Fractals: A Very Short Introduction (Very Short Introductions)
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From the contours of coastlines to the outlines of clouds, and the branching of trees, fractal shapes can be found everywhere in nature. In this Very Short Introduction, Kenneth Falconer explains the basic concepts of fractal geometry, which produced a revolution in our mathematical understanding of patterns in the twentieth century, and explores the wide range of applications in science, and in aspects of economics.
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smaller shape. The Sierpiński triangle of Figure 11 has a template comprising the ‘unit square’, that is the square of side length 1 with corner with coordinates (0, 0), (1, 0), (0, 1), (1, 1), and three smaller squares of side-lengths �. The three functions giving the three similarity transformations that transform the unit square to the bottom left, bottom right, and top left squares respectively, have the formulae: The first function just shrinks everything down towards the origin by a scale
powers are not whole numbers, so as 100.3010 = 2 then log 2 = 0.3010. Here are a few more: Most calculators have a logarithm key, labelled [log], so, for example, the key sequence [log] returns 0.6990. Note that the larger a number, the larger is its logarithm. A fundamental and very useful property of logarithms is that they turn multiplication into addition, that is for any positive numbers a and b, If we multiply two numbers together, the answer is called the product of the numbers. Thus
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
of length 1 and the third side of length and so are congruent. This means that the two triangles have the same angles, and in particular the angle between the lines joining z and 1 to the origin equals the angle between the lines joining z2 and z to the origin. Thus the angle of the complex number z2 is double that of z in the case where z has magnitude 1. Finally, observe that any complex number may be expressed as a real number times a complex number which has magnitude 1 and the same angle,
Garry Flake (MIT Press, 2000), present many approaches to creating fractals on a computer. Many of these books include some historical background and Classics on Fractals, edited by Gerald Edgar (Westview Press, 2003) brings together translations of key mathematical papers, from Weierstrass to Hausdorff to Mandelbrot. A personal view of the development of fractals by Benoit Mandelbrot is contained in The Fractalist: Memoir of a Scientific Maverick (Pantheon Books, 2012), published two years