The Beauty of Tessellations

Tiling is one of the many beautiful patterns that we can see around us. Some of them are man-made and some are created by nature. For example, many modern homes nowadays have tiled floors or even walls. On the other hand, in nature some animals are able to create regular tilings such as shown in the image below. Now why do bees would choose such shape?

Tiling, or popularly known as tessellation in mathematics, is not just beauty for the eyes. It has many interesting mathematical properties. For example, one question that should be asked what are the shapes that can tile a plane? What are the properties of these shapes? What are the properties of polygons that cannot tessellate the plane? Of course these questions had been answered hundreds or perhaps thousands of years ago, but they are a good exercise for the mind to those who haven’t encountered it. They are also good questions for students. 

Here are some facts about tessellations:

  • Tessellations were used in Rome as decorative patterns and they were also seen in early Islamic arts.
  • There are only three regular polygons that can entirely tile a plane — equilateral triangle, square, and regular hexagon. The proof is quite easy.
  • Some irregular polygons such as rectangles and non-equilateral triangles can also tessellate.
  • It is also possible to tessellate using 2 or more polygons.
  • Tessellation is used in computer graphics in order to render realistic effects.
  • Tessellation can be extended in higher dimensions.

In 3 dimensions, the cube is only the regular polyhedron can tessellate alone. The rhombic dodecahedron (shown below) and the truncated octahedron are some of the polyhedra that can be stacked in regular crystal patterns in order to fill the space without gaps or overlaps.

Screen Shot 2014-06-06 at 10.31.26 PM

 

M.C. Escher, a Dutch graphics artist, was well known for his work on tessellations. The three tessellations below are only few of his many stunning work.

tessellation

In addition to Escher’s tessellation on a Euclidean plane, he has also created tessellations on a hyperbolic plane.

hyperbolic tessellation

Indeed, tiling is one of the things that reveals the connection among art, nature, and mathematics.

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