Regular Tessellations (1/4)

 1. Introduction 2. Trial and Error 3. Analysis of Results 4. Naming the Regular Tessellations 5. The Regular Tessellations 6. Hands-On Activities
Introduction
We are now concerned with tessellations of regular polygons, since regular polygons are the simplest types of polygons. Let us start with the simple case of a tessellation using only one type of polygon. Tessellations of only one type of polygon are called regular tessellations.

Trial and Error
First, let us consider the simplest of the regular polygons, the three-sided regular polygon known as the equilateral triangle. Will equilateral triangles tessellate by themselves?

After a little experimentation, it becomes clear that equilateral triangles do tessellate.

 Equilateral triangles do tessellate

Next, it is easy to discover that squares (four-sided regular polygons) also tessellate, in a grid-like fashion.

 Squares also tessellate

However, when you try to put together some regular pentagons, you find that they cannot perfectly fill the space around a point. Thus, regular pentagons cannot tessellate by themselves. The following image explains the situation:

 An additional regular pentagon cannot fill the space marked with "?".

What about regular hexagons? Regular hexagons can tessellate easily, as it turns out.

 Regular hexagons do tessellate

Now consider regular heptagons. When trying to fit regular heptagons around a point or vertex, we see that the situation we had with regular pentagons happens again, overlap. Thus, like pentagons, heptagons do not perfectly fill the space around the point and, therefore, do not tessellate.

 An additional regular heptagon cannot fillthe space marked with "?".

 Real examples of regular tessellations:

 To browse full-page templates of the regular tessellations that are ready to be printed, proceed to the templates page: