Golden ratio

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Corresponding Wikipedia article: Golden ratio


«…the natural sequence, which is essentially a possible, but very artificial mathematical contrivance that has little to do with real nature, has become the basis for those rudiments of mathematics with which the overwhelming majority of Mankind are only familiar»

(Unknown author)

Golden ratio 1.png

The golden section is such division of a line segment into two parts, when the length ratio of whole segment to its long part is equal to the ratio of its long part to its short part. This lengths ratio is called the golden ratio: \[\Phi=\frac{AB}{AC}=\frac{AC}{CB}\] If whole segment \(AB=1\), then \(AC\) is the golden section \(\phi\). \[\Phi=\frac{1}{\phi}=\frac{\phi}{1-\phi}\] \[\phi=\frac{\sqrt{5}-1}{2}\approx0,618\] The relationship between the ratio and the section: \[\Phi=\frac{1}{\phi}=1+\phi=\frac{\sqrt{5}+1}{2}\approx1,618\] The golden ratio sequence is a geometric progression with a common ratio \(\Phi\) or \(\phi\). This sequence has an unique property: \[\Phi^N=\Phi^{N-1}+\Phi^{N-2}\] \[\phi^N=\phi^{N+1}+\phi^{N+2}\] The golden ratio sequence is produced by a sequential division of a line segment and its parts:

Golden ratio 2.png

\(AB=1, AC=DB=\phi, DC=\phi^2\)

The regular pentagon and the pentagram have a golden ratio.

Golden ratio 3.png \(AB=1, AC=DB=AE=EB=\phi, DC=\phi^2, \phi=2sin\frac{\pi}{10}, \Phi=2cos\frac{\pi}{5}\)

Hambidge's golden logarithmic spiral (\(b=\frac{ln\Phi}{\pi}\)). The spiral intersects the orthogonal radial axes in a golden ratio. The intersection points produce the right angles.
The golden logarithmic spiral (\(b=2\frac{ln\Phi}{\pi}\)) with golden rectangles which are dividing into squares.

The human face ratios are close to golden

The human arm has the highest (in comparison with animals) abilities, and theirs bone length ratios are close to a golden ratio:

Hand (long fingers) Hand (thumb) Arm
distal phalanx 1 distal phalanx 1 hand (average) 1
intermediate phalanx \(\Phi\) proximal phalanx \(\Phi\) forearm \(\Phi\)
proximal phalanx \(\Phi^2\) metacarpal \(\Phi^2\) shoulder \(\Phi^2\)
metacarpal \(\Phi^3\)

The arm bones, which bend at the right angles, almost fit to the golden spirals.

Fibonacci sequence

The Fibonacci sequence is an integer approximation of the golden ratio sequence: \[F_N=F_{N-1}+F_{N-2}\;\;\;\;\;F_1=F_0=1\]

\(N\) \(F_N\) \(\Phi^{N-1}\) error
0 1 0,618 62%
1 1 1,000 0%
2 2 1,618 24%
3 3 2,618 15%
4 5 4,236 18%
5 8 6,854 17%
6 13 11,09 17%
7 21 17,94 17%

It’s useful to represent an integer number as the Fibonacci numbers sum. The mankind use such system widely in the sequences of measures, for example, the reference weights, the banknote denominations. But people practically use a sequence of the round numbers:

Practical sequence 1 2 3 5 10 15 20 25 50 100
Fibonacci sequence 1 1 2 3 5 8 13 21 34 55 89

The number represented by the Fibonacci sequence is generally redundant (multimode):

8 5 3 2 1 1
3 1
5 1
6 1
8 2
9 2
10 1
11 1
13 3
14 3
15 2
16 2
17 2
18 1
19 1

The “Golden numbers” are marked yellow and have the unique representation mode. Obviously their sequence is produced by one of the following expressions: \[\sum_{i=0}^{N-1}F_i\;\;\;\;\;N=1,2,3…\] \[F_N-1\;\;\;\;\;N=2,3…\]

See also

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