Many naturally occurring objects have a complicated appearance, and when we look closer the complexity only increases. As we zoom in we see what appear to be miniature replicas of the original object. For instance, a branch of a fir tree might resemble a scaled down version of an entire fir tree, and a broccoli floret might look like a miniature broccoli.
This is true of many organic structures including some within our own bodies, such as the arteries, veins and capillaries of the circulatory system, and the branching components of the respiratory system from our trachea to our lungs. Self-similar objects such as these are known as fractals.
Gaston Julia
It is rather surprising that the idea of fractals can be traced back to the musings of a seriously injured soldier amidst the horrors of the First World War.
That soldier was Gaston Julia who was born in French Algeria in 1893. In his teens he won a scholarship to study in Paris, but within months of completing his maths degree war broke out and he was conscripted into the French army the following day.
Early in 1915 Julia suffered a terrible wound during a German attack. According to the military report, on
January 25, 1915, he showed complete contempt for danger. Under an extremely violent bombardment, he succeeded despite his youth in giving a real example to his men. Struck by a bullet in the middle of his face causing a terrible injury, he could no longer speak but wrote on a ticket that he would not be evacuated. He only went to the ambulance when the attack had been driven back. It was the first time this officer had come under fire.
No Man’s Land by Gregory Manchess
After a series of unsuccessful operations Julia’s injury resulted in the loss of his nose. He was left like a twentieth century counterpart to the sixteenth century Danish astronomer Tycho Brahe who lost his nose to a duelling sword. Whilst Tycho wore a brass prosthesis to hide his loss (with a gold one for special occasions), Julia wore a leather strap to conceal his disfigurement.
Gustav Herglotz and Gaston Julia
During convalescence Julia occupied himself with mathematics. Whereas Tycho investigated outer space, tracking the positions of stars and planets with greater precision than ever before, Julia investigated inner space, plotting the intricate mathematical properties of points in the plane. This involved long sequences of simple calculations, ideally suited to the number-crunching capabilities of a modern computer. But Julia’s work was published long before computers were available.
Gaston Julia with his wife and six children.
In 1917 Julia submitted his doctoral dissertation, much of which had been completed in hospital. It was published as a ground-breaking 200-page paper the following year. That year Julia married Marianne Chausson, daughter of the composer Ernest Chausson, one of the nurses who looked after him while in hospital.
Julia’s work was celebrated when published, but almost forgotten in later years as further investigation was extremely difficult without a computer.
Luckily, Julia lived long enough to see a revival of interest in his research. By the 1970s computers were producing crude low resolution images based on his work. Julia died in 1978 at the age of 85.
Google commemorated the 111th anniversary of Julia’s birth on 3 February 2004.
The illustration below was generated on my PC. It is a colour-coded representations of Julia’s ideas known as a Julia set. Julia sets have a fractal structure and are like relief maps. The colours are arbitrary, but they are determined by a mathematical property of each region of the map as deduced by the computer following Julia’s algorithm and regions sharing the same property are given the same colour.
An Example of a Julia Set.
Computer generated imagery has really taken off in recent decades. It now has many applications, especially in video games and special effects in the film industry. These effects employ sophisticated mathematical algorithms and many involve fractals.
Part of the Mandelbrot set, a close relative of the Julia sets.
Further Information
I have long been interested in the interplay of maths and the arts and the often surprising influences that pass between one and the other. Last year Oxford University Press published the book that I have written on the subject. It is called: Celestial Tapestry: The Warp and Weft of Art and Mathematics.
This is a short video in which I discuss Gaston Julia’s work.
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