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Seamless Patterns

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A Module it s the basic unit that allows to compose 2-D or 3-D structures by repetition" The square, the triangle and the hexagon are the only forms which fill the plane without leaving gaps, in a seamless way.

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We can find several examples of seamless patterns in our everyday life. Cellular structures of living beings. Textile design. Urban patterns

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UNKU Perú, s. XII-XIII. A. GAUDÍ Pabellón Güell, V. VASARELY Tau-Ceti, W. WONG. Too many artists have used the modules and networks to create their works…

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Pentagonal tilings 14 types of pentagonal tilings with irregular pentagons have been discovered Ms. Marjorie Rice discovered four of them. She is not a professional mathematician, but a housewife who makes some very nice quilts!

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A mysterious tessellation: Durero's Pentagons

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Dureros fractals

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The modular space: FACTORS 3.- COLOR CHANGE 1.- CREATION OF THE MODULE 2.- DISTRIBUTION IN A NETWORK

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The modular composition Modular networks are geometric structures that relate modules Grid Repetition of a square Triangular grid Repetition of a equilateral triangle. Hexagonal grid Hexagon recurrence REGULAR : They use a single regular polygon that is repeated. SEMI-REGULAR : They use two or more regular polygons

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Semi-regular Two conditions 1 - All polygons have equal sides 2 - The sum of the angles of polygons around a nodule is worth 360 º

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Rectangul ar. Ravine. Rhomboid Radiated Hexagon COMPOSITION FROM A RED TRIANGLE IRREGULAR: modules disposed in different shapes and varied resources.. Composit e

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Overlapping OVERLAP: This consists of networks or modules mounted on top of each other for more complex structures Super-and sub- modules Kamal Alis Module

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To repeat the modules, we use dynamic geometry based on the composition of motions in the plane: By resources of symmetry. By turns. And so, proceed to fill, or not, all the compositional plane Moving modules. Giro de 30º

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Geometry and Algebra in Moorish art The Mosaics of the Alhambra

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These decorative motifs are found almost everywhere in the Alhambra in Granada

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The main reasons of this explosion of geometry in the Spanish-Muslim art are found in religion The Koran prohibits any iconic depiction of Allah. Divinity is identified with the singularity.

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Y efectivamente comprobamos al observar todos estos mosaicos que ningún punto es singular ni más importante que los demás.

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Lo que se mueve en el plano son polígonos regulares, de tal forma que: - No quede espacio ninguno del plano sin cubrir. - No se superpongan unos polígonos con otros.

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They can cover the plane with figures that are not regular polygons …

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How did they get that? The answer is simple: the figures used come from regular polygons Just turn them properly.

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The Nazari Bone" is obtained by deforming a square:

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The "petal" is obtained by deforming a diamond:

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The " Nazarí bow" is obtained by deforming a triangle:

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The flying fish The Nazarí dove

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Although it seems that there are many structures in these mosaics, everyone adjusts to 17 different models. These models were investigated by Fedorov in the late 19th century, and it was the mathematician who proved that any tiling of the plane is a set of one of these 17 configurations. And here we have them all:

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MOSAICS FOUND IN THE ALHAMBRA

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Patterns in perspective THREE-DIMENSIONAL EFFECTS ADDED SHADE STRUCTURE M.C. ESCHER: Cicle, 1938.

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The Wonderful World of M. Escher

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Penroses Diagrams

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Penrose Universes: A mathematical model for quasicrystals

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