Image by Lasse Rempe-Gillen – click on the image to see a larger version.

To mark Germany's historic win over Brazil in the World Cup semifinal this week, Lasse Rempe-Gillen (from the University of Liverpool) created this beautiful image. It shows the behaviour of a model that describes the phenomenon of phase-locking, something that can be seen in the synchronising flashes of fireflies or when a roaring stadium of football supporters gradually clap or stamp in unison. The image is related to recent research and you can read more in our news story Maths, metronomes and fireflies.

The grey parts of the image show where the model behaves chaotically – here even small changes in where you start can cause drastically different results in the model. The coloured parts of the image show where the model behaves in a more regular fashion where small differences won't dramatically change the results. This is because the model has attractors, special sets of conditions that create similar behaviour, either settling on a single outcome (called a fixed point) or running through a predictable cycle of outcomes. And in honour of the historic 7-1 score from the match, Rempe-Gillen's image has attractors of period 7 (with a repeating cycle of 7 points) and period 1 (a fixed point).

In contrast his image below has no periodic attractors, symbolising the other, goalless, semifinal between Argentina and Holland.

Image by Lasse Rempe-Gillen – click on the image to see a larger version.

You can read more about chaos, fractals and football on Plus!