The article in this week’s That’s Maths column in the Irish Times ( TM035 ) is about the remarkable Christmas Eve journey of Santa Claus.
Dimensions & Fractals
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 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, 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, 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 snow-flakes hanging upon it. Even the broccoli on your dinner plate is fractal. But what of Santa’s route?
How far does Santa Travel?
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 ten metres, we get a length of 10 million km. This is equivalent to about 250 laps of the globe. The long stages, from the North Pole to Kiribati, Santa’s first port of call, and the ocean crossings, contribute very little. It is the small-scale hops from house to house that count for most.
With just two points, let’s try to estimate the fractal dimension of Santa’s route. Taking steps of one Earth radius, he can get around the world in about 6 steps ( 2π ≈ 6 ). So, one point on our log-log plot is at ( S1 = C/6, L1 = C) where C is the circumference of the Earth. With steps of 10 m, he goes about 250 times farther, as seen above, so another point is ( S2 = 10 metres, L2 = 250 C ). Plotting these two points we can draw a line between them. Since C ≈ 40,000 km we have C/6 ≈ 7 x 106 metres,
log( L1 / L2 ) = m log( S1 / S2 ) or log ( C / 250 C ) = m log ( 7 x 106 / 10 )
or log(1/250) = m log( 7 x 105 ) or m = – ( 2 + log (2.5) ) / ( 5 + log 7 ) ≈ – 0.41
which gives a fractal dimension of D = 1 – m = 1.41. This is much higher than the dimension of the Koch snowflake (D = 1.26). It should not be taken too seriously, but its high value suggests that short stages contribute most to the length of the journey, as we have hinted above.
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Peter Lynch’s book about walking around the coastal counties of Ireland is now available as an ebook (at a very low price!). For more information and photographs go to http://www.ramblingroundireland.com/