Tessellations

After completing this section, you should be able to:

1. Apply translations, rotations, and reflections.
2. Determine if a shape tessellates.

A regular tessellation means that the pattern is made up of congruent regular polygons, same size and shape, including some type of movement; that is, some type of transformation or symmetry. Here we consider the rigid motions of translations, rotations, reflections, or glide reflections. A plane of tessellations has the following properties:

• Patterns are repeated and fill the plane.
• There are no gaps or overlaps. Shapes must fit together perfectly. (It was Escher who determined that a proper tessellation could have no gaps and no overlaps.)
• Shapes are combined using a transformation.
• All the shapes are joined at a vertex. In other words, if you were to draw a circle around a vertex, it would include a corner of each shape touching at that vertex.
• For a tessellation of regular congruent polygons, the sum of the measures of the interior angles that meet at a vertex equals \({360}^{∘}.\)

In , the tessellation is made up of squares. There are four squares meeting at a vertex. An interior angle of a square is \({90}^{∘}\) and the sum of four interior angles is \({360}^{∘}.\) In , the tessellation is made up of regular hexagons. There are three hexagons meeting at each vertex. The interior angle of a hexagon is \({120}^{∘},\) and the sum of three interior angles is \({360}^{∘}.\) Both tessellations will fill the plane, there are no gaps, the sum of the interior angle meeting at the vertex is \({360}^{∘},\) and both are achieved by translation transformations. These tessellations work because all the properties of a tessellation are present.

The movements or rigid motions of the shapes that define tessellations are classified as translations, rotations, reflections, or glide reflections. Let’s first define these movements and then look at some examples showing how these transformations are revealed.

A reflection is the third transformation. A shape is reflected about a line and the new shape becomes a mirror image. You can reflect the shape vertically, horizontally, or on the diagonal. There are two shapes in . The quadrilateral is reflected horizontally; the arrow shape is reflected vertically.

Condensed — the full section is in OpenStax Contemporary Mathematics.

The sum of the interior angles of a tessellation is \({360}^{∘}\). In , the tessellation is made of six triangles formed into the shape of a hexagon. Each angle inside a triangle equals \({60}^{∘}\), and the six vertices meet the sum of those interior angles, \(6({60}^{^{\circ}})={360}^{^{\circ}}\).

In , the tessellation is made up of trapezoids, such that two of the interior angles of each trapezoid equals \({75}^{^{\circ}}\) and the other two angles equal \({105}^{^{\circ}}\). Thus, the sum of the interior angles where the vertices of four trapezoids meet equals \({105}^{^{\circ}}+{75}^{^{\circ}}+{75}^{^{\circ}}+{105}^{^{\circ}}={360}^{^{\circ}}\).

These tessellations illustrate the property that the shapes meet at a vertex where the interior angles sum to \({360}^{^{\circ}}\).

We might think that all regular polygons will tessellate the plane by themselves. We have seen that squares do and hexagons do. The pattern of squares in is a translation of the shape horizontally and vertically. The hexagonal pattern in , is translated horizontally, and then on the diagonal, either to the right or to the left. This particular pattern can also be formed by rotations. Both tessellations are made up of congruent shapes and each shape fits in perfectly as the pattern repeats.

We have also seen that equilateral triangles will tessellate the plane without gaps or overlaps, as shown in . The pattern is made by a reflection and a translation. The darker side is the face of the triangle and the lighter side is the back of the triangle, shown by the reflection. Each triangle is reflected and then translated on the diagonal.

Escher experimented with all regular polygons and found that only the ones mentioned, the equilateral triangle, the square, and the hexagon, will tessellate the plane by themselves. Let’s try a few other regular polygons to observe what Escher found.

Just because regular pentagons do not tessellate the plane by themselves does not mean that there are no pentagons that tessellate the plane, as we see in .

Another example of an irregular polygon that tessellates the plane is by using the obtuse irregular triangle from a previous example. What transformations should be performed to produce the tessellation shown in ?

First, the triangle is reflected over the tip at point \(A\), and then translated to the right and joined with the original triangle to form a parallelogram. The parallelogram is then translated on the diagonal and to the right and to the left.

Condensed — the full section is in OpenStax Contemporary Mathematics.

• A tessellation is a particular pattern composed of shapes, usually polygons, that repeat and cover the plane with no gaps or overlaps.
• Properties of tessellations include rigid motions of the shapes called transformations. Transformations refer to translations, rotations, reflections, and glide reflections. Shapes are transformed in such a way to create a pattern.