Each tessellation is based on a mathematical shape repeated over and over again. The pattern begins with a grid. Using regular polygons, there are only three grid patterns that will fit together: equilateral triangle, square, and regular hexagon. You can draw a square grid by using both sides of a ruler to draw parallel lines. Beginning with this square grid, a tessellation can be cut out by the process of cutting and moving: whatever is removed from one side is added to the opposite side .Whatever is taken from the bottom is added to the top. Repetition, of the shape creates a pattern in two colors.
You can draw your own tessellation using the following steps. You need to start by drawing a perfect square. There are two methods that will give you a perfect square:
1) Use a piece of tracing paper, laid over graph paper
2) Use the upper and lower edges of a ruler to draw parallel lines, and your protractor to get a 90 degree angle
Then follow the steps below:
Continue to trace the entire figure, and translate the tracings horizontally and vertically to create a tessellation as large as you like. Don't trace the square. Now you can decorate and color your tessellation!
Using a hexagonal grid, you have more sides to work with, but the process is the same. Whatever you remove from one side by cutting you add to the opposite side. Always be sure to work with a side and its opposite side together, both at the same time. Use different colors for adjoining shapes.
Using a triangle, cut a shape from one side. There is no opposite side, so move it by rotation and then repeat the shape using the same rotation. Repetition creates a pattern. You can create your own rotation tessellation following the steps below:
Repeat step 6 to create a tessellation. This whole figure, if traced and rotated, will tessellate and you can repeat that process to create as large a tessellation as you want.
Regular hexagons can also be used to make grids on which to draw your tessellations, as shown below:
Test Question #10: Can you use a regular 10-sided polygon as the base unit of a tessellation? Explain.
There are many interesting web sites about tessellations. A particularly good one can be found on the Math Forum web site, including tutorials that show you many ways to create tessellations, at the link below.
While you are at the Math Forum website, check out Ask Dr. Math for answers to your math questions, and Problem of the Week for interesting math challenges.