Butterflies, billiards, and pretzels - exploring slow chaos

Chaos is all around us. Typically, we use the term to describe disorder. However, chaos has quite a different meaning in mathematics: what mathematicians call “chaotic systems” are actually defined by clear rules. For researchers, this is not a contradiction. Nevertheless, it is extremely difficult to predict chaotic behavior: just think of the weather, the financial markets, or the orbits of celestial bodies.

Examples of chaotic behaviour: weather, financial markets, asteroid

Chaotic systems are a hot topic in mathematics. To better understand them, researchers deploy innovative approaches and idealizing models. With a few tricks up their sleeves, they combine chaos with something mathematically elegant – whereby the beauty lies in the development of the solution.

“The beauty of mathematics only shows itself to more patient followers.” Maryam Mirzakhani (1977 - 2017), mathematician

A key feature of chaotic behavior is what has been termed the ”butterfly effect” – a small change in an initial condition such as the flapping of a butterfly’s wings results, over time, in a very different outcome (in this example, a tornado).

The game of billiards is often used to demonstrate the butterfly effect. Billiards players know from experience that the slightest change in angle between cue and ball results in the ball coming to rest in dramatically different places once it has hit the rims a few times.

Conventional billiards

The objective of the popular game of billiards is to sink specific balls into the side pockets of the table while bringing the other balls to rest in certain positions. The behavior of the balls is governed by two rules:

Rule 1: Balls move in straight-line trajectories until they hit the rims.

Rule 2: The impact angle of ball against rim is identical to the rebound angle off the rim in a new direction.

Mathematical billiards

Mathematical billiards is an idealization of the game of billiards and acts as a ‘playground’ – model – for mathematicians looking to study chaos. It has two additional rules:

Rule 3: There are no pockets. The objective is not to sink the ball, but to study its trajectory.

Rule 4: The ball is a point with no mass, so there is no friction. Thus, the ball never stops moving – its trajectory continues infinitely.

The trajectories of billiard balls can be used to describe phenomena in both optics and acoustics, because light and sound waves travel in the same way as a billiard ball by following rules 1 and 2 (see previous table). Perhaps more surprisingly, these trajectories also appear in the description of numerous other systems: e.g. mechanics, the kinetic theory of gases, and solid state physics.

The ‘tables’ in mathematical billiards used for modeling many natural systems are not like ordinary rectangular tables: they can be unusual shapes and can contain obstacles and barriers. Different shapes lead to different chaotic characteristics; this is why billiards makes an ideal ‘playground’ for mathematicians studying aspects of chaos.

With rectangular (oblong or square) mathematical billiards, only two types of behavior are possible:

1. The ball can keep repeating the same trajectory (periodic trajectory), or

2. The ball’s trajectory can fill the table area (dense trajectory).

When the shape of the mathematical billiard table is changed, more complicated trajectory patterns can appear. Non-rectangular tables allow for a greater number of trajectory patterns. A trajectory can also

a) be neither periodic, nor dense, or

b) fill the area unevenly.

Finding such patterns is like looking for a needle in a haystack. Only the application of powerful mathematical tools for dealing with surfaces makes their prediction and explanation possible.

To answer many questions about mathematical billiard trajectories on polygonal tables – such as “At what angle should you hit a billiard ball to achieve a periodic trajectory?” – mathematicians use a clever trick called ‘Unfolding’.

Unfolding is an imaginary process and – from a mathematical point of view – an extremely elegant solution that involves reflecting a square table over and over, allowing the ball’s trajectory to be a straight line. This may seem strange, but as often happens in mathematical research, it offers a new perspective that simplifies the problem. Now that the trajectory is just a straight line crossing a succession of squares, one can, for example, see precisely which ball trajectories are periodic.

Try it out for yourself by playing the Unfolding game on the table in the exhibition.

The ball’s trajectory on a square table (represented here as a tile) becomes a straight line through unfolding. The unfolded trajectory runs in a straight line over the edges of the squares, which are reflected repeatedly. The colors of the edges make it clear that the squares are reflections of each other. Four tables (corresponding to the number of sides of a square) are sufficient to represent all possible trajectories as a straight line, since the orientation of the fifth table is identical to the first (Fig. C). In the next section, read how a 3D surface can be created from such a plane (a plane is a flat two-dimensional surface that extends indefinitely).

Unfolding square and rectangular tables into a plane helps answer many questions about trajectories. But this trick works for very few polygonal-shaped tables.

