2 The square, the triangle and the hexagon 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 hexagonare the only forms which fill the plane without leaving gaps, in a seamless way.Miguel Sánchez. Graphic Design for 2º ESO
3 We can find several examples of seamless patterns in our everyday life. Cellular structures of living beings.Textile design.Urban patterns
4 Too many artists have used the modules and networks to create their works… V. VASARELYTau-Ceti,W. WONG.UNKUPerú, s. XII-XIII.A. GAUDÍ Pabellón Güell,
5 Pentagonal tilings14 types of pentagonal tilings with irregular pentagons have been discoveredMs. Marjorie Rice discovered four of them. She is not a professional mathematician, but a housewife who makes some very nice quilts!Miguel Sánchez. Graphic Design for 2º ESO
8 The modular space: FACTORS 1.-CREATION OF THE MODULE2 .- DISTRIBUTION IN A NETWORK3.- COLOR CHANGE
9 The modular composition Modular networks are geometric structures that relate modulesREGULAR: They use a single regular polygon that is repeated.Triangular grid Repetition of a equilateral triangle.GridRepetition of a squareHexagonal gridHexagon recurrenceSEMI-REGULAR: They use two or more regular polygons
10 Semi-regular Two conditions 1 - All polygons have equal sides 2 - The sum of the angles of polygons around a nodule is worth 360 ºMiguel Sánchez. Graphic Design for 2º ESO
11 IRREGULAR: modules disposed in different shapes and varied resources. Rectangular.Ravine. RhomboidRadiatedCOMPOSITION FROM A RED TRIANGLE. CompositeHexagon
12 Super-and sub-modules OVERLAP: This consists of networks or modules mounted on top of each other for more complex structuresOverlappingKamal Ali’s ModuleSuper-and sub-modules
13 By resources of symmetry. By turns. To repeat the modules, we use dynamic geometry based on the composition of motions in the plane:By resources of symmetry.By turns.Giro de 30ºMoving modules.And so, proceed to fill, or not, all the compositional planeMiguel Sánchez. Graphic Design for 2º ESO
14 Geometry and Algebra in Moorish art The Mosaics of the AlhambraGeometry and Algebra in Moorish art
15 These decorative motifs are found almost everywhere in the Alhambra in Granada
16 The Koran prohibits any iconic depiction of Allah. The main reasons of this explosion of geometry in the Spanish-Muslim art are found in religionThe Koran prohibits any iconic depiction of Allah.Divinity is identified with the singularity.
17 Y efectivamente comprobamos al observar todos estos mosaicos que ningún punto es singular ni más importante que los demás.
18 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.
19 They can cover the plane with figures that are not regular polygons…
20 The answer is simple: the figures used come from regular polygons How did they get that?The answer is simple: the figures used come from regular polygonsJust turn them properly.
21 The “Nazari Bone" is obtained by deforming a square:
22 The "petal" is obtained by deforming a diamond:
23 The " Nazarí bow" is obtained by deforming a triangle:
24 The flying fish The Nazarí dove Miguel Sánchez. Graphic Design for 2º ESO
25 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: