There are several comparable spirals that approximate, but do not exactly equal, a golden spiral. For example, a golden spiral can be approximated by first starting with a rectangle for which the ratio between its length and width is the golden ratio. This rectangle can then be partitioned into a square and a similar rectangle and this newest rectangle can then be split in the same way. After continuing this process for an arbitrary number of steps, the result will be an almost complete partitioning of the rectangle into squares. The corners of these squares can be connected by quarter-circles. The result, though not a true logarithmic spiral, closely approximates a golden spiral. Another approximation is a Fibonacci spiral, which is constructed slightly differently. A Fibonacci spiral starts with a rectangle partitioned into 2 squares. In each step, a square the length of the rectangle's longest side is added to the rectangle. Since the ratio between consecutive Fibonacci numbers approaches the golden ratio as the Fibonacci numbers approach infinity, so too does this spiral get more similar to the previous approximation the more squares are added, as illustrated by the image.
Spirals in nature
Approximate logarithmic spirals can occur in nature, for example the arms of spiral galaxies - golden spirals are one special case of these logarithmic spirals, although there is no evidence that there is any general tendency towards this case appearing. Phyllotaxis is connected with the golden ratio because it involves successive leaves or petals being separated by the golden angle; it also results in the emergence of spirals, although again none of them are golden spirals. It is sometimes stated that spiral galaxies and nautilus shells get wider in the pattern of a golden spiral, and hence are related to both and the Fibonacci series. In truth, spiral galaxies and nautilus shells exhibit logarithmic spiral growth, but at a variety of angles usually distinctly different from that of the golden spiral. This pattern allows the organism to grow without changing shape.
Mathematics
A golden spiral with initial radius 1 is the locus of points of polar coordinates satisfying The polar equation for a golden spiral is the same as for other logarithmic spirals, but with a special value of the growth factor : or with e | being the base of natural logarithms, being the initial radius of the spiral, and such that when is a right angle : Therefore, is given by The numerical value of depends on whether the right angle is measured as 90 degrees or as radians; and since the angle can be in either direction, it is easiest to write the formula for the absolute value of : An alternate formula for a logarithmic and golden spiral is: where the constant is given by: which for the golden spiral gives values of: if is measured in degrees, and if is measured in radians. With respect to logarithmic spirals the golden spiral has the distinguishing property that for four collinear spiral points A, B, C, D belonging to arguments ,,, the point C is the projective harmonic conjugate of B with respect to A, D, i.e. the cross ratio has the singular value −1. The golden spiral is the only logarithmic spiral with =.
Polar slope
In the polar equation for a logarithmic spiral: the parameter is related to the polar slope angle : In a golden spiral, being constant and equal to , the slope angle is: Its complementary angle is the angle the golden spiral arms make with a line from the center of the spiral.
