Don't Miss

# Plus Magazine Monthly Column – December 2016

By on 05/12/2016

Plus magazine is very proud to start a monthly column for Mathematics in Europe!

Plus (http://plus.maths.org) is a free online magazine about mathematics aimed at a general audience. It is part of the Millennium Mathematics Project, based at the University of Cambridge and our aim is to open a door on the world of maths for everyone. We run articles, videos and podcasts on all aspect of mathematics, from pure maths and theoretical physics to mathematical aspects of art, medicine, cosmology, sport and more. Plus has a news section, covering news from the world of maths as well as the maths behind the mainstream news, reviews of books, plays and films, as well as puzzles for you to sharpen your wits.

Credits: Tony Phillips

In this column we’ll bring you some of our favourite recent content. This month, topology has been looming large. A beautiful article by Tony Phillips (courtesy of the AMS) explores the topology of music. A one-voice musical score has two dimensions: time, running on a horizontal axis, and pitch, running on a vertical axis. If the score repeats, that is, plays the same sequence of notes over and over, then it has the topology of a cylinder: the score is represented by a two-dimensional strip, and the fact that it repeats means you can glue together the ends of that strip to get a cylinder.

Some pieces of music, however, conceal more complicated shapes. Phillips’ article (based on work with Eric Altschuler) describes a piece by Johann Sebastian Bach in which four voices sing two canons in parallel. The piece is periodic (it repeats), but also contains the symmetry of a glide reflection: when you shift the score along and then flip it vertically, it looks the same; and if you shift in the same direction and flip again, you’re back where you started. This means that the score can be encoded on a Möbius strip, as shown in the image above. As you can see, two copies of each note now match exactly (except for their tails). You can find out more here.

The shape of things to come

Our second helping of topology comes from a very different area. In October David Thouless, Duncan Haldane and Michael Kosterlitz were awarded the 2016 Nobel Prize in Physics for their work using methods from topology to understand strange phenomena in materials. Plus editor Rachel Thomas spoke to Fiona Burnell, a mathematical physicist at the University of Minnesota, to explore the maths and science behind the work.

Topology is the area of maths that describes the properties of objects that remain unchanged (called invariant properties) when you are allowed to bend, stretch or squash the object, but not cut or tear it. The most famous example is that, topologically speaking, a doughnut is the same as a coffee cup. If a doughnut was made out of clay you could bend and stretch it to form a coffee cup. But you can’t form a clay ball into the shape of a coffee cup without tearing a hole – they are topologically different shapes.

Thouless, Haldane and Kosterlitz haven’t been recognised for their ability to squash or stretch physical materials or rubber bands. Instead they used the topology of the states of quantum particles to understand surprising observations and predict strange new states of matter. For example, Thouless realised that a phenomenon called the quantum hall effect, is explained by the topology of shapes related to the equation that describes the behaviour of electrons in a solid. Kosterlitz and Thouless discovered and explained a previously unknown phase transition using topology, and Haldane predicted that edges or surfaces of some materials have fundamentally different properties to the rest of the material. The mathematical methods developed by Thouless, Haldane and Kosterlitz have gone on to have huge impacts in condensed matter physics and have uncovered a wealth of new topological materials that could result in smaller, faster conventional computers, or even lead the way to elusive quantum computers. You can find out more here.

Coffee, doughnuts and cake

Doughnuts and coffee cups bring us neatly to a favourite news story we ran recently on Plus. It covers a new result in the theory of cake cutting. This isn’t as frivolous as it seems: a cake can represent any continuous object, be it a piece of land, broadcasting time, or oil. Cake cutting theory is inspired by problems that come up in all sorts of contexts, from divorce proceedings to political conflicts.

The problem of dividing a cake fairly is quite simple when there are only two people involved. Get the first person to cut the cake and the second to choose the piece they want. The first person will make sure they cut the cake so that he or she is content with either of the two pieces. Depending on preference, and what’s on the cake, he or she might cut it into two equal halves, or into a smaller piece that comes with a particular bonus (a strawberry, say) and a larger one. Whatever piece the second person chooses, the first won’t be left envious. The second person might not like the way the cake was cut, but, since he or she got to pick first, also won’t be envious of the other person’s piece. Generally, a division method is called envy-free if no person would prefer another person’s allocation to their own.

When more than two people are involved, the problem of dividing the cake in an envy free way becomes tricky. In 1995 Steven J. Brams and Alan D. Taylor came up with a method that works for any number of people, but has a drawback. Even when only four cake eaters are involved, the number of cuts required to make the fair division could be arbitrarily large. The number of questions you need to ask cake eaters in order to find the division — which relates to the time it takes the method to complete — could also be arbitrarily large. This is particularly frustrating because mathematicians know for sure that for cake eaters there is an envy-free division that requires only n-1 cuts. It’s finding that division which is the problem.

The new method, devised by Haris Aziz and Simon MacKenzie, does comes with a bound on the number of cuts needed and the number of questions you need to ask of the cake eaters to find the division, and is also discrete: it could be implemented on a computer. This doesn’t mean that conflicts over cake-like goods can now be solved in a flash. The bound on the number of questions you need to ask when there are n  cake eaters is

This can lead to unimaginably large numbers: even for n=2 standard calculators will give up. The result is of theoretical interest only, but from the theoretical viewpoint it counts as a breakthrough.