Golden Triangles and Plane non Periodical Penrose Tilings




An aperiodic Penrose tiling of the plane

Jean-François COLONNA
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CMAP (Centre de Mathématiques APpliquées) UMR CNRS 7641, École polytechnique, Institut Polytechnique de Paris, CNRS, 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:

 The Golden Rectangle



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
 The two subdivisions of the 'flat' Golden Triangle
 The 'flat' Golden Triangle = One of the two subdivisions of the 'flat' Golden Triangle + One of the two subdivisions of the 'flat' Golden Triangle  


 Slim
 The two subdivisions of the 'sharp' Golden Triangle
 The 'sharp' Golden Triangle = One of the two subdivisions of the 'sharp' Golden Triangle + One of the two subdivisions of the 'sharp' Golden Triangle  



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:

 A random triangular tiling of the plane



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

 A pseudo-periodical triangular tiling of the plane



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:

 An aperiodic Penrose tiling of the plane An aperiodic Penrose tiling of the plane
 An aperiodic Penrose tiling of the plane An aperiodic Penrose tiling of the plane An aperiodic Penrose tiling of the plane -a Tribute to Piet Mondrian and Roger Penrose- An aperiodic Penrose tiling of the plane -the extended paradoxal demoniac Rubik's Cube-
 An aperiodic Penrose tiling of the plane
 An aperiodic Penrose tiling of the plane An aperiodic Penrose tiling of the plane An aperiodic Penrose tiling of the plane



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

 An aperiodic Penrose tiling of the plane An aperiodic Penrose tiling of the plane
 An aperiodic Penrose tiling of the plane An aperiodic Penrose tiling of the plane
 An aperiodic non linear Penrose tiling of the plane An aperiodic non linear Penrose tiling of the plane An aperiodic non linear Penrose tiling of the plane An aperiodic non linear Penrose tiling of the plane
 A tridimensionally distorded aperiodic Penrose tiling of the plane A tridimensionally distorded aperiodic Penrose tiling of the plane
 An aperiodic Penrose tiling of the Golden Decagon A tridimensionally distorded aperiodic Penrose tiling of the Golden Decagon
 A tridimensionally distorded -by means of a fractal bidimensional height field- aperiodic Penrose tiling of the Golden Decagon A tridimensionally distorded aperiodic Penrose tiling of the Golden Decagon
 Untitled 0153 Untitled 0154
 An aperiodic Penrose tiling of the Golden Decagon Artistic view of an aperiodic Penrose tiling of the Golden Decagon Artistic view of an aperiodic Penrose tiling of the Golden Decagon Tridimensional visualization of an aperiodic Penrose tiling of the Golden Decagon


 A 'triple stack' of two aperiodic Penrose tilings of the plane



Copyright © Jean-François COLONNA, 2012-2026.
Copyright © CMAP (Centre de Mathématiques APpliquées) UMR CNRS 7641 / École polytechnique, Institut Polytechnique de Paris, 2012-2026.