Totally Tessellated: Tessellations of Pentagons, page 1/3

Tessellations of Pentagons (1/3)

1. Introduction
2. Limited Technique 1
3. Limited Technique 2
4. Limited Technique 3
5. Hands-On Activities

Introduction
We have already investigated quadrilaterals, so the logical next step is to consider pentagons. A interesting point to ponder is whether all pentagons will tessellate by themselves. We have already found techniques for any triangle or quadrilateral to tessellate by itself, so it is natural to believe that perhaps techniques for tessellating any pentagon exist.

We ask the same question as we have for quadrilaterals, triangles, and regular polygons: Is there a way to arrange a pentagon around a point to perfectly fill 360 degrees?

Recall that the sum of the interior angles of any pentagon is 540 degrees. Therefore it is impossible to fit all the angles of pentagon into 360 degrees.

Angles of a Pentagon will not Fit into 360 Degrees

The technique of rotating about the midpoints of the sides will not work because the angles of a pentagon do not total 360 degrees.

Suppose we restricted the type of pentagon allowed. Then we will be able to fit some of the pentagon's angles into 360 degrees?

1. Introduction
2. Limited Technique 1
3. Limited Technique 2
4. Limited Technique 3

Limited Technique 1
If two adjacent angles of a pentagon total 180 degrees, then the pentagon will tessellate in one of two ways. Translations or reflections can be used to extend the pentagon. The following example illustrates this idea. Notice how the requirement makes it easy for the pentagon to be repeated.

How to create a tessellation from a pentagon with two adjacent angles that sum to 180 degrees

Additional examples:

Examples using the translation version of the technique

Example of the Translation Version of the Technique

An application of the translation version of the technique; the shaded portion is the pattern generated in the animated example shown above Example of the Translation Version of the Technique

An application of the translation version of the technique; the shaded portion is the pattern generated in the animated example shown above

Examples using reflection version

Example of the Reflection Version of the Technique

An application of the reflection version of the technique; the shaded portion is the pattern generated in the animated example shown above Example of the Reflection Version of the Technique

An application of the reflection version of the technique; the shaded portion is the pattern generated in the animated example shown above

In summary, any pentagon with two adjacent angles that sum to 180 degrees will tessellate in two different ways.

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