Golden spiral
In geometry, a golden spiral is a logarithmic spiral whose growth factor is Template:Math, the golden ratio.<ref>Chang, Yu-sung, "Golden Spiral Template:Webarchive", The Wolfram Demonstrations Project.</ref> That is, a golden spiral gets wider (or further from its origin) by a factor of Template:Math for every quarter turn it makes.
Approximations of the golden spiralEdit
There are several comparable spirals that approximate, but do not exactly equal, a golden spiral.<ref name=madden>Template:Cite book</ref>
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.<ref name=madden/>
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 natureEdit
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 Template:Math and the Fibonacci series.<ref> For example, these books: Template:Cite book, Template:Cite book, Template:Cite book, Template:Cite book, Template:Cite book</ref> 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.<ref> Template:Cite book</ref><ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref><ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> 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.<ref>Template:Cite journal</ref> 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,<ref>Template:Cite journal</ref> these are more closely related to Fermat spirals than logarithmic spirals.<ref>Template:Cite journal</ref>
MathematicsEdit
A golden spiral with initial radius 1 is the locus of points of polar coordinates <math>(r,\theta)</math> satisfying <math display=block>r = \varphi^{2\theta/\pi},</math> where <math>\varphi</math> 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 Template:Mvar:<ref>Template:Cite book</ref> <math display="block">r = ae^{b\theta}</math> or <math display=block>\theta = \frac{1}{b} \ln(r/a),</math> with Template:Mvar being the base of natural logarithms, Template:Mvar being the initial radius of the spiral, and Template:Mvar such that when Template:Mvar is a right angle (a quarter turn in either direction): <math display=block>e^{b\theta_\mathrm{right}} = \varphi.</math>
Therefore, Template:Mvar is given by <math display=block>b = {\ln{\varphi} \over \theta_\mathrm{right}}.</math>
The numerical value of Template:Mvar depends on whether the right angle is measured as 90 degrees or as <math>\textstyle\frac{\pi}{2}</math> radians; and since the angle can be in either direction, it is easiest to write the formula for the absolute value of Template:Mvar (that is, Template:Mvar can also be the negative of this value): <math display=block>|b| = {\ln{\varphi} \over 90} \doteq 0.0053468</math> for Template:Mvar in degrees, or <math display=block>|b| = {\ln{\varphi} \over \pi/2} \doteq 0.3063489</math> for Template:Mvar in radians.<ref>Template:Cite OEIS</ref>
An alternate formula for a logarithmic and golden spiral is<ref>Template:Cite book</ref> <math display=block>r = ac^{\theta}</math> where the constant Template:Mvar is given by <math display=block>c = e^b</math> which for the golden spiral gives Template:Mvar values of <math display=block>c = \varphi ^ \frac{1}{90} \doteq 1.0053611</math> if Template:Mvar is measured in degrees, and <math display=block>c = \varphi ^ \frac{2}{\pi} \doteq 1.358456</math> if Template:Mvar is measured in radians.<ref>Template:Cite OEIS</ref>
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 Template:Math, Template:Math, Template:Math, Template:Math 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).
Polar slopeEdit
In the polar equation for a logarithmic spiral: <math display=block>r = ae^{b\theta}</math> the parameter Template:Mvar is related to the polar slope angle <math>\alpha</math>: <math display=block>\tan\alpha=b. </math>
In a golden spiral, <math>b</math> being constant and equal to <math>|b| = {\ln{\varphi} \over \pi/2} </math> (for Template:Mvar in radians, as defined above), the slope angle <math>\alpha</math> is <math display=block>\alpha = \arctan(|b|) = \arctan\left({\ln{\varphi} \over \pi/2}\right),</math> hence <math display=block>\alpha \doteq 17.03239113</math> if measured in degrees, or <math display=block>\alpha \doteq 0.2972713047</math> if measured in radians.<ref>Template:Cite OEIS</ref>
Its complementary angle <math display=block>\beta = \pi/2 - \alpha \doteq 1.273525022</math> in radians, or <math display=block>\beta = 90 - \alpha \doteq 73</math> in degrees, is the angle the golden spiral arms make with a line from the center of the spiral.