Darts and kites in the sky of mathematics
A 16 minutes animation on the celebrated Penrose
Alessandro Musesti (
Maurizio Paolini (
Although it might look strange, mathematics also studies the placement of
tiles on a floor! If the floor is an infinite plane, we have a tiling of the plane.
A tiling is a way to cover the plane using nonoverlapping
copies from a small set of geometric shapes.
In a periodic tessellation there is a region that repeats
itself by translations.
Mathematicians discovered that there are exactly 17 different kinds
of periodic tessellations (the well known tessellations that cover the
Alhambra walls in Granada contain examples of each of the existing 17
On the contrary the Penrose tessellation is a nonperiodic way
to fill the plane, therefore there is no region that continually
In a sense one could say that the Penrose tessellation is very varied,
never repeating itself.
To construct it, we use two forms: darts (dark colored tiles)
and kites (light colored tiles).
The fascinating thing is that the Penrose tessellation is not only
non-periodic, but there is no way to place the tiles in a periodic
way: like the pieces of a jigsaw puzzle, the way to fit darts and
kites invariably leads to a non-periodic tessellation of the plane.
This short movie (16 minutes long) explains the construction
of the Penrose tessellation and shows some of its properties.
Content of the dvd...
It is a DVD-VIDEO, then you can play it with any DVD player;
in the menu you can choose the language (there are three choices at the moment)
and show some quick informations.
EXTRA of the DVD, which can be viewed on a PC,
contains the movie in a higher resolution and some additional material.
A few comments on the soundtrack
The Variations on a Rococo Theme, by Tchaikovsky, were chosen as soundtrack in the early versions of the animation.
The result was good, but there was a nasty problem related to the copyright (of the performance).
It is a shame that many artistic products cannot be used by others,
only because the copyright holders are unavailable.
Fortunately, nowadays many authors choose to make their works available without restrictions.
The website www.jamendo.com has a huge collection
of free music (usually published under Creative Commons license, the same of this DVD).
On that website we found the Ambient Symphony album
from which we took the songs for the current soundtrack.
What can you do with DVD?
Everything you want! Indeed, the license we have chosen for this work
is the Creative Commons. It allows people to share, study and adapt
the work, provided that they respect the original attribution
and that they use it for non-commercial purposes.
Sharing is not only allowed, it is encouraged.
Maurizio Paolini and Alessandro Musesti teach Mathematics at the
Department of Mathematics and Physics
Niccolò Tartaglia at the Catholic University of the Sacred Heartk
in Brescia, Italy.
You can contact them by email: