Today, we can talk about tessellation art of the great M. Escher and a number of contemporary artists who are using the concepts of tiling to create artworks in a variety of media. Perhaps the most interesting examples of tessellation technique can be found in Islamic art and architecture, in particular of the Muslim inhabitants of the Maghreb, North Africa, the Iberian Peninsula, Sicily and Malta during the Middle Ages. The concept of coloured geometric shapes fit in perfectly, as Islam forbids the living object as a representation in art, so they embraced the abstract characteristics of a tessellation in a spectacular manner.
Apart from the Alhambra palace, geometric patterns and tiling were used in decorative arts as well, like textiles and pottery. Traces of zillij tessellation art can still be found in Morocco and Algeria , on the walls and floors of homes, mosques, public water fountains, tombs etc.
Tessellation patterns have been widely used in art and architecture since ancient times, but what lies under it is mathematics. Tessellation theory is extensive and complex, but we will explain some basics in order to bring you closer to what is behind these beautiful works of art. When it comes to tessellation in mathematics , also known as tiling , it is necessary to explain several technical terms that geometry operates with.
A fundamental region is a shape that is repeated in order to form a tessellation. It is also called the tile. An edge is an intersection between two bordering tiles that is often a straight line. A vertex is the point of intersection of three or more bordering tiles. A polygon is a plane figure with at least three straight sides and angles, and typically five or more. One shape of a tile in a tessellation is called a prototile.
In terms of the number of prototiles used, the tiling that has only one prototile is named monohedral tiling. This type of tiling is composed of a single shape, meaning that all tiles used are congruent to one another. At first, you might think that this type of tiling is simple and rather boring, but it is definitely not so.
A particularly interesting type of monohedral tiling is the spiral monohedral tiling that was first discovered by Heinz Voderberg in The tiling can also be dihedral. In this case, every tile is congruent to one or the other of two distinct prototiles, meaning that the tiling is composed of two different shapes.
Similarly, there are trihedral , tetrahedral or n-hedral tilings that signify the involvement of three, four or 'n' prototiles. Based on the types of polygons, tessellations are classified as regular, semi-regular and non-regular or irregular.
A highly symmetric one, a regular tessellation is made up of regular polygons that are all of the same shape and all meeting vertex to vertex. A regular polygon is one where all the sides and angles are equivalent.
There are only three regular tessellations and they are made of a network of equilateral triangles, squares and hexagons. The sum of the angles of polygons in a regular tessellation forms degrees around each vortex.
A semi-regular tessellation is made of two or more types of regular polygons. These regular polygons are arranged in a way that every vertex point is identical, meaning that each vertex is surrounded by the same polygons arranged in the same cyclic order. There are eight semi-regular tessellations which comprise different combinations of equilateral triangles, squares, hexagons, octagons and dodecagons. Non-regular tessellation is the one in which there are no restrictions regarding the shapes used or their arrangement around vertices.
It is believed there is an infinite number of irregular ones. When observing a completed tessellation, you could notice that the original tile, or a motif, repeats in a pattern. One mathematical idea that can be emphasized through tessellations is symmetry.
Yet, a line symmetry that could be found when two halves of a certain figure are congruent should not be confused with the type of symmetries that could be found in the plane of a tessellation. In terms of an infinite plane, these symmetries are referred to as plane symmetries or geometric transformations.
There are three types of symmetry in a plane that move the motif in a way that it still matches the original one exactly: the translational one, the rotational one and a glide reflection. A translational symmetry is a result of moving a figure a certain distance in a certain direction. These tessellation patterns could also be rotated around a certain point with a certain angle of rotation and that is called a rotational symmetry.
A glide reflection involves the reflection of a motif across a mirror line and its translation. It is the only type that involves two steps. These three types of symmetry are called isometric , meaning that tiles don't change size. But, there are also patterns consisted of tiles gradually getting smaller or larger in an expanding circle. Many mathematicians wouldn't call these tessellations since the tiles are not of the same size and could not entirely fill a plane in the centermost point.
Tessellations can be made with different combinations of transformations. There are actually 17 possible ways that a pattern can be used to tile a flat surface. These are called wallpaper groups and it is a mathematical classification based on the symmetries in the two-dimensional motif. It is believed that the Alhambra palace in Granada contained examples of all 17 groups. Tessellations may be further classified according to how unit cells containing one or more tiles are arranged.
If the unit cells are arranged such that a regular repeating pattern is produced, the tessellation is termed periodic. Periodic tessellations repeat the tile or a motif in two separate directions forever and they form motifs with symmetry given by one of the seventeen wallpaper groups. If the arrangement produces an irregular or random pattern, the tessellation is then termed aperiodic. These tessellations have no translation symmetry and the pattern cannot be repeated periodically only covering a portion of the plane.
Some of the best-known examples of aperiodic tessellation patterns are Penrose tilings that employ two different quadrilaterals or Pinwheel tilings where tiles appear in infinitely many orientations. Even though aperiodic tessellations look random, they do have rules that generate them such as the substitution rule or a Fibonacci word. Another type of aperiodic tessellations are Wang tiles consisted of squares coloured on each edge and placed in a way that abutting edges of adjacent tiles have the same colour.
Sometimes called Wang dominoes, these tessellation patterns can tile a plane, but only aperiodically. When considering the use of colour in tessellation technique, it needs to be specified whether colours are part of the tiling or just part of its illustration added randomly.
This is important in terms of understanding whether tiles of the same shape but different colours are considered identical or not. They were used to make up 'tessellata' - the mosaic pictures forming floors and tilings in Roman buildings The term has become more specialised and is often used to refer to pictures or tiles, mostly in the form of animals and other life forms, which cover the surface of a plane in a symmetrical way without overlapping or leaving gaps.
If you look at a completed tessellation, you will see the original motif repeats in a pattern. One mathematical idea that can be emphasized through tessellations is symmetry. There are 17 possible ways that a pattern can be used to tile a flat surface or 'wallpaper'. Between and Escher produced 43 colored drawings with a wide variety of symmetry types while working on possible periodic tilings.
He adopted a highly mathematical approach with a systematic study using a notation which he invented himself. There are 4 ways of moving a motif to another position in the pattern. These were described by Escher. A translation is a shape that is simply translated, or slid, across the paper and drawn again in another place. The translation shows the geometric shape in the same alignment as the original; it does not turn or flip.
A reflection is a shape that has been flipped. Most commonly flipped directly to the left or right over a "y" axis or flipped to the top or bottom over an "x" axis , reflections can also be done at an angle.
Also, quilt patterns many times have tessellations that can be found in them, as shown in this Lady of the Lake quilt pattern. Islamic Architecture is a good place to find tessellations. Architects could not depict any animals or humans on any buildings because people thought this might lead to idol worship.
So, Islamic art utilized geometric, floral, arabesque, and callig raphic primary forms, which are often interwoven into the architecture. Tessellations can be found in the hobby or art of origami. Paper is folded into triangles, hexagons, and squares to form many different patterns and shapes. This tessellation is called the honeycomb, another place to find tessellations in the real world. Another hobby of sorts are puzzles.
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