Everything you always wanted to know
about Penrose tiling
(but were afraid to ask)
The Penros tiling is a NON-PERIODIC way to fill the plane
using two tiles: KITE (bright tiles) and DART (coloured tiles)
Actually the main feature of this tiling is the property of
APERIODICITY of the two tiles: it is NOT POSSIBLE using them to perform PERIODIC tilings
In the following we will see the meaning of APERIODIC TILING
and we will have to introduce ADJACENCY CONSTRAINTS between the tiles
in order to assure the aperiodicity property
The two tiles, 'kite' and 'dart', can be built
starting from a regular decagon with side 1 and radius 1.618 (the golden ratio)
according to the rules described in the animation
Curiously, we meet the ubiquitous GOLDEN RATIO
this is the DART
and this is the KITE
It is quite simple to fill all the plane with the two tiles:
by construction, they can be easily joined together
forming a rhombus (with side 1.618...)
But in this way we obtain a PERIODIC tessellation:
as you can see, the tiles can be shifted in many directions
without affecting the initial pattern.
A periodic tessellation requires translations in at least two different directions
with such an invariance property.
There are others symmetries for a rhomboidal tiling, e.g. rotations by 180 degrees
In order to forbid the periodic tiling, we can shape the tiles as in a jigsaw puzzle
as you can see in the animation
From now on, we will suppose the tiles with such a shape, even if they will not be shown
It is no longer possible to match kites and darts to obtain a rhombus,
however is seems still possible (at least by hand) to fill the entire plane.
How can we be sure that it is REALLY possible to fill the entire plane without holes?
Well, we are going to describe an operational procedure which leaves no holes.
In the first step we SPLIT the tiles, obtaining two isosceles triangles
KITE: two acute isosceles triangles
DART: two obtuse isosceles triangles
The animation shows that the sides of the triangles are in the golden ratio
Acute triangles: base = 1, side = 1.618...
Obtuse triangles: side = 1, base = 1.618...
The main idea is to split up the triangles in smaller copies of themselves
Acute triangles: two acute triangles and one obtuse triangle
Obtuse triangles: one obtuse triangle and one acute triangle
Let us carry out the procedure...
We begin with whatever initial configuration we want (the axiom)
for instance, a decagon made up by five kites
Then we apply the subdivision procedure on each triangle, obtaining a more refined subdivision
Look: the new triangles can piece together again kites and darts
Finally, we blow up all the tiles to the original size
and repeat once more the procedure
Now we have to split also some obtuse triangles
The procedure obeys the adjacency constraints between the tiles
again we blow up the tiles to the original size
Third step...
Fourth step...
A central decagon exactly alike the initial configuration has appeared!
Fifth step...
Sixth step...
Seventh step...
Eight step...
We can go on, tiling a bigger and bigger part of the plane
This is the result after nine steps
In conclusion, we have a procedure which tiles all the plane obeying the adjacancy constaints.
Due to the initial configuration (a decagon made up by five kites), the resulting tiling
will have a rotational symmetry (by 72 degrees)
and also axial symmetries (reflections with respect to lines by the center)
We can also modify a little the silhouette of the DART and the KITE, obtaining new tiles
in the spirit of Escher's artworks
Le we employ the new forms in the tessellation...
The result has the same structure as the KITE and DART tiling
Here is the result after five steps of deflation/inflation
A remarkable option can be found by a suitable choice of two different rhombi
However, there is a close relation with Kite and Dart, as you can see in the animation
The rhombi are split in two isosceles triangles
which are equal to those we have already seen, apart from the proportions
We can perform a procedure of deflation/inflation starting from a suitable subdivision of the golden triangles
similarly to the subdivision of Dart and Kite
After six steps of the new procedure of deflation/inflation
starting from a suitable initial configuration
we obtain the tiling with rhombi...
Roger Penrose tried to split a large regular pentagon into six smaller pentagons
which side lenght is the inverse square of the golden ratio of the large side
Repeating the subdivision a few times, some holes remain
Some of them have the shape of a regular pentagon, hence they can be included in the subdivision procedure
on the contrary, some other holes have different shapes: rhombus, crown, star
The three different colors of the pentagons are useful to mark adjacency constraints
which induce a non periodic tiling.
In such a way we get a set of six tiles which have the aperiodicity property
Up to now it is not known if there exists a single tile which has the aperiodicity property!
