Tessellations

by Jillian Gregory

Jigsaw puzzles consist of pieces that interlock with each other. The puzzle pieces can be linked together in various shapes, sizes, and angles. A simpler form of a jigsaw puzzle is a puzzle that consists of one repeating pattern. A repeating geometric pattern is called a tessellation. Tessellations combine mathematics and art.

Tessellations are all around us. You have seen them before, but may not have realized that they derive from mathematics. Tessellations are commonly found on fabrics, buildings, and in artwork. Buildings in your town may contain designs that consist of a repeating pattern. For example, a façade of a building may have an interlocking repeating pattern of triangles or of human faces. The possibilities for the patterns are endless. Architects and Interior Designers employ the use of tessellations every day. Can you think of repeating patterns in your home? The diamond shaped tiles in your bathroom are a great example.

What is the mathematical background behind tessellations? Tessellations are created by starting with a regular polygon. Picture a square piece of cardboard. You cut pieces from the cardboard that can then be rotated, reflected, or translated to another location.

What do these words mean? Let's take a look at a simple translation. Translation means that you will "slide" a piece of the polygon. For example, take the piece of square cardboard and cut a small triangle on the left side. Remove the piece of triangle. Translate this small triangle by sliding it horizontally to the other side of the square. Make sure that you slide it directly across from its original starting position. Match up the edges so that the triangle sticks passed the edge of the right side. Then tape the small triangle piece to the square cardboard piece.

Sound confusing? Think of the square pizzas that many pizza places have to offer. Start with a whole pizza that has been cut into 4 pieces in a 2 by 2 matrix. Imagine taking the piece of pizza in the bottom left corner and moving it horizontally to the right. Place it just to the right of the original bottom right piece of pizza. It is helpful to draw a picture in order understand the effect of the translation.

Now back to the square piece of cardboard and the small triangle piece cut out. Once you have translated the triangle, you are ready to create the tessellation. Take a piece of paper and outline the new object with pencil. Move your object to the right and you will see that it fits like a puzzle piece to the original drawing. Outline the object again with pencil. Continue to do this until you have gone across the page completely. You will have a repeating pattern or tessellation.

You can color in your pattern and add texture to it. Adding more translations in both the horizontal and vertical directions can add to the complexity of your pattern. If you are an expert with translations then you can work on rotations and reflections. Rotations involve rotating a shape around a specific point or axis. Reflections are slightly more complicated. Reflections are the mirror images of a shape. Shapes can be reflected over a line or axis. It is recommended that you stick to translations when first working on tessellations.

Now that you are familiar with tessellations they will pop out at you when you walk down the street. Check out various walk, brick, and tile patterns. They are a delightful mix of art, math, and puzzles. Tessellations are not only a joy to look at, but are fun to create.