7) Tessellations

Fitting geometric shapes into each other to fill a plane surface is called tiling. A tessellation is the name given to a type of pattern made up of congruent shapes which interlock without overlapping or leaving any gaps. Which of the regular polygons can be used as the base unit of a tessellation? The question becomes, then, which ones will fit together to "fill the plane". This depends on their angles. Lets look at the regular polygons, one by one to see which ones will work in a tessellation.

The interior angles of an equilateral triangle are each 60 degrees. If we put 6 equilateral triangles together, the three 60 degree angles add up to 360 degrees, and the triangles do "fit together perfectly:

We already know that quadrilaterals, with their 90 degree angles, fit together perfectly. Five doesn't fit well, as 108 degrees doesn't divide evenly into 360.

360 degrees is perfectly divisible by 3, and the hexagon is one of the most common tiling patterns: tiles, fences, a patchwork quilt ...even those natural mathematicians, the honeybees, know what a perfect geometric figure the hexagon is.

The exterior angles of a seven-sided figure are found by dividing 360 by 7, which is approximately 51.43 degrees, so the interior angles are approximately 128.57 degrees. A seven-sided polygon will not tile:

The interior angles of an octagon are 135 degrees. Regular octagons will not fit together:

Of all the regular polygons, there are only three that will fit together perfectly: the equilateral triangle, square, and regular hexagon, because theirs are the angles that divide evenly into 360 degrees.

This tessellation is based on squares:

There are an infinite number of irregular polygons and figures that will interlock and fill the plane, and therefore can be used to create tessellations.The green and pink figures in this simple example of a geometric tessellation below are said to "tessellate". The tessellation below is based on parallelograms. In the pages that follow, you will find step-by -step constructions that will show you how to make tessellations based on parallelograms and on other polygons.

Some tessellations are made of abstract shapes, while others can be made to look like animals, birds, fish or other creatures. The tessellation above looks a little bit like a rabbit, and with eyes, a nose and a fluffy tail, would look even more so!

We see examples of tessellations and otehr geometric patterns in fabric designs and in ornamental tilework. Some very beautiful examples of this can be seen in the tilework of the Alhambra palace in Spain, at the following website:

http://www.punahou.edu/sanders/SpanishGeometry.html

The Dutch artist, M.C. Escher, visited the Alhambra and was inspired by the patterns there. He is well known for his fascinating tessellations. Using regular polygons for a framework, he did many fascinating drawings.. He first became interested in tessellations after a visit to the Alhambra in Spain. He studied the designs and did many drawings of them. He read books about ornamentation and mathematics, and said: "I can rejoice over this perfection and bear witness to it with clear conscience, for it was not I who invented it. The laws of mathematics are not merely human inventions or creations. They simply 'are': they exist quite independently of the human intellect. The most that any man with a keen intellect can do is to find out that they are there and to take cognizance of them."

There are many web pages showing his work, and books about him with beautiful illustrations. You can learn more about Escher, and see some of his beautiful work, at the following website:

http://www.etropolis.com/escher/

Although Escher's drawings are extremely complex, you can learn to do ones like them. Students like yourselves drew these tessellations using grids made from polygons. In the next section on this page you'll learn how to do these interesting drawings.

Go to Topic 8 Creating Tessellations