"Elementary functions, such as trigonometric functions and rational functions, have their roots in Euclidean geometry. They share the feature that when their graphs are 'magnified' sufficiently, locally they 'look like' straight lines. That is, the tangent line approximation can be used effectively in the vicinity of most points. Moreover, the fractal dimension of the graphs of these functions is always one. These elementary 'Euclidean' functions are useful not only because of their geometrical content, but because they can be expressed by simple formulas. We can use them to pass information easily from one person to another. They provide a common language for our scientific work. Moreover, elementary functions are used extensively in scientific computation, computer-aided design, and data analysis because they can be stored in small files and computed by fast algorithms." (Michael Barnsley, "Fractals Everwhere", 1988)
"For a map to be useful it must have information marked on it, such as heights above sea-level, population densities, roads, vegetation, rainfall, types of underlying rock, ownership, names, incidence of volcanoes, malarial infesta- and so on. A good way of providing such information is with colors. For example, if we use blue for water and green for land then we can 'see' the land on the map and we can understand some geometrical relationships. We can estimate overland distances between points, land areas of islands, the shortest sea passage from Llanellian Bay to Amylwch Harbour, the length of the coastline, etc. All this is achieved through the device of marking some colors on a blank map!" (Michael Barnsley, "Fractals Everwhere", 1988)
"Fractal geometry is concerned with the description, classification, analysis, and observation of subsets of metric spaces (X, d). The metric spaces are usually, but not always, of an inherently 'simple' geometrical character; the subsets are typically geometrically 'complicated'. There are a number of general properties of subsets of metric spaces, which occur over and over again, which are very basic, and which form part of the vocabulary for describing fractal sets and other subsets of metric spaces. Some of these properties, such as openness and closedness, which we are going to introduce, are of a topological character. That is to say, they are invariant under homeomorphism." (Michael Barnsley, "Fractals Everwhere", 1988)
"How big is a fractal? When are two fractals similar to one another in some sense? What experimental measurements might we make to tell if two different fractals may be metrically equivalent? [...] There are various numbers associated with fractals which can be used to compare them. They are generally referred to as fractal dimensions. They are attempts to quantify a subjective feeling which we have about how densely the fractal occupies the metric space in which it lies. Fractal dimensions provide an objective means for comparing fractals." (Michael Barnsley, "Fractals Everwhere", 1988)
"In deterministic fractal geometry the focus is on those subsets of a space which are generated by, or possess invariance properties under, simple geometrical transformations of the space into itself. A simple geometrical transformation is one which is easily conveyed or explained to someone else. Usually they can be completely specified by a small set of parameters." (Michael Barnsley, "Fractals Everwhere", 1988)
"In deterministic geometry, structures are defined, communicated, and analysed, with the aid of elementary transformations such as affine transfor- transformations, scalings, rotations, and congruences. A fractal set generally contains infinitely many points whose organization is so complicated that it is not possible to describe the set by specifying directly where each point in it lies. Instead, the set may be defined by "the relations between the pieces." It is rather like describing the solar system by quoting the law of gravitation and stating the initial conditions. Everything follows from that. It appears always to be better to describe in terms of relationships." (Michael Barnsley, "Fractals Everwhere", 1988)
"In fractal geometry we are especially interested in the geometry of sets, and in the way they look when they are represented by pictures. Thus we use the restrictive condition of metric equivalence to start to define mathematically what we mean when we say that two sets are alike. However, in dynamical systems theory we are interested in motion itself, in the dynamics, in the way points move, in the existence of periodic orbits, in the asymptotic behavior of orbits, and so on. These structures are not damaged by homeomorphisms, as we will see, and hence we say that two dynamical systems are alike if they are related via a homeomorphism." (Michael Barnsley, "Fractals Everwhere", 1988)
"The observation by Mandelbrot of the existence of a "Geometry of Nature" has led us to think in a new scientific way about the edges of clouds, the profiles of the tops of forests on the horizon, and the intricate moving arrangement of the feathers on the wings of a bird as it flies. Geometry is concerned with making our spatial intuitions objective. Classical geometry provides a first approximation to the structure of physical objects; it is the language which we use to communicate the designs of technological*products, and, very approximately, the forms of natural creations. Fractal geometry is an extension of classical geometry. It can be used to make precise models of physical structures from ferns to galaxies. Fractal geometry is a new language. Once you can speak it, you can describe the shape of a cloud as precisely as an architect can describe a house." (Michael Barnsley, "Fractals Everwhere", 1988)
"We must be careful how we interpret a map. Geographical maps are complicated by the real number system and the unphysical notion of infinite divisibility. Mathematically, the map is an abstract place. A point on the map cannot represent a certain physical atom in the real world, not just because of inaccuracies in the map, but because of the dual nature of matter: according to current theories one cannot know the exact location of an atom, at a given instant." (Michael Barnsley, "Fractals Everwhere", 1988)
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