TESSELLATIONS

MATH MEETS ART...

What is a tessellation?

A tessellation is a repeating pattern of shapes. In order for a pattern to tessellate all the shapes have to be touching each other with no overlapping and no gaps. The shapes lock into each other in an infinitely repeating tile pattern. 

A right triangle is a shape that tessellates. All you have to do is trace the triangle and flip it over on the hypotenuse (the side opposite the right angle). Trace a mirror image of the first triangle,

rotate the shape, match the sides up, trace the shape again, flip again and continue the pattern. If you follow the trace, flip, trace, rotate, trace, flip pattern exactly, over and over again, you can create a repeating tile pattern that continues indefinitely with no beginning and no end.

Now, imagine for a minute, you cut a bit off of one side of this shape and taped it to another side. It would take up the exact same area that it did before. The side with the piece cut away from it would be an exact negative of the side with the piece taped on. Nothing would have been added or taken away. When you tried to tessellate this new shape the negative and positive shapes would fit together exactly, like puzzle pieces.

To make it easier, do not cut or tape the hypotenuse.

This student is drawing a squiggly line from one corner of the triangle to the other.

She then cuts out the shape exactly along the line. DO NOT trim your shape once you are done. If you have any scraps at all, the new shape you create will not take up the exact same area as your triangle and the newly created shape will not tessellate.

Now, she tapes the shape on to the other side. Be careful to match the shape up exactly to the edge of the triangle. Do not tape the shape to the hypotenuse. You want to leave the longest side of the triangle straight so that you can flip the shape and create a mirror image.

See how your newly created shape tessellates, just like the original triangle did? Pretty cool huh?

Now it's time to get down to real fun. Take a look at your shape. What does it look like? This student thought her shape looked like fall leaves floating on the water.

The finished tessellation

Here are some other designs students in the same class came up with using the same formula:

Gospel singers and clouds

Red and Green Devils

This student duplicated, flipped and rotated the tiles of his tessellation design using Adobe  PhotoShop.
As he built his design he periodically saved the images. When he was done he uploaded the images to Microsoft PowerPoint. Here is the result:

Pink and Blue Angels

The artist M.C. Escher invented this technique of taking a geometric shape, altering it, tessellating it and then turning it into a recognizable picture. He wrote in his journal about how he was inspired to come up with this idea while visiting the Alhambra in Spain.

Below are examples of some of the tile work that Escher saw on his visit to Spain.

You can learn more about M.C. Escher by clicking here

You can learn more about the  Alhambra by clicking here
You can learn more about how to create your own tessellations by clicking here

Standards
Common Core Standards for Math:
Geometry


8.G.1
Verify experimentally the properties of rotations, reflections, and translations:
8.G.1.a
Lines are taken to lines, and line segments to line segments of the same length.
8.G.1.b
Angles are taken to angles of the same measure.
8.G.1.c
Parallel lines are taken to parallel lines.
8.G.2
Understand that a two-dimensional figure is congruent to another if the second can be obtained from the first by a sequence of rotations, reflections, and translations; given two congruent figures, describe a sequence that exhibits the congruence between them.
8.G.3
Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates.
8.G.4
Understand that a two-dimensional figure is similar to another if the second can be obtained from the first by a sequence of rotations, reflections, translations, and dilations; given two similar two-dimensional figures, describe a sequence that exhibits the similarity between them.

Reason with Shapes and their attributes
1.G.2. Compose two-dimensional shapes (rectangles, squares, trapezoids, triangles, half-circles, and quarter-circles) or three-dimensional shapes (cubes, right rectangular prisms, right circular cones, and right circular cylinders) to create a composite shape, and compose new shapes from the composite shape

2.G.1. Recognize and draw shapes having specified attributes, such as a given number of angles or a given number of equal faces.1 Identify triangles, quadrilaterals, pentagons, hexagons, and cubes

Consortium of National Arts Education Associations
In grades K-12 all students should-
• understand the visual arts in relation to history and cultures.