![]() 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: The most famous pair of such tiles are the dart and the kite.Ĭlick here for the lesson plan of non-periodic Tessellations.Figure 10.101 Penrose tiling represents one type of tessellation. The pattern of shapes still goes infinitely in all directions, but the design never looks exactly the same. In the 1970s, the British mathematician and physicist Roger Penrose discovered non-periodic tessellations. Whatever direction you go, they will look the same everywhere. They consist of one pattern that is repeated again and again. It may be better to show a counter-example here to explain the monohedral tessellations.Īll the tessellations mentioned up to this point are Periodic tessellations. All regular tessellations are also monohedral. If you use only congruent shapes to make a tessellation, then it is called Monohedral Tessellation no matter the shape is. You can use Polypad to have a closer look to these 15 irregular pentagons and create tessellations with them. ![]() Among the irregular pentagons, it is proven that only 15 of them can tesselate. We can use any polygon, any shape, or any figure like the famous artist and mathematician Escher to create Irregular tessellationsĪmong the irregular polygons, we know that all triangle and quadrilateral types can tessellate. The good news is, we do not need to use regular polygons all the time. If one is allowed to use more than one type of regular polygons to create a tiling, then it is called semi-regular tessellation.Ĭlick here for the lesson plan of Semi - Regular Tessellations. If you try regular polygons, you ll see that only equilateral triangles, squares, and regular hexagons can create regular tessellations.Ĭlick here for the lesson plan of Regular Tessellations. the most well-known ones are regular tessellations which made up of only one regular polygon. There are several types of tessellations. ![]()
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