Teaching an old spiral new tricks

by Burkard Polster and Marty Ross

The Age, 27 March 2009

Beautiful and baffling pictures, such as that of our infinitely repeating Lara, have become very popular. These pictures are based on logarithmic spirals, and Lara is actually holding such a spiral in her hands. In fact, such spirals can be found everywhere: in cyclones and galaxies, in nautilus shells, in flower heads, and on and on.

All spirals catch the eye, but logarithmic spirals are notable for what is called their self-similarity. Take a logarithmic spiral and blow it up, expanding away from the centre. Then the resulting spiral will be exactly the same as the original, just rotated through a certain angle. For a nautilus, this is extremely handy. It means that it needn’t change the shape of its body as it grows its spiral shell.

To create our repeating picture, we start with a photo of Lara holding her favourite pretty plate. We then cut the photo open and deform it into a nautilus shape. The distorted photo now fits inside part of a logarithmic spiral, with a hole in the middle. Next, a shrunken copy of the distorted photo fits snugly into the hole, leaving a still smaller hole. Repeating this over and over, the picture will be filled.

This distortion of a picture is called the Droste effect. It was invented by the brilliant artist M. C. Escher, who in 1956 incorporated it into his stunning work, Print Gallery. This picture was recently scrutinised by a team of mathematicians, and they revealed the logarithmic key to the magic. Check out http://escherdroste.math.leidenuniv.nl for more information, and for some amazing animations of Escher’s original picture.

By the way, little Lara is very happy with her Droste picture, and has been proudly showing it to all her friends. Like her Maths Master father, she believes that anyone can see the magic in mathematics.