Seamless Patterns. A Module it s the basic unit that allows to compose 2-D or 3-D structures by...

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Seamless Patterns

Transcript of Seamless Patterns. A Module it s the basic unit that allows to compose 2-D or 3-D structures by...

Page 1: Seamless Patterns. 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.

Seamless Patterns

Page 2: Seamless Patterns. 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.

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.

Page 3: Seamless Patterns. 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.

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

beings.

Textile design. Urban patterns

Page 4: Seamless Patterns. 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.

UNKU

Perú, s. XII-XIII.

A. GAUDÍ Pabellón Güell, 1884-1887.

V. VASARELY

Tau-Ceti, 1955-1965W. WONG.

Too many artists have used the modules and networks to create their works…

Page 5: Seamless Patterns. 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.

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!

Page 6: Seamless Patterns. 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.

A mysterious tessellation: Durero's Pentagons

Page 7: Seamless Patterns. 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.

Durero’s fractals

Page 8: Seamless Patterns. 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 modular space: FACTORS

3.- COLOR CHANGE

1.-CREATION OF THE MODULE

2 .- DISTRIBUTION IN A NETWORK

Page 9: Seamless Patterns. 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 modular The modular compositioncompositionModular networks are geometric structures that relate modules

GridRepetition of a square

Triangular gridRepetition of a equilateral

triangle.

Hexagonal gridHexagon recurrence

REGULAR: They use a single regular polygon that is repeated.

SEMI-REGULAR: They use two or more regular

polygons

Page 10: Seamless Patterns. 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.

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

Page 11: Seamless Patterns. 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.

Rectangular.

Ravine

.

Rhomboid

Radiated

Hexagon

COMPOSITION FROM A RED TRIANGLE

IRREGULAR: modules disposed in different shapes and varied resources.

.

Composite

Page 12: Seamless Patterns. 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.

Overlapping

OVERLAP: This consists of networks or modules mounted on top of each other for more complex structures

Super-and sub-modules

Kamal Ali’s Module

Page 13: Seamless Patterns. 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.

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

Page 14: Seamless Patterns. 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.

Geometry and Algebra in Moorish art

The Mosaics of the Alhambra

Page 15: Seamless Patterns. 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.

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

Page 16: Seamless Patterns. 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 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.

Page 17: Seamless Patterns. 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.

Y efectivamente comprobamos al observar todos estos mosaicos

que ningún punto es singular ni más importante que los demás.

Page 18: Seamless Patterns. 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.

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.

Page 19: Seamless Patterns. 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.

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

Page 20: Seamless Patterns. 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.

How did they get that?

The answer is simple: the figures used come from regular polygons

Just turn them properly.

Page 21: Seamless Patterns. 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 “Nazari Bone" is obtained by deforming a square:

Page 22: Seamless Patterns. 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 "petal" is obtained by deforming a diamond:

Page 23: Seamless Patterns. 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 " Nazarí bow" is obtained by deforming a triangle:

Page 24: Seamless Patterns. 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 flying fish

The Nazarí dove

Page 25: Seamless Patterns. 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.

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:

Page 26: Seamless Patterns. 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.

MOSAICS FOUND IN

THE ALHAMBRA

Page 27: Seamless Patterns. 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.
Page 28: Seamless Patterns. 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.

Patterns in perspective

THREE-DIMENSIONAL EFFECTS

ADDED SHADE STRUCTURE

M.C. ESCHER: Cicle, 1938.

Page 29: Seamless Patterns. 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 Wonderful World of M. Escher

Page 30: Seamless Patterns. 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.
Page 31: Seamless Patterns. 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.
Page 32: Seamless Patterns. 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.
Page 33: Seamless Patterns. 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.
Page 34: Seamless Patterns. 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.

Penrose’s Diagrams

Page 35: Seamless Patterns. 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.

Penrose Universes: A

mathematical model for quasicrystals

Page 36: Seamless Patterns. 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.

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