21. Variants on the von Koch curve with the dimensions depending on the angles used in the constructions
A similar approach enables the box counting dimension of many other fractals to be found. The snowflake and spiral in Figure 12 have dimensions about 1.535 and 1.585, respectively. Fractals in the plane can have any dimension between 0 and 2. By varying the angles in the motif used in the von Koch curve construction, all dimensions between 1 and 2 can be achieved, see Figure 21—as the irregularities of the curve becomes more pronounced the dimension increases.
Practical box-counting
Box-counting is a natural way to estimate the dimension of real fractals such as a fern, a forest skyline, a coastline, a crystal growth in a shallow dish, or the perimeter of a ‘fluffy’ cumulus cloud. Take a photograph of the fern, draw on it square grids of a range of sizes, and for each side length count the number of squares that overlap the fractal to get a list of numbers N(r) and apply definition (3).
However, caution is needed. Mathematical fractals, such as the von Koch curve, are defined by a procedure that, in principle, goes on for ever, specifying the fractal in exact detail. They are fractal at all scales and the power law for the box-counts remains valid at arbitrarily fine scales. Real objects, however, only behave as fractals over a restricted range of scales. If the box sizes are too big then hardly any boxes will overlap the fractal and we get no useful information from a count. If the box sizes are too small then other factors dominate. A fern looks like the fractal in Figure 17 when viewed at scales between about 10cm and 5mm, but if inspected more closely solid stubby teeth are observed rather than scaled down replicas of the whole so the fractal assumption breaks down at fine scales. In reality, box-counts of real objects can only be expected to satisfy a power law over a limited range of scales at which the object displays a reasonably consistent degree of irregularity; we call this the range of fractality.
This poses a problem for estimating dimension. The von Koch curve is defined with complete precision at arbitrarily fine scales and we could, in principle, find box-counting numbers at scales as small as we like, either by photo-enlarging the curve and boxes to a size suitable for careful and tedious counting, or by doing some calculations. The definition of dimension (3) depends on the limit of what happens as the box sizes become infinitesimally small—the large scale box-counts don’t matter. But for real fractals we cannot do this—we can only take box-counts over a limited range of scales and we have to harness this information to estimate the dimension as best we can.
Equation (2) expressed the power law in terms of logarithms as
If box-counts are available for very fine grids, so that r can be taken very small, then log N(r) and log(1/r) will be large and will dominate this equation, so log c can be neglected, leaving logN(r)/log(1/r) as a fair estimate for d. However, if we only have box-counts N(r) available for a range of relatively large box sizes when log(1/r) and log N(r) remain fairly small, then the term log c may be significant and we cannot ignore it.
Fortunately there’s a way around this difficulty. If we draw the graph of the equation y = 3x + 2, so for each x we plot the point with coordinates (x, 3x + 2), we get a straight line. The number 3 in this equation is the slope or gradient of this line, meaning that if we increase the value of x by a certain amount, the corresponding value of y given by the graph will increase by 3 times as much. More generally, for fixed numbers d and a, the graph of the equation y = dx + a is a straight line with slope d; the slope does not depend at all on the number a. Thus if we know the (x, y)-values of a few points satisfying some straight line equation and draw a straight line through the points with these coordinates, the line will have slope d. This is the situation in equation (4). If, for the values of r at which we have taken box-counts, we plot the points with coordinates (log(1/r), log N(r)) on graph paper and draw the best straight line that we can through these points then, regardless of what log c is, the slope of this line will be d which we can take to be the dimension of the object regarded as a fractal over its range of fractality.
Let’s look at the classic example of the coastline of Britain. The closer the coastline is examined, the more detail becomes apparent, so the coastline might be expected to behave like a fractal perhaps over a range of scales 100m–100km. By superimposing grids on suitable maps, see Figure 22(a), (showing squares of 60km side), we get the following box-counts:
The points with coordinates (log(1/r), log N(r)) are plotted for each of these r on the graph in Figure 22(b) and the ‘best’ straight line, that passes as closely as possible to these points, is drawn (taking more account of the points corresponding to finer scales r where the box-counts provide more detailed information). The dimension is given by the slope of this line, which we measure by dividing a vertical increase v across part of the graph by the corresponding horizontal increase h">12, This procedure enables the dimension of many physical fractals to be found. For example, the dimension of a fern, estimated using boxes from 1 to 20cm is about 1.5, and the dimension of copper deposition by electrolysis in a flat dish (see Chapter 6) estimated on scales from 2mm to 5cm is about 1.7.
