"Golden" spirals

The spiral is a plane line derivated by a driving point, which moves away according to the definite law from the beginning of the ray uniformly rotated around of the beginning. If we select the beginning of the spiral as the pole of the polar coordinate system then mathematically the spiral can be presented with the help of some polar equation r = f(j), where r is the radius-vector of the spiral, j is an angle put aside on the polar axis, f(j) is some monotonically increasing or decreasing positive function.

If the point moves away from the beginning uniformly (r = aj) we have Archimedes spiral. If the point moves away according to the exponential law (r = aemj where a where a is an arbitrary positive number we have an equiangular spiral (Fig.1).

Equiangular spiral
Figure 1. Equiangular spiral.

The equiangular spiral has a number of interesting properties:

  1. In the equiangular spiral the line segments OA, OB, OC, OD, ... derivate geometrical progression, that is


    where m is a denominator of the progression.
  2. The radius-vector and tangent to any point of the equiangular spiral derivate a constant angle b, that is the curve intersects all rays coming out of poles O under the same angle.
  3. The equiangular spiral is degenerated accordingly in a straight line and circumference at values of angles b = 0 and b = 90°. This means that the spiral has properties of both straight line and circumference.

Any equiangular spiral represents the scheme of growth or ascending and can be expressed by geometrical progression. Here the "golden" equiangular spiral is of a special interest. In this spiral the terms of geometrical progression corresponding to the spiral are the degrees of the golden proportion {tn}(n = 0, ±1, ±2, ±3, ...). Such spiral has property to be simultaneously by geometrical and arithmetic progression. This means that its exponential growth is provided by simple addition of two adjacent terms. In opinion of many researchers, this remarkable property (a possibility of implementation of ascending by simple addition) enables to explain many phenomena and processes in botanic and biology.

Note also, that the "golden" spiral by a natural mode is connected to the "golden" rectangle (Fig.2).

'Golden' spiral
Figure 2. "Golden" spiral.

If as the beginning of the spiral we select the point, to which "golden" rectangles sequentially converge, the "golden" spiral will pass through three of four topics of each "golden" rectangles sequentially constructed on Fig. 2.