Английская Википедия:Golden spiral
In geometry, a golden spiral is a logarithmic spiral whose growth factor is Шаблон:Math, the golden ratio.[1] That is, a golden spiral gets wider (or further from its origin) by a factor of Шаблон:Math 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.[2]
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.[2]
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[3] – 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 Шаблон:Math and the Fibonacci series.[4] 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.[5][6][7] This pattern allows the organism to grow without changing shape.Шаблон:Citation needed 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.[8]
Mathematics
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 Шаблон:Mvar:[9] <math display="block">r = ae^{b\theta}</math> or <math display=block>\theta = \frac{1}{b} \ln(r/a),</math> with Шаблон:Mvar being the base of natural logarithms, Шаблон:Mvar being the initial radius of the spiral, and Шаблон:Mvar such that when Шаблон:Mvar is a right angle (a quarter turn in either direction): <math display=block>e^{b\theta_\mathrm{right}} = \varphi.</math>
Therefore, Шаблон:Mvar is given by <math display=block>b = {\ln{\varphi} \over \theta_\mathrm{right}}.</math>
The numerical value of Шаблон: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 Шаблон:Mvar (that is, Шаблон:Mvar can also be the negative of this value): <math display=block>|b| = {\ln{\varphi} \over 90} \doteq 0.0053468</math> for Шаблон:Mvar in degrees, or <math display=block>|b| = {\ln{\varphi} \over \pi/2} \doteq 0.3063489</math> for Шаблон:Mvar in radians.[10]
An alternate formula for a logarithmic and golden spiral is[11] <math display=block>r = ac^{\theta}</math> where the constant Шаблон:Mvar is given by <math display=block>c = e^b</math> which for the golden spiral gives Шаблон:Mvar values of <math display=block>c = \varphi ^ \frac{1}{90} \doteq 1.0053611</math> if Шаблон:Mvar is measured in degrees, and <math display=block>c = \varphi ^ \frac{2}{\pi} \doteq 1.358456</math> if Шаблон:Mvar is measured in radians.[12]
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 Шаблон:Math, Шаблон:Math, Шаблон:Math, Шаблон: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 slope
In the polar equation for a logarithmic spiral: <math display=block>r = ae^{b\theta}</math> the parameter Шаблон: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 Шаблон: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.[13]
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.
See also
References
Шаблон:Metallic ratios Шаблон:Spirals
- ↑ Chang, Yu-sung, "Golden Spiral Шаблон:Webarchive", The Wolfram Demonstrations Project.
- ↑ 2,0 2,1 Шаблон:Cite book
- ↑ Шаблон:Cite book
- ↑ For example, these books: Шаблон:Cite book, Шаблон:Cite bookШаблон:Self-published inline, Шаблон:Cite book, Шаблон:Cite book, Шаблон:Cite book, Шаблон:Cite book
- ↑ Шаблон:Cite book
- ↑ Шаблон:Cite web
- ↑ Шаблон:Cite web
- ↑ Шаблон:Cite journal
- ↑ Шаблон:Cite book
- ↑ Шаблон:Cite OEIS
- ↑ Шаблон:Cite book
- ↑ Шаблон:Cite OEIS
- ↑ Шаблон:Cite OEIS
