Where Discovery Begins

# Oxford Mathematics Online Exhibition 2021 – the videos

We asked our Oxford Mathematicians, young and less young, to come up with art that expresses a mathematical idea in the form of their choice. Several people chose video. Here they are in order of appearance and with explanation:

Siddiq Islam – Love-Heart-Shaped Curve Song:
A maths song. Siddiq is an undergraduate at Oriel College, Oxford.

Romy Williamson – Rubik’s Cube:
This is a Rubik’s cube of my own design. It’s made from 156 squares of paper, and other things I had on my desk. The Rubik’s cube is a popular example of Group Theory in real life.

The set of all ways to assemble a cube from the Rubik’s cube pieces has cardinality (12!)(2^12)(8!)(3^8) because there are 12 edge pieces each with 2 possible orientations, and 8 corner pieces each with 3 possible orientations. However, many of these configurations are impossible to reach by turning a solved cube. So the actual Rubik’s cube group has “only” (12!)(2^10)(8!)(3^7)=43,252,003,274,489,856,000 elements.

Romy Williamson is an undergraduate at Merton College, Oxford.

Martin Parker – Diffusion limited aggregation:
Diffusion Limited Aggregation (DLA) is a random fractal object created by repeatedly simulating random walks (‘diffusing particles’). It is a topic of great mathematical and physical interest. For this multimedia piece, DLA was simulated on a line and the heights of the particles in the object were then converted into musical notes which can be heard on a synthesiser. The video is a visualisation of the music (it does not represent how the object was grown). The work is inspired by the piece ‘dendritic’ (https://www.multiverseseries.org/blog/dendritic).

Martin Parker is an undergraduate at St Catherine’s College, Oxford.

Aidan Strong – Bach Crab Canon
Here I play one of the canons from Bach’s ‘Musical Offering’, which is one of Bach’s most complex and mathematical works. The musical offering is filled with devices such as canons, augmentations, and retrogrades, each of which can be thought of as translations, stretches, and reflections of the music in the time axis respectively. In this canon, the 18 bar theme is played against itself but in reverse, so when the full 18 bars are played in reverse, it sounds the same – i.e. there is symmetry about the time axis!

Aidan Strong is an undergraduate at Lincoln College, Oxford.

Sam Palmer – Polyrhythms
Polyrhythms are a rare gem in music. They appear in a range of contexts from Queen’s Bohemian Rhapsody to tribal African music. Polyrhythm refers to multiple rhythms happening at the same time, for example having one melody playing three beats per bar while another melody plays four beats in the same bar. This is a little melody I wrote where most of the time the bass notes are playing in three time and the high notes are in four time.

Sam Palmer is a Postdoctoral Research Associate at the Mathematical Institute in Oxford.

Andrew Krause – Pattern Forming Snake
James Murray’s Mathematical Biology textbook discussed a model of cell movement (chemotaxis) plausibly underlying the formation of patterns on the skin of snakes. In the embryonic stage when these patterns form, the snakes are already coiled. The book states, “although it would be more realistic to study the model mechanism on the surface of a coiled cylindrical domain, the numerical simulation difficulties were already considerable even on a plane domain.”

This is such a simulation not feasible in the 1980s. Rather than stationary patterns, we see continuous movement of cells. It took about an hour of computing in 2021.

Andrew Krause is a Postdoctoral Research Associate at the Mathematical Institute in Oxford.