AcCosted at Healesville

 

One of Victoria’s fabulous family offerings is Healesville Sanctuary. It’s a great place to view native Australian wildlife, and one of your Maths Masters and his very willing family just made the trip for the umpteenth time. However, on this occasion there was also a wonderful surprise: amidst the exotic Australian wildlife was hiding a rare and beautiful mathematical creature.

Located at Healeseville Sanctuary is the Australian Wildlife Health Centre. It's a very impressive training and research hospital, as well as being a fascinating educational component of the Healesville tour. Visitors are able to talk to the vets and nurses, and to watch operations being performed on the animal patients.

On his recent trip, your Maths Master was witness to some involved wombat dentistry, and his squeamishness encouraged him to search around for more pleasing sights. That was very fortunate, because he happened to glance up at the roof of the Health Centre: it was curved in a beautiful and elaborate manner, and seemed strangely familiar. And then he figured it out.

 

The Health Centre roof is a close mathematical relative of Costa’s Surface, discovered by Brazilian mathematician Celso José da Costa in 1982. Costa's Surface created a huge buzz, and was named Time Magazine's Surface of the Year in 1985. (Well, it would have been if Time had a maths section.) We’ll try to explain the reason for all the excitement.

A computer rendering of the main part of Costa's Surface appears below. However, unlike the computer graphic (and the roof of the Health Centre), Costa's Surface continues beyond the orange boundary curves, forming an infinite surface without edges  – in the lingo, Costa's Surface is said to be complete. 

So what's so special about Costa's Surface? To begin, it is what is known as a minimal surface, which is the mathematical model of a soap film. Imagine a small loop embedded into Costa's Surface, as pictured below. 

 

Now, think of the loop being made of wire and imagine dipping the wire in a soap solution. Then the soap film that would form is exactly the piece of Costa's Surface within the loop. Mathematically, this is saying that little pieces of Costa's Surface have least area amongst all imaginable surfaces with the same edge. 

Mathematicians had been discovering and studying minimal surfaces for centuries, but until recently they knew of very few complete minimal surfaces. In fact, amongst complete minimal surfaces with the extra property of finite topology (meaning roughly that the surface has a finite number of holes), only three types were known: a boring flat plane; the staircase-shaped helicoid; and the tube-shaped catenoid. 

And then came Costa's Surface. But if Costa discovered his surface in 1982, why did it take until 1985 for all the cheering to begin?

Costa used a technical method (involving complex numbers and integrals) to write down the equations for his surface. Consequently, Costa's Surface was very difficult to analyse and to picture; the tricky question was whether Costa's Surface would curve around and run into itself. Such self-intersecting minimal surfaces are quite common and are much less interesting; a surface which avoids these self-intersections is referred to as embedded.

In 1985, mathematicians David Hoffman and Bill Meeks employed (very early) computer graphics to try to understand Costa's Surface. Their graphics were incredibly grainy, but Hoffman and Meeks saw enough to be convinced that Costa's Surface was indeed embedded. The graphics also indicated that Costa's Surface had a number of symmetries, which were not obvious from Costa's equations.

Hoffman and Meeks then returned to take a careful look at Costa's equations. With the symmetries to guide them, they were able to prove conclusively that Costa's Surface was embedded, that it was indeed a new complete minimal surface.

The discovery and analysis of Costa's Surface opened the floodgates. There has since been discovered an incredible zoo of minimal surfaces. The Health Centre roof is modelled on what is known as the Meeks-Hoffman-Costa Surface, and features an order three rotational symmetry; it is closely related to Costa's original surface, and was proved to be embedded soon after. 

Not surprisingly, turning Costa's beautiful surface into the Health Centre's stunning roof was anything but straight-forward. For the fascinating details, see the excellent article in Award Magazine.

Of course, nowadays computer graphics are much easier and cheaper, and beautiful graphics of minimal surfaces (and everything else) are very easy to find. In fact, the images for this column were created with the excellent free software 3D-XplorMath. We urge you to download the program and explore for yourself! (Within the program, the Costa Surface and the Costa-Hoffman-Meeks Surface can be found in the "Minimal (H=0)" submenu of surfaces.)

So, there you have it. Healesville's Health Centre is yet another beautiful addition to our growing and growing and growing and growing and growing and growing and growing collection of Melbourne's mathematical architecture. What an amazing city!

 

Burkard Polster teaches mathematics at Monash and is the university's resident mathemagician, mathematical juggler, origami expert, bubble-master, shoelace charmer, and Count von Count impersonator.

Marty Ross is a mathematical nomad. His hobby is smashing calculators with a hammer.

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