What if you were given a hexagon and asked to tile it over a plane such that it would fill the plane with no overlaps and no gaps? Could you do this?

### Tessellations

You have probably seen tessellations before, even though you did not call them that. Examples of tessellations are: a tile floor, a brick or block wall, a checker or chess board, and a fabric pattern. A **tessellation** is a tiling over a plane with one or more figures such that the figures fill the plane with no overlaps and no gaps. Here are a few examples.

Notice the hexagon (cubes, first tessellation) and the quadrilaterals fit together perfectly. If we keep adding more, they will entirely cover the plane with no gaps or overlaps. The tessellation pattern could be colored creatively to make interesting and/or attractive patterns. To tessellate a shape it must be able to exactly surround a point, or the sum of the angles around each point in a tessellation must be \begin{align*}360^\circ\end{align*}. Therefore, every quadrilateral and hexagon will tessellate. For a shape to be tessellated, the angles around every point must add up to \begin{align*}360^\circ\end{align*}. A regular pentagon does not tessellate by itself. But, if we add in another shape, a rhombus, for example, then the two shapes together will tessellate.

Tessellations can also be much more complicated. Here are a couple of examples.

#### Tessellating Quadrilaterals

Tessellate the quadrilateral below.

To tessellate any image you will need to reflect and rotate the image so that the sides all fit together. First, start by matching up each side with itself around the quadrilateral.

This is the final tessellation. You can continue to tessellate this shape forever.

Now, continue to fill in around the figures with either the original or the rotation.

#### Determining if an Object Tessellates

Does a regular pentagon tessellate?

First, recall that there are \begin{align*}(5 - 2)180^\circ = 540^\circ\end{align*} in a pentagon and each angle is \begin{align*}540^\circ \div 5 = 108^\circ\end{align*}. From this, we know that a regular pentagon will not tessellate by itself because \begin{align*}108^\circ \times 3 = 324^\circ\end{align*} and \begin{align*}108^\circ \times 4 = 432^\circ\end{align*}.

#### Applying Knowledge about Tessellations

How many squares will fit around one point?

First, recall how many degrees are in a circle, and then figure out how many degrees are in each angle of a square. There are \begin{align*}360^\circ\end{align*} in a circle and \begin{align*}90^\circ\end{align*} in each interior angle of a square, so \begin{align*}\frac{360}{90}=4\end{align*} squares will fit around one point.

#### Earlier Problem Revisited

You could tessellate a regular hexagon over a plane with no overlaps or gaps because each of its interior angles is \begin{align*}120^\circ\end{align*}. Three hexagons whose angles sum to \begin{align*}360^\circ\end{align*} surround each point in the tessellation.

### Examples

#### Example 1

How many regular hexagons will fit around one point?

First, how many degrees are in a circle, and then figure out how many degrees are in each angle of a regular hexagon. There are \begin{align*}360^\circ\end{align*} in a circle and \begin{align*}120^\circ\end{align*} in each interior angle of a hexagon, so \begin{align*}\frac{360}{120}=3\end{align*} hexagons will fit around one point.

#### Example 2

Does a regular octagon tessellate?

First, recall that there are \begin{align*}1080^\circ\end{align*} in a pentagon. Each angle in a regular pentagon is \begin{align*}1080^\circ \div 8 = 135^\circ\end{align*}. From this, we know that a regular octagon will not tessellate by itself because \begin{align*}135^\circ\end{align*} does not go evenly into \begin{align*}360^\circ\end{align*}.

### Review

Will the given shapes tessellate? If so, how many do you need to fit around a single point?

- A regular heptagon
- A rectangle
- A rhombus
- A parallelogram
- A trapezoid
- A kite
- A regular nonagon
- A regular decagon
- A completely irregular quadrilateral
- In general, which regular polygons will tessellate?
- Use equilateral triangles and regular hexagons to draw a tessellation.
- The blue shapes are regular octagons. Determine what type of polygon the white shapes are. Be as specific as you can.
- Draw a tessellation using regular hexagons.
- Draw a tessellation using octagons and squares.
- Make a tessellation of an irregular quadrilateral using the directions from Example A.

### Review (Answers)

To view the Review answers, open this PDF file and look for section 12.7.