22. (a) Box-counting to estimate the dimension of the coastline of Britain, (b) a ‘log-log’ plot of the box counts—the slope v/h gives an estimate of the dimension
So far, our discussions have been restricted to fractals in the plane—this really is just for the convenience of drawing or visualizing them. In reality, objects belong to our three-dimensional world. There is no problem in principle in defining the dimension of fractals in space in the same way as in the plane. The dimension is given by the same formula (3) corresponding to the power that occurs in the power law behaviour of the box-counts, the difference is that now we superimpose a grid of cubes of side r and take N(r) to be the number of these cubes that overlap the object. In practice, it is almost impossible to count the overlapping cubes. Dividing the space enclosing a plant, say, up into small cubes and counting how many cubes contain parts of the plant would be a considerable challenge! Nevertheless, there are ways around this which will be indicated shortly.
Dimension of self-similar fractals
To find the dimension of self-similar fractals we can utilize the self-similarity directly. When we found the dimension of the Sierpiński triangle by box-counting, we noted that each time we halved the side-length of the grid, the box-counting number was multiplied by 3, that is, for small r,
3N(r) = N(r/2),
see Figure 20. Assuming that the box-counting numbers satisfy a power law N(r) = c(1/r)d, this means
Cancelling c(1/r)d on each side of this equation leaves
3 = 2d.
To find d we take logarithms of both sides and use the power rule, so
log3 = log (2d) = dlog2,
and dividing by log 2 gives that d = log3/log2. The dimension of the Sierpiński triangle is the exponent in the power law for its box-counting numbers, so its dimension is log3/log2.
This little argument depends on the fact that the Sierpiński triangle comprises 3 similar copies of itself each scaled by a factor ½, and consequently that on halving the grid size the box-counting number is multiplied by 3. The dimension of other self-similar fractals may be found using a similar approach:
A fractal that is made up of a similar copies of itself at scale 1/b has dimension log a/log b.
For instance, the von Koch curve consists of four copies of itself at scale , so has dimension log4/log3 = 1.2618…, as before.
Measurement within dimension
Answering the (somewhat hackneyed) question ‘How long is a piece of string?’ with the words ‘it’s 1-dimensional’, is of little help for most practical purposes. To know whether the string is any use for tying a parcel or hanging a picture, ‘50 centimetres’ or ‘30 inches long’ would be more helpful. When dealing with something that we know to be 1-dimensional, such as string, cotton, or wire, length provides further information on its size. Similarly, for a 2-dimensional object such as a sheet of paper or pane of glass, area is the appropriate measurement, whilst for a block of wood or glass of beer, volume, the 3-dimensional measure, is what is needed. There is a method of measurement appropriate for use within each of these dimensions—length, area, volume. Analogously, for each fractal dimension, there is a natural way of measuring objects within that dimension. To get the rough idea, let’s return to the power laws—they contain information that we have so far ignored. Recall that the box-counts of the grid squares of side r that overlap solid squares of side 1, 2, and 3 are near enough 1(1/r)2, 4(1/r)2 and 9(1/r)2 respectively when r is very small. The multipliers 1, 4, and 9 are recognizable as the areas of the squares of sides 1, 2, and 3. Thus if the box-counts for a square gives a power law of c(1/r)2, this not only says that it is 2-dimensional, but also suggests that its area, that is its 2-dimensional measurement, is given by the multiplier c. In fact it turns out that any shape in the plane satisfies a power law c(1/r)2 for small r, where the multiplier c is its area.
In the same way, if the box-counts for a fractal using fine grids satisfy a power law N(r) = c(1/r)d for some c and d, not only does this tell us that the dimension of the fractal is d but also indicates that it is of ‘size’ c within the collection of d-dimensional fractals. We can think of c as a ‘d-dimensional measurement’ of the fractal, draw a straight line through theNix in the sense that area is a 2-dimensional measurement.
Now consider the way measurements change under scaling. If we scale by a factor 2 (that is, do a 200 per cent enlargement on a photocopier) the lengths of lines and curves are multiplied by 2, but the area of squares, circles, etc., are multiplied by 22 = 4, see Figure 23. Similarly scaling by a factor 3 (or 300 per cent) multiplies lengths by 3 and areas by 32 = 9. Our d-dimensional measurement scales in the same sort of way, but the power involved is just the dimension d. Recall that the box-count for the von Koch curve satisfies an (approximate) power law N(r) = (1/r)1.26. So if we consider the square grid of side r/2 (as opposed to r), about (2/r)1.26 of the boxes overlap the curve. Enlarging this whole picture by a factor 2 enlarges the grid boxes to side length r, of which the same number (2/r)1.26 = 21.26(1/r)1.26 overlap the enlarged fractal. Thus the enlarged fractal has box-counting numbers N(r) = 21.26(1/r)1.26 The multiplier in the power law is now 21.26 so scaling the von Koch curve by a factor 2 has multiplied its 1.26-dimensional measurement by 21.26 = 2.39. Similarly, scaling by a factor 3 multiplies the measurement by 31.26 = 3.99, and so on. The same holds for any fractal of any dimension d: scaling by 2 multiplies the d-dimensional measurement by 2d, scaling by 3 multiplies it by 3d, etc.
