# Tessellations

You may have seen many tessellations before without fully realizing it. Mathematicians use the word "tessellation" to describe all kinds of common patterns, such as grids. But what are the mathematical principles behind tessellations? Let''s find out:

## What is a tessellation?

A tessellation is a repeating geometric pattern. Just like a reflection or a dilation, it is a type of transformation. The key thing to remember about tessellations is that their patterns must repeat without leaving any gaps and be able to "tile the plane". Only certain shapes can do this. For example, try creating a tessellation by laying out a dozen pentagons on a table without leaving any gaps. You can''t!

The above is a very simple tessellation made with repeating squares.

Here''s another tessellation. This one is made up of a repeating pattern of triangles.

This is not a tessellation. As you can see, there is no way to lay out these regular pentagons without leaving any spaces in between.

Here''s a tessellation made from a repeating pattern of rhombi (the plural term for rhombus).

Notice anything familiar about this tessellation? That''s right -- the same rhombus can be tessellated in a different manner to form a new pattern! This one also passes the rotational symmetry test.

We can even tessellate the same shape in a third completely unique way!

Check out this tessellation: It''s particularly complex because it involves repeating patterns of several shapes, including trapezoids, triangles, and squares. As long as there are no spaces between the shapes, it still counts as a tessellation.

## Topics related to the Tessellations

Reflections

Trapezoid

Rotational Symmetry