02. Sectio Aurea – Michel Tombroff

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Michel Tombroff

Sectio Aurea

Texts by Simone Gutt & Michel Cahen

Short approach

The golden ratio is equal to \(\frac{1+\sqrt{5}}{2}\). It is an “irrational” number, which means it cannot be written as a fraction \(\frac{a}{b}\) with a and b as integers. It is the unique positive solution resulting from the following equation: \(x^2-x-1=0\). Its approximate value is \(1,61803398……\).

It is said to be harmonious or aesthetically pleasing. It is expressed as  (phi) in honour of Greek sculptor Phidias (5th century BC). At the end of the 15th century, Luca Pacioli named it the “divine proportion” in a book illustrated by Leonardo da Vinci. Johannes Kepler saw it as a “precious jewel”, a treasure of geometry. Terms such as “golden section” and “golden number” appeared in the 19th century.  German philosopher Adolphe Zeising  argued that the golden ratio was key to understanding numerous realms (architecture, painting, music, biology, anatomy). Despite questionable scientific claims, his ideas became very popular throughout the 20th century.

The golden ratio also influenced work by composer Iannis Xenakis, architect Le Corbusier, poet Paul Valery, and painter Salvador Dali.

Around 300 BC, Euclid provides the first definition as the ratio between two integers: the ratio between two positive integers \(L\) and \(\ell\) is equal to the golden ratio positives (\(\frac{L}{\ell}=\phi\))  if and only if \(\frac{L+\ell}{L}=\frac{L}{\ell}\).
((It follows that: \(\frac{L+\ell}{L}=\frac{L}{\ell}\) ssi \(1+\frac{\ell}{L}=\frac{L}{\ell}\) iff \(1+\frac{1}{x}=x\) with \(x=\frac{L}{\ell}\) iff \(x^2-x-1=0\) and \(x=\frac{L}{\ell}>1\) iff \(x=\frac{L}{\ell}=\frac{1+\sqrt{1+4}}{2}=\phi\).)

A rectangle is a golden rectangle if the ratio between its length and its width is equal to \(\phi\). To confirm that a rectangle is golden, take two copies of the rectangle and place them perpendicular to each other:

If the line segment \(d\) intersects \(O\) and \(S\) and summit \(A\), then it is a golden rectangle.

(In this case, the Thales theorem states that the two right-angle triangles with a horizontal base and whose hypotenuse intersects\(d\) are homothetic. It follows that: \(\frac{L+\ell}{L}=\frac{L}{\ell}\)).

(En effet, le théorème de Thalès nous dit dans ce cas que les deux triangles rectangles debase horizontale et d’hypoténuse suivant la droite \(d\) sont homothétiques, donc \(\frac{L+\ell}{L}=\frac{L}{\ell}\).

Try this with your ID card or credit card…

In a similar manner, we can define acute or obtuse golden triangles; they appear in pentagons

… and are the base for Roger Penrose’s (2020 Nobel prize) aperiodic tiling of a plane.

The golden ratio is closely related to Fibonacci sequence.
It is also possible to build a golden spiral.

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