How far must Santa travel on Christmas Eve? At a broad scale, he visits all the continents. In more detail he travels to every country.
Zooming in, he goes through each city, town and village and ultimately to every home where children are asleep. The closer we examine the route, the longer it seems. This is a characteristic of paths or graphs called “fractals”.
A straight line, like a road, has dimension 1 and a plane surface, like a field, has dimension 2. But some curves are so wiggly that they have a dimension between 1 and 2. Such curves, with fractional dimensions, are called fractals.
Fractals were first considered by the English Quaker mathematician Lewis Fry Richardson. He was studying the factors contributing to warfare, and he thought that the length of the borders between neighbouring countries might be a relevant measure. But when he investigated the frontier between France and Spain, he found that widely different lengths were reported. Similarly, when he measured the west coast of Britain, the length varied with the scale of the map he used.
Larger scale maps include finer detail, resulting in a longer coastal length. As we may choose to include variations corresponding to every rock, pebble or grain of sand, it is impossible to assign a length unequivocally. All we can do is to describe how the length varies with our “ruler” or unit of measure. This is determined by the fractal dimension. Mathematicians can assign a numerical dimension D to an irregular curve or surface, and it need not be a whole number.
Classical mathematics treated the regular geometric shapes of Euclid and the smoothly evolving dynamics of Newton. Fractal structures were regarded as pathological but, following the inspiration of Benoit Mandelbrot, we now realise that they are inherent in many objects of nature, like clouds, ferns, lightning bolts, our blood vessels and lungs, and even the jagged price-curves of the stock market.
Recently a group of students in the school of physics at Trinity College Dublin, under the supervision of Stefan Hutzler, studied Ireland's coastline using Google maps and a measure called the box dimension, and obtained a value D = 1.2. Independently, a group in the school of computing and mathematics at the University of Ulster used printed maps and dividers of varying length, and found a value D = 1.23. The close agreement confirms the fractal nature of the coastline. Not surprisingly, the ragged Atlantic shore has a higher fractal dimension (D = 1.26) than the relatively smooth east coast (D = 1.10).
Fractals are all around us at Christmas. Just consider the branching structure of the Christmas tree, or the snowflakes hanging upon it. Even the broccoli on your dinner plate is fractal. But what of Santa’s route?
If we assume the route is fractal, most of the distance is due to the small segments from one house to the next. With a billion houses to visit, and the typical distance between neighbouring houses being 10m, we get a length of 10 million kilometres. This is equivalent to about 250 laps of the globe. The long stages, from the North Pole to Kiribati, Santa’s first point of call, and the ocean crossings, contribute very little. It is the small-scale hops from house to house that count for most.
Peter Lynch is professor of meteorology at University College Dublin. He blogs at thatsmaths.com