Leading the fractal revolution

20 January 2014

There is a fractal revolution underway and Professor Michael Barnsley is leading the charge.

"In my bones I am a fractal geometer," says Barnsley, who attributes his passion to a childhood gift of a microscope.

"The first thing I did was make copper sulfate solution with warm water and put it on the slide. As it gets cooler, crystals form. To me, it was this secret world of magnification. You could change the power of the lens and you could see this extraordinary blue world of crystals forming, and you could get lost in the magnified world!

"Type fractals into Google and you can feast on beautiful images like these," says Barnsley, who these days studies fractals at the ANU Mathematical Sciences Institute.

Barnsley says these self-referencing, repetitive images inspire fractal geometers to develop new, beautiful mathematics, which then takes on a life of its own.

"Geometers, such as Pythagoras, look at straight lines, circles - they make observations. But it was Euclid who made it mathematics which stood on its own, independent of the observations. In the same way, fractal geometers are inspired by these fractal images to take it further."

"Fractal geometry is now at a very interesting stage. Thirty years ago to say you could use it for image compression was to be laughed out of town. But now fractal geometry is used in image processing in digital cameras, fractal antennas used in the latest Boeing Dreamliner, and to design new materials that may be printed using a 3D printer."

As one of the few fractal geometers in Australia, Barnsley can see a fractal revolution happening.

"The fractal revolution is happening and it's a big deal," he says. "Don't confuse the pictures with the mathematics; the revolution is in mathematics itself."

An aspect of the revolution, he says, is in the kinds of manifolds and surfaces that mathematicians are looking at.

"Historically, we have devoted huge effort to the study of smooth manifolds, like spheres and tori. But now there is a growing interest in singular manifolds: these can be used to model such things as fractal antennas.

"For example, the surface of the Earth is a manifold. If you observe it globally it is very round and spherical. But locally, you can model it with a collection of maps, an atlas. If you have an atlas with lots of maps, and glued them all together right they would make re-make the sphere of the Earth."

Fractal geometry deals with structures that, unlike a sphere, look different when you blow them up at different places.

"Take a fractal manifold such as a Sierpinski triangle. On a Sierpinski triangle, space is not uniform; everywhere is not like everywhere else in the triangle.

"Say you were walking on the surface of this Sierpinski triangle and wanted to know how to get somewhere, you would go to your atlas which will tell you where you are. The problem is, from what things look like locally, you could be in many different places.

"You keep wandering then go back to the atlas and the map looks different from this point on the triangle - the detail is different. Things you couldn't see from the first point, you can see now you have 'gone around the corner'.

"So you have all these charts, with extra details - how do they glue together? The answer is provided by what I call a fractal manifold."

This novel way of thinking about space and shapes is leading to some novel technologies.

"I think it's important for Australia to be playing in this area in a big way. I think we already are; some fundamental mathematical research is funded, but it will be very exciting to see some more activity."

The passionate mathematician hopes to inspire the next generation of fractal geometers in his public lecture How to tile the Moon and other fractal manifolds in Manning Clarke Theatre 1 at 6pm Tuesday 21 January 2014. The lecture is run in conjunction with the Australian Mathematical Sciences Institute Summer School and bookings can be made on the AMSI website.