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Darts and kites in the sky of mathematics

A 16 minutes animation on the celebrated Penrose tiling

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 structures).

On the contrary the Penrose tessellation is a nonperiodic way to fill the plane, therefore there is no region that continually repeats itself. 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. The folder 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 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 by zero-project, 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.

The authors

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: