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Golden Section

24 25 exPonenTials anD sPirals an extended family of wonderful curves In nature gnomonic growth occurs through simple accretion. It produces the beautiful logarithmic spiral growth we see in mollusks, which constantly add new material at the open end of their shells. Importantly, the shell grows in size, increasing in length and width, without varying its proportions. This accretive process, also used by crystals, is the simplest law of growth. The golden spiral, derived from Fibonacci numbers as shown on the cover, and from the arm of a pentagram below, is a member of the family of logarithmic spirals. These also go by the name of growth spirals, equiangular spirals, and sometimes spira mirabilis, wonderful spirals. When a spiral is logarithmic the curve appears the same at every scale, and any line drawn from the centre meets any part of the spiral at exactly the same angle for that spiral. Zoom in on a logarithmic spiral and you will discover another spiral waiting for you. They are to be contrasted with Archimedean spirals, which have equalspaced coils, like a coiled snake or hose. Nature uses numerous different logarithmic spirals in leaf and shell shapes, cacti and seedhead phyllotaxis, whirlpools and galaxies. Many can be approximated using a family of golden spirals derived from equal divisions of a circle see opposite after Coates Colman.
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