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__** Tessellations Arts & Mathematics​ **__

The Tessellations refer to a partition on the same plane by polygons or group of polygons identical entirely appropiate groupings covering the plane. They are the designs of geometric figures that for if same or in combination they cover a flat surface without leaving hollows to superpose, or, the coverage of the plane with juxtaposed figures. Therefore, the tessellations have two conditions, the first one it is that the plane does not have blanks or emptinesses and the second one is that the figures are not superposed.

They can qualify according to it figure in regular, semi-regular, demi-regular and irregular and in addition they can qualify according to it periodicity in non-periodic and periodic. The regular tessellations are a highly symmetric tessellation made up of congruent regular polygons. Only three regular tessellations exist for what the union of every vertex are 360 º and there do not stay blanks or emptinesses, which are: 1. Tessellation of equilateral triangles.

2. Tessellation of squares, for example the chessboard.

3. Tessellation of hexagons, for exameple the honeycomb. The semi-regular tessellations contain a variety (two or more) of regular polygons between which 6 different polygons are outlined to combine and they are dodecagons, octagons, squares, rhombuses, equilateral triangles and hexagons. This type of tessellations fulfill the following properties: 1. They are formed only by regular polygons. 2. The arrangement of polygons is identical in every vertex. 3. Eight exist only tessellations you will semi-regulate.



The demirregulares tessellations, unlike the semirregulares, are achieved combining several types of regular polygons and fulfill that the vertexes have the same distribution.

The irregular tessellations are formed by polygons you will not regulate. These tessellations are constructed from the squares, triangles and hexagons, deforming them for the removes and puts method, that is to say him extracting parts of a side to a regular polygon, then to put them in the opposite side. Then this image repeats itself n times and they are placing so that they fit perfectly, using the isometric transformations (adjournment, rotation and symmetry).



The tessellation is non-periodic, when he lacks symmetry translacional some. Also when it is not possible to define a fundamental region or when the tessellation cannot move in two directions independent from the plane since it would coincide with itself. Inside these "less symmetrical" tessellations, there exist some of them to which they are appreciated by "certain regularity". They are that we name quasiperiodic tessellations. This type was proposed by Roger Penrose who on having tried to fill with tiles the plane with regular pentagons demonstrated that it is necessary to leave hollows. Penrose found a particular form of tessellation in which the hollows could be filled by other forms like a star, boat, diamond and pentagons. Later he thought that with the diamond it could create two tessellations more, commit one and an arrow.



The periodic tessellation would be those in the which it is possible to identify a fundamental region, the only piece (or block of pieces), with that to cover the whole plane with adjournments. They are the most habitual, very symmetrical mosaics, and are the tessellations that more used we are to seeing.In addition it is possible to displace in two directions independent from the plane and do that it coincides with itself.

The tessellations can be constructed across adjournment, rotation and axial symmetry. The first one, is achieved copying, moving and sticking the figure. The second one, taking a point and turning around it. Finally, the axial symmetry is the inverse movement in the plane in order that a figure belongs superimposable on his counterpart since for this it is not enough to slide it on the plane.

The tessellations come from the former civilizations that were using them for the construction of houses and temples near the year 4000 A.C. In this time the sumerian realized decorations with mosaics that geometric models were forming. They were using the cooked clay that they were coloring and enameling. Then, the Persian ones, Moors and Muslims demonstrated mastery in this type of work. Later, the mathematical group of the Pythagorean ones analyzed such constructions and probably these he has led them to the famous theorem that establishes that the sum of the interior angles is equal to a flat angle. The word tessellation comes of "tessellae". This way the Romans were calling to the constructions and pavements of his city. Nowadays, the mosaics and his symmetries are not exclusive of the artistic and handcrafted field. The teselados are in the habit of being used in Physics to search models of structure of the matter.

On the other hand, two methods exist to create tessellations, one it is the method of isommetry in which from the movements or transformations in the plane diverse designs can be achieved. And other one is the method there removes and puts, which consists of drawing a geometric figure that for itself tessellate the plane, as a parallelogram or a triangle. Then, him parts of a side are extracted, to put them in the opposite side. Later, this image repeats itself n times and they are placing so that they fit perfectly, using the isometric transformations (adjournment, rotation and symmetry). Escher became famous by his pictures of tessellations constructed by this method.

Finally, Maurits Cornelius Escher was born in 1898 in Leeuwarden (Netherlands). It tried to study Architecture, for paternal influence, but it left these studies to study Decorative Arts, taking as a teacher S. J. de Mesquita (for the one that felt great admiration and with the one that supported contacts, until the year 1944). As reports of the epoch reflect, Escher was not considered to be an authentic artist, for his lack of ideas and spontaneity. Between 1922 and 1923 he travelled for Italy and Spain settling itself in Siena. Of here it went to Ravello, where it knew his wife, Jetta Uniker, that he married in 1924 going to live through Rome. In 1926 his first son was born and until 1935 it travelled and lived for the whole Italy, fundamentally for the South. From 1935, the political climate forces him to go out of of Italy and Oex (Switzerland) happened to Chateaux-d ', but the lack of the Sun and sea makes to him be embarked by them in a cruise for the Mediterranean. He visited Granada, studying the ornaments of the Alhambra that so much had impressed him in a previous trip. In 1937 it moved to Ukkel (Belgium) and in 1941 it went to Baarn (Holland) where it settled itself and his production was more abundant. In 1970 it moved to an artists' residence in the North of Holland and there it expired in 1972. The interest for his work was big on the part of mathematicians and physicists, having illustrated multitude of scientific publications, spending for against unnoticed and ignored for the artistic world largely of his life. In 1937, the symmetry and the perspective, the continuity and the infinite, they constitute his major worry and though the critique of art it was not treating. Some of his works are: Reptiles (1943), Three worlds (1955), Relativity (1953), Day and night (1938), Circle limit the IVth (Sky and Hell) (1960), etc.