#
Golden Triangles and Plane non Periodical Penrose Tilings

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Jean-François COLONNA

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jean-francois.colonna@polytechnique.edu

CMAP (Centre de Mathématiques APpliquées) UMR CNRS 7641, Ecole Polytechnique, CNRS, 91128 Palaiseau Cedex, France

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**[en français/in french]**

**Keywords**: Golden Ratio, Nombre d'or, Golden Rectangle, Rectangle d'or, Golden Triangle, Triangle d'or, Plane Non Periodical Tiling, Pavage Non Périodique du Plan, Penrose Tiling, Pavage de Penrose.

The **Golden Ratio** (**phi**) is a very famous number as well known as pi. It is the positive solution of the second degree equation:
2
x = x + 1

and has the value:
___
1 + \/ 5
phi = ----------- ~ 1.6180339887498949
2

The **Golden Rectangle** is a rectangle whose edge ratio is equal to phi.
It is known as being the most pleasant rectangle to watch and appears white in the following picture:

It is possible to define two **Golden Triangles** (dubbed *Flat* et *Slim* respectively).
They are isosceles and their edge ratios are equal to phi and 1/phi respectively.
It is worth noting that they can be subdivided using two different symetrical ways
in a certain number of smaller triangles of the same nature:

**Flat**
= +

**Slim**
= +

This subdivision process can be repeated again and again at smaller and smaller scales.
It gives birth to a non periodical tiling of the plane (inside the first triangle, the bigger one) in particular
when using a random choice between the *red* subdivision and the *green* one.
Here is an example:

With a more complex choice between the two subdivisions, one can associate systematically two triangles of the same kind:

Then, erasing the common edge inside each pair of the preceding triangles one obtains a non periodical tiling
of the plane as devised by Roger Penrose:

At last, one can "play" with these elements:

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Copyright © Jean-François Colonna, 2012-2021.

Copyright © CMAP (Centre de Mathématiques APpliquées) UMR CNRS 7641 / Ecole Polytechnique, 2012-2021.