Speedtape
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- Oct 10, 2004
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Logarithmic Spirals
The spiral shapes of a nautilus are called Equiangular or Logarithmic spirals. If P is any point on the spiral then the length of the spiral from P to the origin is finite. In fact, from the point P which is at distance d from the origin measured along a radius vector, the distance from P to the pole is d sec a. 
The Fibonacci sequence relates closely to the golden ratio and to logarithmic spirals. Logarithmic spirals are simply spirals that increase at a logarithmic rate. The golden ratio, however, is a special fraction equivalent to about 0.618. A logarithmic spiral can be generated by subdividing a golden rectangle into increasingly smaller squares and golden rectangles. This subdivision begins by fitting a square within the golden rectangle. The remaining space forms a new, smaller golden rectangle. By repeating this process, the spiral form soon becomes evident. Furthermore, the subdivided sections can be thought about as Fibonacci numbers.
One reason why the logarithmic spiral appears in nature is that it is the result of very simple growth programs such as:
Any process which turns or twists at a constant rate but grows or moves with constant acceleration will generate a single logarithmic spiral.
