the sidebar onto the canvas. Which shapes tessellate well? Are there any shapes that don’t tessellate at all? Try to create interesting patterns!

#### Examples of other students’ tessellations

## Tessellations from regular polygons

You might have noticed that some

This has to do with the size of their

You can similarly check that, just like pentagons, any regular polygon with 7 or more sides doesn’t tessellate. This means that the only regular polygons that tessellate are triangles, squares and hexagons!

Of course you could combine different kinds of regular polygons in a tessellation, provided that their internal angles can add up to 360°:

## Tessellations from irregular polygons

We can also try making tessellations out of

It turns out that you can tessellate not just equilateral triangles, but *any triangle*! Try moving the vertices in this diagram.

The sum of the internal angles in a triangle is

More surprisingly, *any quadrilateral* also tessellates! Their internal angle sum is

Pentagons are a bit trickier. We already saw that *regular* pentagons

Here are three different examples of tessellations with pentagons. They are not *regular*, but they are perfectly valid 5-sided polygons:

Over time, mathematicians have only found 15 different kinds of tessellations with (convex) pentagons – the most recent of which was discovered in 2015.

Two years later, in 2017, *Michaël Rao* published a proof that there are no other possibilities, except the 15 that had already been found. Can you make a tessellation using all of them?

## Tessellations in Art

Many artists, architects and designers use tessellations in their work. One of the most famous examples is the Dutch artist

These artworks often look fun and effortless, but the underlying mathematical principles are the same as before: angles, rotations, translations and polygons. If the maths isn’t right, the tessellation is not going to work!

## Penrose Tilings

All the tessellations we saw so far have one thing in common: they are **periodic**. That means they consist of a regular pattern that is repeated again and again. They can continue forever in all directions and they will look the same everywhere.

In the 1970s, the British mathematician and physicist *non-periodic* tessellations – they still continue infinitely in all directions, but *never* look exactly the same. These are called **Penrose tilings**, and you only need a few different kinds of polygons to create one:

Penrose was exploring tessellations purely for fun, but it turns out that the internal structure of some real materials (like aluminium) follow a similar pattern. The pattern was even used on toilet paper, because the manufacturers noticed that a non-periodic pattern can be rolled up without any bulges.