# Dancing wallpaper patterns: the beautiful mathematics of tango

Dancing Wallpaper Patterns: 2019 Festival of Creative Learning

by Imogen Morris

I started dancing Argentine tango 2 years ago, when I began my PhD on the formalisation of mathematical proofs. At first, I thought tango was just going to be a relaxing change from the intense alternation of frustration and eureka! that my PhD research involved. But I began to realise that logic, symmetry, permutations and topology weren’t just concepts that I was thinking about when sitting at my desk. I was also thinking about them, in a kinetic way, an intuitive way, an alive way, when dancing tango! I was also surprised to find that over half of my fellow tangueras were studying maths, engineering, informatics or some kind of maths-heavy science. Mathematics permeates tango, and especially the kind of mathematics which deals with infinite 2 dimensional symmetries named wallpaper patterns.

This workshop on dancing wallpaper patterns for the Festival of Creative Learning was designed to share some of the discoveries of the connections between mathematics and tango which we have worked out. It was also designed to help the people who attended find their own connections, by drawing and dancing their own symmetric patterns. Here, I’d like to share with you some of the wallpaper patterns they created and explain a little of the maths behind them, as well as how you can dance tango on them!

All wallpaper patterns are made out of four kinds of symmetries: translations, reflections, rotations and miracles. Each of these symmetries is embodied by a tango step. In tango, you always stand on one foot. That means we can easily represent reflection by changing the foot you are standing on. I am sure you can work out steps that represent translations and rotations. You are probably wondering ‘what are miracles?’. In fact, they are the easiest to understand in terms of tango! They are just a step forward. When you step forward, you change the foot you were standing on. That means a reflection happened. But you have also moved forwards. So a miracle is a reflection plus a translation.

Wallpapers from the FCL Workshop

In this wallpaper the thin blue lines are lines of reflection and the pinwheel designs are rotation points that have 4 separate rotations around them. These are the symmetries that describe the design, but the wallpaper also has miracles and translational symmetries. You can dance the miracles by simply walking along one of the blue lines of reflection.

On this wallpaper, there are three types of points where three reflection lines meet. There is one with dots around it, one with snakes emanating from it and one with curls around (these are always at the edge). If you look closely, you can see all the points where three lines meet are one of these three kinds. The people who designed this wallpaper were more interested in the symmetries of the shapes than the colours: so if you take into account the colours, not many symmetries are left!

This wallpaper is also entirely made of reflections, however the lines of reflection meet in fours or pairs. Most wallpaper patterns that you see around you will be made of only reflections, if they are not simply translations. Far fewer have rotations. I believe there is something about reflectional symmetries that are more pleasing to the human eye. In fact, even our faces and bodies have reflectional symmetry, but no rotational symmetry. In tango, every single step you take includes a reflection, because you change the foot you are standing on. Rotations are only made out of sequences of steps. So tango is an inherently reflectional dance, and this wallpaper reflects that!

Now that you have seen these examples, why not have a go of making your own danceable wallpaper pattern? And if you want to learn the tango steps to go with it, you are welcome at the Edinburgh University Tango Society!