Why is that? Play the Laying Tiles game on the table in the exhibition.

Trajectories on other polygonal tables can be unfolded to become trajectories on surfaces such as those of a donut or a pretzel. This new trick helps to better understand billiards, because mathematicians over the ages have developed very powerful techniques for studying and understanding motion on surfaces.

From billiards to surfaces: if you stick the opposite sides of a paper square together, you get a cylinder (Figs. A & B). If the cylinder were made of rubber and you could bend it round and stick its two ends together, the result would be the surface of a donut (Figs. B & C). The ball’s trajectory on the formerly unfolded square looks like a line moving in and out. Once the square has changed shape through gluing, the trajectory now wraps around the donut (Fig. C).

Different polygons unfold into different surfaces. For example, S-shaped or diamond-shaped billiard tables can unfold into pretzels with three “holes”, known as a “genus 3” surface in mathematics. The donut, on the other hand, would be a genus 1 surface, as it has only one hole.

Mathematicians distinguish between fast and slow chaos according to how quickly the butterfly effect occurs, i.e. the speed with which an initially small change (e.g. in the initial position of the ball) propagates and magnifies. Different shapes of obstacles (or tables) lead to different chaotic behaviors, fast or slow.

The parallel trajectories of two billiard balls will diverge on hitting a round obstacle: because the impacts are in different places on the obstacle’s surface, the balls will scatter in very different directions. This mechanism is an example of fast chaos.

By contrast, if the obstacle is rectangular, only hitting (or missing) the corner of the obstacle makes the ball trajectories diverge. Here, the butterfly effect is much slower to become apparent. This is an example of slow chaos.

Over 100 years ago, physicists proposed infinite billiard tables as simplified models of gases and metals. In some, the billiard balls represent electrons traveling on an infinite plane, encountering an infinite series of obstacles (e.g. atoms). If these are round, this is an example of fast chaos (see the “Lorentz model” below). A few years later, the Ehrenfest model with its rectangular obstacles was presented as a simplified variation of the Lorentz model – an example of slow chaos.

The Lorentz model (1905) is an example of fast chaos. Advances in mathematical understanding were already being achieved in the 1970s.

By contrast, the Ehrenfest model (1912) with its rectangular obstacles was introduced at the time as a simplified form of the Lorentz model.

Fast chaotic systems bear similarities to random behavior, because they can be understood and described by probabilistic predictions. It turned out that the Ehrenfest model is incomparably more difficult to describe mathematically.

Recent progress has also been made in the research of infinite, slow chaotic billiards. However, many questions are still not fully understood and remain unanswered.

UZH Professor Corinna Ulcigrai and her co-author Fraczek demonstrated that billiard ball trajectories, unexpectedly and unlike fast chaotic models, are usually not dense on the Ehrenfest billiard table. UZH Professor Artur Avila and his collaborator Hubert showed that, nevertheless, most trajectories in an Ehrenfest model come back arbitrarily close to their starting point.

Prof. Corinna Ulcigrai and her research team at UZH are at the forefront of mathematical research into slow chaotic systems, particularly on surfaces. Many physical phenomena including the movement of electrons in gases and metals can be reduced to the description of the motion of points on surfaces, such as the motion of bodies in celestial mechanics, or (in solid state physics) electrons in metals in magnetic fields.

Prof. Corinna Ulcigrai

Mathematician

Origin: Italy, at UZH since 2018

“My quest is to uncover the subtle chaotic features and fundamental mechanisms that generate slow chaos, and, on the way, to discover more beautiful mathematics…”

Example from solid state physics: electrons in metals. The Novikov model describes the movement of electrons in metals in an external magnetic field. It leads to the study of the motion of a point on an infinite periodic surface (Fig.: An example of the Fermi level of a metal surface).

You can find out more about the research of Prof. Corinna Ulcigrai in the videos on the tochscreen in the exhbtion.

Prof. Artur Avila

Mathematician

Origin: Brazil, at UZH since 2018

Artur Avila has worked on many aspects of both fast and slow chaotic systems, including billiards and fractals. A recipient of the Fields Medal for some of his groundbreaking discoveries.

Prof. Claire Burrin

Mathematician

Origin: Switzerland, at UZH since 2022

Claire Burrin’s work deals primarily with number theory. Using novel techniques in this field, she has been able to provide highly accurate descriptions of periodic trajectories in polygonal billiards.