58 Geometry in Art and Science
Tessellations
A tessellation[1] is a design using one ore more geometric shapes with no overlaps and no gaps. The idea is that the design could be continued infinitely far to cover the whole plane (though of course we can only draw a small portion of it).
Many tessellations have translational symmetry, but it’s not strictly necessary. The Penrose tiling shown below[2] does not have any translational symmetry.
It’s actually much harder to come up with these “aperiodic” tessellations than to come up with ones that have translational symmetry. So we’ll focus on how to make symmetric tessellations.
The first two tessellations above were made with a single geometric shape (called a tile) designed so that they can fit together without gaps or overlaps. The third design uses two basic tiles. Tessellations are often called tilings, and that’s what you should think about: If I had tiles made in this shape, could I use them to tile my kitchen floor? Or would it be impossible?
On Your Own
Work on these exercises on your own or with a partner. You will need lots of copies (maybe 10–15 each) of each shape below. In each problem, focus on just a single tile for making your tessellation.
- Start with the square tile. Can you fit the squares together in a pattern that could be continued forever, with no gaps and no overlaps? Can you do it in more than one way?
- Now try one of the triangular tiles. Can you use many copies of a single triangle to tessellate the plane?
- Repeat this process with each of the other tiles. Keep track of your findings.
Think / Pair / Share
- Which of the tiles given above tessellate, and which do not?
- Do you have any conjectures based on this experience, about which shapes will tessellate and which will not?
Escher Drawings
The artist M.C. Escher created many works of art inspired by mathematics, including some very beautiful tessellations. Below you will see some images[3] inspired by his work. You can view the real thing at http://www.mcescher.com/ in the “Symmetry” gallery.
You can make your own Escher-like drawings using some facts that you learned while studying tessellations.
Theorem: Tessellations
Any triangle will tessellate. So will any quadrilateral.
The explanation for this comes down to what you know about the sums of angles. The sum of the angles in a triangle is 180°.
So if you make six copies of a single triangle and put them together at a point so that each angle appears twice, there will be a total of 360° around the point, meaning the triangles fit together perfectly with no gaps and no overlaps.
You can then repeat this at every vertex, using more and more copies of the same triangles.
Think / Pair / Share
- Use the fact that the sum of the angles in any quadrilateral is 360° to explain why every quadrilateral will tessellate.
- Use angles to explain why regular hexagons will tessellate.
- Explain why regular pentagons will not tessellate.
On Your Own
Work on the following exercises