Golden section

definition and geometric construction

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Costruzioni geometriche

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29 June 2023

Golden section

Its geometric and mathematical properties and its frequent re-presentation in various natural and cultural contexts, apparently unrelated to each other, have for centuries aroused in the mind of man the confirmation of the existence of a relationship between macrocosm and microcosm, between God and the man, the universe and nature: a relationship between the whole and the part, between the largest and the smallest part that is repeated endlessly through infinite subdivisions. [1] Over time, various philosophers and artists have come to grasp an ideal of beauty and harmony, pushing themselves to search for it and, in some cases, to recreate it in the anthropic environment as a canon of beauty; testimony of this is the history of the name which in more recent times has taken on the names of golden and divine ….. Wikipedia … >>

 A     M        B
 
| 1-x |    x    |

 
The segment AB is divided by the point M in such a way that the ratio of the two parts, the smallest to the largest (AM and MB), is equal to the ratio of the largest part (MB) to all AB.

If AB is of length 1, and we call x the length of the segment MB, then the above definition gives rise to the following equation:
 
     1 – x    =   x   , e cioè   1-x = x2
       x            1

which has two solutions for x, (-1-√5) / 2 and (√ 5-1) / 2.

The first is negative, so it does not satisfy the conditions of the problem. The second represents the golden section ratio and is an irrational number corresponding to about 0.618.
The reciprocal of x (1 / x) is denoted by Ø and corresponds to 1 + x, i.e. about 1.618. Most often this ratio is referred to as the golden ratio and is used in the construction of the golden rectangle.
The construction of the auras section suggests the possibility of realizing a growth process in which relationships are constantly preserved, that is, growth gives rise to organisms that always remain similar to themselves.
 

Si consideri un segmento iniziale e si divida in modo che la proporzione di (B) con (A) sia la stessa della proporzione di (C) con (B):

Sectioning a line to form the Golden Section or Divine Proportion

Si divida il segmento di nuovo e poi ancora sempre allo stesso modo:

Successive Golden Sections of a Line

Combinando i segmenti si ottiene una sorta di "regolo aureo".

The "Golden Ruler" - a Fibonacci Measuring Stick (copyright EOT 1997)
 

This seems to give rise to the proportions of many life forms:


Given a segment, the geometric construction of its golden section is easily obtained with a ruler and compass. >>

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