Why Do Some Shapes Tessellate and Others Not?

Andrew Malone/CC-BY 2.0

Tessellations occur when a shape is repeated in an interlocking pattern that fully covers a flat surface, or plane, like the pieces of a puzzle. Some shapes cannot tessellate because they are not regular polygons or do not contain vertices (corner points). They therefore cannot be arranged on a plane without overlapping or leaving some space uncovered. Due to its rounded edges and lack of vertices, the circle is normally not tessellated.

Three main categories of tessellations exist: regular, semi-regular and demi. The three regular tessellations use one repeating polygon that produces an identical pattern at each vertex. Regular tessellations are constructed with triangles, squares or hexagons. The tessellations are named according to the number of sides in each intersecting shape. For example, a pattern of four lines intersecting at right angles is a tessellation.

The eight semi-regular tessellations are composed of two or more regular polygons. They use the following combinations of shapes:

  • Triangles and hexagons
  • Triangles and squares
  • Triangles, squares and hexagons
  • Triangles and dodecagons
  • Squares, hexagons and dodecagons
  • Squares and octagons

Like regular tessellations, the pattern at each vertex is the same. However, because more than one shape is used, the pattern will contain more than one number. A pattern using triangles and hexagons is a tessellation because four triangles and one hexagon meet at each vertex. Controversy exists between mathematicians regarding the definition of demi tessellations. These patterns do not follow normal rules of constructing a tessellation and may contain irregular polygons, curved shapes, and non-identical vertices.