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Symmetry

28 29 spirAls And helices natures favourite structures Of all the regular curves, spirals and helices are probably the most common. They are found throughout the natural world, in many forms, at every scale of existencein spiderwebs 1, galaxies 2 and particle tracks 3 in animal horns 4, seashells 5, plant structures, and DNA 6. It is clearly one of natures favourite patterns. In purely geometric terms, the common planar spirals are of three principle types below the Archimedean a, the Logarithmic b, and the Fermat c. The Archimedean spiral is perhaps the simplest, consisting of a series of parallel, equidistant lines as in old vinyl records. Logarithmic or growth spirals are the most intriguing and complex of all, particularly the golden spiral 8 that is associated with the Fibonacci series see next page. Logarithmic spirals in general have the property of selfsimilarity, i.e., of looking the same at every scale. In the Fermat or parabolic spiral, successive whorls enclose equal increments of area, which accounts for its appearance in phyllotaxis, the arrangements of leaves and florets on a stem and in coffeecups. Helices are symmetrical about an axis, so always have a particular handedness d . Dilation symmetry can apply to helices, gradually increasing their width e, and of course they may be expressed in any number of strands, in the way that ropes are laid f . a. b. c. d. e. f. 5. 4. 1. 2. 3. 6. 7. An Evolute spiral. 8. The Golden logarithmic spiral.
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