Golden spiral


In geometry, a golden spiral is a logarithmic spiral whose growth factor is, the golden ratio. That is, a golden spiral gets wider by a factor of 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.
A study explored the aesthetic preference between different spirals, focusing on the golden spiral and the Fibonacci spiral. In an online experiment comprising 106 participants, the golden spiral was found to be significantly preferred over the Fibonacci spiral, primarily due to the continuity of its curvature.

Spirals in nature

It is sometimes erroneously 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, 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. 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, and also at variance with the other mathematical spirals used to model them.
Phyllotaxis, the pattern of plant growth, is in some case connected with the golden ratio because it involves successive leaves or petals being separated by the golden angle. Although this can sometimes be associated with spiral forms, such as in sunflower seed heads, these are more closely related to Fermat spirals than logarithmic spirals.

Mathematics

A golden spiral with initial radius 1 is the locus of points of polar coordinates satisfying
where is the golden ratio.
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 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 :
for in degrees, or
for in radians.
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
hence
if measured in degrees, or
if measured in radians.
Its complementary angle
in radians, or
in degrees, is the angle the golden spiral arms make with a line from the center of the spiral.