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__** Wow nothing to say Jessy and Bertha... Great job. What about Diego?? 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 his figure in regular, semi-regular, demi-regular and irregular and in addition they can qualify according to his 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 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 ** teselado ** (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 ** teselados ** (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.



Correction: Jessica De Sousa