Totally Tessellated: Unit Cells, page 1/2

Unit Cells (1/2)

1. Introduction
2. Unit Squares
3. Unit Triangles
4. Unit Hexagons
5. Hands-On Activities

Introduction
Unit cells can be used to create new tilings. Strictly speaking, a unit cell is a square with sides measuring one unit. However, for the purposes of tessellations, unit cells can either be a square, equilateral triangle, or regular hexagon. Notice that these are the polygons involved in regular tessellations which is not a coincidence. The technique of unit cells takes advantage of this property. Patterns are drawn inside unit cells so that when the unit cells are repeated in the manner of the regular tessellations, the patterns repeat to form a tessellation.

Unit Squares
Here is an example of a simple unit square tiling:

Example of a Simple Unit Tiling

A unit cell is duplicated, and then the original square lines are removed

Not all unit squares produce a valid tiling, however.

For example, the following unit cell produces a pattern that is not enitrely made of polygons. Look at the white areas. There are actually no white shapes, because the white areas are continuous in the vertical direction. Thus, this pattern does not fit the definition of tiling. A true tiling would have distinct shapes arranged in a repeating pattern.

An example of a unit cell that does not produce a valid tiling

The cause of the problem is that the points on the red polygons do not line up. In other words, the points on the left side of the square need to line up with the points on the right side of the square; and the points on the top side of the square need to line up with the points on the bottom side of the square.

How do we solve this problem? One clever solution involves reflections.

First, one corner of a unit square is drawn. (Let us assume for simplicity that the upper-left corner is drawn.) The corner is then reflected across the vertical axis of the unit square. Next, the resulting pattern is reflected across the horizontal axis to complete the unit square. These reflections ensure that points on opposites will line up.

Example (animated)

How to create a unit cell that will produce a pleasing tessellation

Example Tiling

The tessellation that results when the unit cell shown to the left is repeated

Here are two examples of unit cell tilings created using the reflection technique:

Example of Reflection Technique

The gray dotted lines indicate the portion of the unit cell that was reflected twice

Real examples of the unit cells technique:

You can find templates of squares, equilateral triangles, and equilateral hexagons at the templates page. Use these templates as a basis for creating new tessellations with the unit cells technique.

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