Darts and kites in the sky of mathematics
The Penrose tessellation
(http://frecceaquiloni.dmf.unicatt.it/)
A. Musesti - M. Paolini
Department of Mathematics and Physics
Catholic University - Brescia
Made with www.povray.org
Penrose tiling is a NON-PERIODIC way to fill a plane
using two tiles: the KITE (light tiles) and the DART (dark tiles)
Actually the main feature of this tiling is that
the two tiles are APERIODIC: they cannot be used to perform PERIODIC tilings
Now we will see the meaning of APERIODIC TILING
and we will have to introduce ADJACENCY CONSTRAINTS between the tiles
in order to ensure that they are indeed aperiodic
the making of darts and kites
The two tiles, the 'kite' and the 'dart', can be built
starting from a regular decagon with side of 1 and radius of 1.618 (the golden ratio)
according to the rules described in the animation
Surprisingly, we meet the ubiquitous GOLDEN RATIO
this is the DART
and this is the KITE
periodic tessellation
It is quite simple to fill the whole plane with the two tiles:
by construction, they can be easily joined together
forming a rhombus (with a side of 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 the initial pattern maintained.
There are other symmetries for a rhomboidal tiling, e.g. rotations by 180 degrees
by the way, rotations are not symmetries of our periodic tessellation
indeed, the colors move and the initial pattern is not recovered
Note that the symmetries have to move ALL the tessellation, not only a few tiles
In this case we obtain the symmetry group named "cm" in crystallography
We can obtain all possible symmetries by combining two translations
and a reflection about a vertical axis
Of course, other "generators" can be chosen
A rotation of 180 degrees about a rhombus does not work!
Unless we consider a tessellation where only rhombi are present,
achieving the group "cmm"
enforcing adjacency constraints
In order to prevent periodic tiling, we can shape the tiles as in a jigsaw puzzle
as you can see in the animation
From now on, we will presume that the tiles are this shape, even if they will not be shown
trial and error
It is no longer possible to match kites and darts to obtain a rhombus,
however it still seems 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.
breaking the tiles
In the first step we SPLIT the tiles, obtaining two isosceles triangles
KITE: two acute isosceles triangles
DART: two obtuse isosceles triangles
golden 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...
subdivision
The main idea is to split up the triangles into smaller copies of themselves
Acute triangles: two acute triangles and one obtuse triangle
Obtuse triangles: one obtuse triangle and one acute triangle
deflation inflation
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 pair together to form kites and darts
Finally, we blow up all the tiles to the original size
and repeat the procedure once more
Now we also have to split some obtuse triangles
The procedure complies with the adjacency constraints between the tiles
we blow up the tiles again to the original size
Third step...
Fourth step...
A central decagon exactly alike the initial configuration has appeared!
Fifth step...
Sixth step...
Seventh step...
Eighth step...
We can go on, tiling a larger and larger part of the plane
This is the result after nine steps
In conclusion, we have a procedure which tiles the whole plane and respects the adjacency constaints.
tessellation symmetries
Given 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 about five axes through the center)
as suggested by some of the movements that you can see
There are precisely ten symmetries, forming the dihedral group "D5" (symmetries of a pentagon)
The group can be generated by a rotation of 72 degrees
...and a reflection about a suitable axis through the center
birds... and reptiles
We can modify the silhouette of the DART and the KITE, obtaining new tiles
in the spirit of Escher's art
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
fat rhombi thin rhombi
A remarkable option can be found by a suitable choice of two different rhombi
However, there is a close relationship with the Kite and Dart, as you can see in the animation
The rhombi are split into two isosceles triangles
which are like those we have already seen, apart from the proportions
We can perform a deflation/inflation procedure starting from a suitable subdivision of the golden triangles
similar 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...
the original Penrose tiles
Roger Penrose tried to split a large regular pentagon into six smaller pentagons
scaled by the square of the inverse of the golden ratio
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
Some other holes have different shapes: rhombus, crown, star
The three different colors of the pentagons are useful to mark adjacency constraints
that induce a non periodic tiling.
In such a way we get a set of six tiles which are aperiodic
To date it is not known if there is a single tile having this property of aperiodicity!
Darts and kites in the sky of mathematics
In 1974 Roger Penrose conceived the fascinating tilings named after him
In them we find a mix of mathematics, geometry, physics and art in a surprising blend
Making this short animation was a demanding yet enjoyable challenge.