geometry, a golden spiral is a
logarithmic spiral whose growth factor is φ, the
golden ratio. That is, a golden spiral gets wider (or further from its origin) by a factor of φ for every quarter turn it makes.
Approximations of the golden spiral
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 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
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 (necessarily) 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 φ and the Fibonacci series.
In truth, many
mollusk shells including 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. Although spiral galaxies have often been modeled as logarithmic spirals,
Archimedean spirals, or
hyperbolic spirals, their pitch angles vary with distance from the galactic center, unlike logarithmic spirals (for which this angle does not vary), and also at variance with the other mathematical spirals used to model them.
A golden spiral with initial radius 1 is the locus of points of
polar coordinates satisfying
with e being the base of
natural logarithms, a being the initial radius of the spiral, and b such that when θ is a
right angle (a quarter turn in either direction):
Therefore, b is given by
The numerical value of b 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 b (that is, b can also be the negative of this value):
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
θ, θ + π, θ + 2π, θ + 3π
the point C is the
projective harmonic conjugate of B with respect to A, D, i.e. the
cross ratio (A,D;B,C) has the singular value −1.
The golden spiral is the only logarithmic spiral with (A,D;B,C) = (A,D;C,B).