These tessellations work because all the properties of a tessellation are present.įigure 10.127 Tessellating with Obtuse Irregular Trianglesįirst, the triangle is reflected over the tip at point A A, and then translated to the right and joined with the original triangle to form a parallelogram. Both tessellations will fill the plane, there are no gaps, the sum of the interior angle meeting at the vertex is 360 ∘, 360 ∘, and both are achieved by translation transformations. The interior angle of a hexagon is 120 ∘, 120 ∘, and the sum of three interior angles is 360 ∘. There are three hexagons meeting at each vertex. In Figure 10.103, the tessellation is made up of regular hexagons. An interior angle of a square is 90 ∘ 90 ∘ and the sum of four interior angles is 360 ∘. There are four squares meeting at a vertex. In Figure 10.102, the tessellation is made up of squares. For a tessellation of regular congruent polygons, the sum of the measures of the interior angles that meet at a vertex equals 360 ∘.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. All the shapes are joined at a vertex.Shapes are combined using a transformation.(It was Escher who determined that a proper tessellation could have no gaps and no overlaps.) Patterns are repeated and fill the plane.A plane of tessellations has the following properties: Here we consider the rigid motions of translations, rotations, reflections, or glide reflections. Escher Tessellation Properties and TransformationsĪ 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. When a number divides another number evenly, there are no remainders, like there are no gaps when a shape divides or fills the plane.Įscher went far beyond geometric shapes, beyond triangles and polygons, beyond irregular polygons, and used other shapes like figures, faces, animals, fish, and practically any type of object to achieve his goal and he did achieve it, beautifully, and left it for the ages to appreciate. The idea is similar to dividing a number by one of its factors. He experimented with practically every geometric shape imaginable and found the ones that would produce a regular division of the plane. Escher became obsessed with the idea of the “regular division of the plane.” He sought ways to divide the plane with shapes that would fit snugly next to each other with no gaps or overlaps, represent beautiful patterns, and could be repeated infinitely to fill the plane. The Dutch graphic artist was famous for the dimensional illusions he created in his woodcuts and lithographs, and that theme is carried out in many of his tessellations as well. These movements are termed rigid motions and symmetries.Ī good place to start the study of tessellations is with the work of M. The topic of tessellations belongs to a field in mathematics called transformational geometry, which is a study of the ways objects can be moved while retaining the same shape and size. We will explore how tessellations are created and experiment with making some of our own as well. There are countless designs that may be classified as regular tessellations, and they all have one thing in common-their patterns repeat and cover the plane. These two-dimensional designs are called regular (or periodic) tessellations. It may be a simple hexagon-shaped floor tile, or a complex pattern composed of several different motifs. Repeated patterns are found in architecture, fabric, floor tiles, wall patterns, rug patterns, and many unexpected places as well. In this section, we will focus on patterns that do repeat. What's interesting about this design is that although it uses only two shapes over and over, there is no repeating pattern. Notice that there are two types of shapes used throughout the pattern: smaller green parallelograms and larger blue parallelograms. The illustration shown above ( Figure 10.101) is an unusual pattern called a Penrose tiling. Apply translations, rotations, and reflections.(credit: 'Penrose Tiling' by Inductiveload/Wikimedia Commons, Public Domain) Learning ObjectivesĪfter completing this section, you should be able to: Figure 10.101 Penrose tiling represents one type of tessellation.

Tessellation shapes