41. A computer simulation of fractal fingering
Similar patterns can occur when bacteria grow on food. Nutrients, which encourage bacterial growth, diffuse across a suitable surface and are most likely to meet a bacterial colony near its extremities and so encourage growth in a branching, fingering manner. Under suitable conditions, the growth takes the form of fractal fingers. In a geological setting, fractal fingers of manganese oxide crystals sometimes form on rock surfaces when water rich in manganese flows through limestone or other rock and deposits tend to favour existing protrusions.
Lightning, which may be forked or fingered, is the massive electrical discharge that occurs when the electric field becomes strong enough to break down the normal insulating property of air. Branching electrical discharges with a fractal fingering form can also occur on the surface or interior of other insulating materials—these are known as Lichtenberg figures, named after the German physicist Georg Lichtenberg (1742–99) who observed them in 1777. They can be created experimentally by applying a high voltage source through a spark gap to a needle touching an insulating surface such as ebonite or glass. The discharge creates a fingering charge distribution across the surface which may be revealed by sprinkling flowers of sulphur and lead tetroxide onto the surface. A similar effect is obtained if a CD i is completely defined by the ppes put in a microwave cooker for three to four seconds (though this is not advised since it may damage the cooker!). Recently, spectacular fractal Lichtenberg figures have been created in blocks of clear acrylic by injecting a beam of electrons at high speed (over 95 per cent of the speed of light). This type of fractal fingering is also sometimes observed on the skin of people who have been struck by lightning, as a result of rupture of blood vessels under the skin as the lightning current spreads through the body.
Fractal viscous fingering may result from the interaction of two liquids under pressure. A Hele-Shaw cell (named after English engineer Henry Hele-Shaw (1854–1941)) consists of two glass plates placed about 0.5mm apart with the gap in between filled with a viscous liquid such as oil or glycerol. If water is injected through a small hole in one of the plates, then the water spreads through the oil in highly branched fingers as a result of the balance between the pressure and surface tensions of the liquids.
Although these settings may seem unconnected, they have much in common mathematically. Laplace’s equation, named after the French mathematician Pierre-Simon Laplace (1749–1827), is a fundamental equation which describes many physical situations. The local density of copper ions in the copper sulphate solution, the pressure at points in the oil between the plates of a Hele-Shaw cell, and electrostatic potential all satisfy Laplace’s equation away from the fingering itself, and, combined with the inherent randomness of the processes, this underlies the similarities between fractal growth patterns in different settings.
Fractal time records
As well as financial data, many other quantities vary with time in an erratic manner, for example, temperature and windspeed, discharge rates of rivers, levels of reservoirs, and internet traffic. Graphs of such quantities plotted over a long time range may display fractal behaviour, although the graphs may appear somewhat different from those of Brownian motion.
A method for analysing time records, known as rescaled range analysis, was introduced by the British hydrologist Harold Hurst (1880–1971) who made detailed observations over many years on the water levels of rivers and lakes, in particular on the Nile river system. His ultimate aim was to design reservoirs so that they neither overflowed nor dried up; this needed to take into account not only the normal variation in the water level, but also the extreme levels that might occur on rare occasions following extended periods of rain or drought. Hurst calculated the ratio over various time periods between the greatest variation in level and the typical variation, and noted that, to a good approximation, this ratio obeyed a power law in time. More precisely, for observations over a period of time t, let R(t) denote the range (i.e. greatest minus the least values) and let S(t) be the standard deviation, or typical variation, of the values. Hurst found that
for some power H between 0 and 1, where H is now termed the Hurst index or Hurst exponent of the observations. He also found that, as well as for water levels, this law held for many other classes of time-dependent data. Figure 42 shows graphs of simulated processes with Hurst indices of various values. For small H, the variation of the graph over short time intervals is almost A function moves points to points and so transforms sets of points to sets of pointsin6 as much as for much longer intervals, whereas for H near 1 there is little short-term variation and it is a long time before a significant change builds up.
