Soap bubbles used to explore mathematical singularities

Films of soap with complex shapes can be used to model and understand mathematical singularities across a range of fields.

soap.jpg
Professor Raymond Goldstein, University of Cambridge

In mathematical terms, a singularity is the point at which any given mathematical object -- such as an equation or surface -- breaks down and "explodes", no longer defined. Sometimes this can mean becoming a separate form -- predictable in theoretical situations, but not necessarily in nature, where singularities can occur in such phenomena as solar flares and ocean waves.

Using films of soap, however, may help change this.

According to a new study, identifying a type of curve in a film of soap can help predict where singularities are likely to occur in other films of soap -- which, in turn, could help researchers understand real-world natural singularities.

Researchers at the University of Cambridge in the UK have shown that the way in which films of soap stretch, collapse, and re-form may be able to help predict singularities that occur in nature.

A film of soap is created by dipping a looped wire into a soap solution. These are "minimal surfaces" -- they always have the smallest area of all possible shapes that could span the frame. The loop itself is the simplest of all shapes -- but, by using different wire configurations, more complex shapes can be achieved, such as Mobius strips.

When the shape of the supporting wire is changed, this destabilises the surface of the bubble. This occurs in just a fraction of a second, during which time the film forms a new shape. The team had previously determined that a Mobius strip singularity occurred at the wire frame, where the surface would rearrange itself, and not between the wire, as occurs between two separate wire loops.

The key lay in something called the systole. In geometry, the systole refers to the shortest closed curve around the surface of a 3D object. The researchers found that the systole's properties can determine where on the surface the singularity will occur. If the systole loops around the wire frame, for instance, the singularity will occur at the wire frame -- as in the Mobius strip -- but if it does not, then the singularity will occur at the surface's bulk.

"This is an example of experimental mathematics, in the sense that we are using laboratory studies to inform conjectures on mathematical connections," said study co-author Professor Raymond Goldstein. "While they are certainly not rigorous, we hope they will stimulate further research into this new, developing area."

Now, who's up for blowing bubbles?

 

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