7. The Hйnon attractor with a portion enlarged to display its banded structure

The structure of this attractor merits closer examination. On magnifying a small portion, the curves apparently forming the attractor are seen to comprise many closely spaced (almost) parallel lines. Further magnification shows that these lines themselves are made up of many more closely spaced lines, and so on—the attractor has a fine structure formed by nearly parallel lines. Thus iteration of the Hйnon function yields a fractal attractor, sometimes called a strange attractor. Although the attractor is highly intricate, it is completely defined by the Hйnon function given by a formula of" height="26" alt="image"/>8941 just a few symbols. Once again, repeated application of a simple operation gives rise to a complex fractal form.

A nice feature of iteration is that it is very easy to realize on a computer. Computers are very good at performing the same operation over and over again, and a few lines of code are enough to calculate the itinerary of a function from a given starting point, with each point obtained by applying the function to its predecessor, and this may be plotted on a screen to give a picture of the attractor. It is easy to enlarge parts of the attractor so that any fractal structure can be investigated visually. This contrasts markedly with mathematical analysis of many attractors. Even for simple functions, explanations of why an attractor has a particular fractal structure can be beyond the reach of current mathematics.

What can be done with fractals?

It is natural to query the role of highly irregular fractal objects within mathematics and science. The clue to this comes from classical geometry—the questions that have been addressed for centuries concerning traditional geometrical shapes are precisely those that more recently have come to be asked about fractals.

• Description A circle consists of those points that lie at some constant distance from a given centre, and an ellipse comprises the points such that the sum of their distances from two fixed points (the ‘focii’) is constant. Are there such concise descriptions of fractals? We will see how simple ‘templates’ can encode intricate fractal forms.

• Measurement Almost the first thing one asks about any mathematical object is ‘how big is it?’. For a circle or rectangle we can measure the area or the perimeter length. When considering the size of a fractal the notion of ‘dimension’ turns out to be central.

• Geometrical properties If a circular ring is held up under a light or in the sun, the shadow will be an ellipse, which may be highly elongated or almost circular, depending on its angle to the light. Do fractals have any such geometrical properties?

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Chapter 2 Self-similarity

Self-similar fractals and their templates the powers are not wholeRing a simple step over and over again

The concept of similarity features prominently in classical geometry, notably in the writings of the Greek mathematician Euclid (c.300 BC), the ‘Father of Geometry’, who laid the foundations of rigorous geometrical argument. Two figures in the plane are similar if they have the same shape, but not necessarily the same size, so that one may be obtained from the other first by scaling and then sliding it around and rotating it, perhaps flipping it over. In modern parlance, two objects in the plane are similar if it is possible to make a reduced or enlarged photocopy of one and position it to coincide exactly with the other, perhaps turning it over first. If this can be done without scaling, in other words, with a same size photocopy, the objects are called congruent. Otherwise, the reduction or enlargement factor required to produce the similar copy is called the scale or scaling ratio of the copy, which may be expressed as either a fraction or percentage. Thus if a figure is a ј scale (or 25 per cent scale) copy of another then all the lines in the copy are ј of the length of the corresponding lines in the original.

All circles are similar to each other, as are all squares. However, two triangles are similar precisely when the three angles of one triangle are the same as the three angles of the other. Two rectangles are similar just when the length of the longer side divided by that of the shorter side is the same for both rectangles. Figure 8 shows several figures that are all similar to each other.