1. Seamless Patterns

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 hexagon
are 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. VASARELY
Tau-Ceti,
W. WONG.
UNKU
Perú, s. XII-XIII.
A. GAUDÍ Pabellón Güell,

5. 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!
Miguel Sánchez. Graphic Design for 2º ESO

6. A mysterious tessellation: Durero's Pentagons

7. Durero’s fractals

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

9. The modular composition
Modular networks are geometric structures that relate modules
REGULAR: They use a single regular polygon that is repeated.
Triangular grid Repetition of a equilateral triangle.
Grid
Repetition of a square
Hexagonal grid
Hexagon recurrence
SEMI-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 º
Miguel Sánchez. Graphic Design for 2º ESO

11. IRREGULAR: modules disposed in different shapes and varied resources.
Rectangular.
Ravine
. Rhomboid
Radiated
COMPOSITION FROM A RED TRIANGLE
. Composite
Hexagon

12. Super-and sub-modules
OVERLAP: This consists of networks or modules mounted on top of each other for more complex structures
Overlapping
Kamal Ali’s Module
Super-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 30º
Moving modules.
And so, proceed to fill, or not, all the compositional plane
Miguel Sánchez. Graphic Design for 2º ESO

14. Geometry and Algebra in Moorish art
The Mosaics of the Alhambra
Geometry 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 religion
The 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 polygons
Just 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:

26. MOSAICS FOUND IN THE ALHAMBRA

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

29. The Wonderful World of M. Escher

34. Penrose’s Diagrams

35. Penrose Universes: A mathematical model for quasicrystals