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6 7 Geometric selfsimilArity gnomons and other selfsimilar figures Symmetry is an invariable characteristic of both growth and form, whether in simple or complex, living or nonliving, systems. The gnomon demonstrates one of the simplest examples of geometrical growth below. The principle is this when a gnomon is added to another figure that figure is enlarged but retains its general shapeand this can be carried on indefinitely. This is essentially what happens in the elaborate forms created by shells and horns, where new growth is added to dead tissue. Dilation symmetries also produce figures that are geometrically similar to an original. These derive from the enlargement or reduction of a form by way of lines radiating from a centre. Dilation symmetries, which may extend from the infinitely small to the infinitely large, can use any angle from a centre 1, or any regular division of the circle 2, or its entirety 3. Dilation may also be linked to rotation, producing continuous symmetries that can give rise to equiangular spirals 4 of which more later, or discontinuous symmetries 5, in which case the increments are not necessarily a submultiple of a complete turn. Dilation symmetries also occur in threedimensional space. As can be seen, spiral symmetries are intimately connected with the movements of rotation and dilation, and tend to emerge whenever these are combined. 1. Dilation symmetries involve regular increase or decrease 2. Pointcentred dilation 3. Dilation over 360 o 4. Dilation combined with rotation 5. Discontinuous rotated dilation 6. Similarity symmetries arising from the regular arrangement of figures
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