The X+Y Files

Issue 1

Patterns with Polygons

All around us we see geometrical designs – on wallpaper, caprpets and fabrics for instance. Some of the simplest patterns are constructed using regular polygons, which are straight line shapes with equal sides and equal angles. If these are fitted togther on a flat surface, edge to edge with no gaps, to make a repeating pattern which could be continued indefinitely in any direction, the result is called a plane tessellation.

Regular tessellations are made of only one type of regular polygon. As only equilateral triangles, squares and hexagons fit together in the right way, there are only three regular tessellations. More interesting are tessellations constructed using two or more types of regular polygon. Patterns made so that every vertex is the same are called semi-regular tessellations.
(A vertex is a point where two or more lines meet.)

This example of a semi-regular tessellation has 2 squares and 3 equilateral triangles at each vertex – always arrnaged in the same order:
2 triangles, then a square, a triangle and a square.
A neater way of writing down this vertex pattern uses index notation:


A different vertex pattern using the same polygons may produce a different tessellation. Try it out – and then try to find all the semi-regular tessellations. There are only 8 different types, but one of these has 2 forms which are mirror images of each other.

Drawing out tessellations using ruler, compasses and protractor for each shape can take a very long time, so you might like to experiment with other methods. You could, for instance, carefully cut out accurate polygon shapes in cardboard, then draw round them. (Your school might have ready-made shapes or templ,ates for this kind of activity.)

If you enjoy working with computers, you could use Logo or a graphics package to construct tessellations. Some of you may even have access to software specially designed for exploring tessellations.

What about repeating patterns using regular polygons which have two or more types of vertex? We illustrate two of these here – the one above uses two different shapes and three different vertex patterns. The design below involves three shapes and three vertex patterns:,  3.4².6  and 3.6.4².

Borrowing from the language of music, we will call these patterns demi-semi-regular tesselations. Can you find more – or any other interesting patterns with polygons?