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 only forms which fill the plane without leaving gaps, in a seamless way.

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

beings.

Textile design. Urban patterns

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…

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!

A mysterious tessellation: Durero's Pentagons

Durero’s fractals

The modular space: FACTORS

3.- COLOR CHANGE

1.-CREATION OF THE MODULE

2 .- DISTRIBUTION IN A NETWORK

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

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

Rectangular.

Ravine

.

Rhomboid

Radiated

Hexagon

COMPOSITION FROM A RED TRIANGLE

IRREGULAR: modules disposed in different shapes and varied resources.

.

Composite

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

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

Geometry and Algebra in Moorish art

The Mosaics of the Alhambra

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

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.

Y efectivamente comprobamos al observar todos estos mosaicos

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

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.

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

How did they get that?

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

Just turn them properly.

The “Nazari Bone" is obtained by deforming a square:

The "petal" is obtained by deforming a diamond:

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

The flying fish

The Nazarí dove

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:

MOSAICS FOUND IN

THE ALHAMBRA

Patterns in perspective

THREE-DIMENSIONAL EFFECTS

ADDED SHADE STRUCTURE

M.C. ESCHER: Cicle, 1938.

The Wonderful World of M. Escher

Penrose’s Diagrams

Penrose Universes: A

mathematical model for quasicrystals