Art, Architecture, Nature and Mathematics: The Golden Ratio and the Modulor

The "golden ratio" evokes a mysterious universe... It is even referred to as the "divine proportion" for this number: 1.6180339887. What is special about this number? Why should it be of interest to biologists, mathematicians, artists, and of course architects? This article offers some answers.

An Invitation To Cross Disciplines

The golden ratio is obtained by looking for a particular rectangle. Let us imagine a rectangle of length b and width a. Inside this rectangle, we draw a square (an x a). We are left with a rectangle of length a, and width a - b... Hope you can follow?

A diagram will help us.

A gold rectangle is one in which the proportions of the new rectangle are the same as those of the original rectangle. As a result, one can go on indefinitely.

And this rectangle is found everywhere. The face of the Mona Lisa, the Parthenon, also on snail shells... The proportion b/a which is therefore 1.618 is found in rose petals, the organization of sunflowers, and a whole range of natural elements.

The calculation of the golden ratio goes back to Fibonacci, a traveling mathematician, who helped introduce modern numeration to Europe.

Two hundred years later, Luca Pacioli published "De Divina proportione". Leonardo da Vinci illustrates this work and is inspired by it in his designs. The "divine proportion" partly explains the harmony that emerges from Vitruvian Man.

For Architects: Harmonious Buildings, and Adapted to Man

In his work, Le Modulor, the architect Le Corbusier synthesizes the work that led him to become interested in the golden ratio. Le Corbusier believes that the metric system has dehumanized objects. A meter corresponds to nothing in particular, whereas the ancient measures (the foot, the cubit, the hand-span...) referred to the human body.

Le Corbusier evokes the articulated arm that architects used to measure lengths. With meters, decimeters, and centimeters. Architects, surveyors, and urban planners rationalized, but above all dehumanized, again according to Le Corbusier.

The Modulor

Le Corbusier set out to build a more human system, based on the proportions of a body and the golden ratio.

The Modulor is conceived based on a human of 1 meter and 83 centimeters. All you have to do is divide 1.83 by the golden ratio to get the first set. By multiplying by two the first series, we obtain a second one. These series of lengths will then serve as a reference for all constructions, from the width of corridors to the height of stools. The Cité Radieuse is a practical implementation of this model that Le Corbusier used throughout his life.

The work of Le Corbusier also provides an opportunity to consider the role of architecture and urbanism. Making architecture to the scale of a human being necessitates a strong picture of the people who will live there. Buildings become "living machines," modeling human behavior and societal organizations, and turning the architect into a demiurge.

Recently, the newspaper le Monde echoed the debates on the occasion of the exhibition dedicated to this architect at the Centre Pompidou, in Paris.

Art, architecture, nature, mathematics, and even ethics and politics.

NB:  The golden ratio can take us very far!

Illustrations: Frédéric Duriez

Resources

The resources presented are numerous, but this is only a small part of the work you can find on the Internet. Most of them are conducted in interdisciplinary work, and the resources are most often student work or teacher developments that present their common pedagogical progression. They are therefore mostly intended to stimulate pedagogical activities.

Mathematics Portal in Geneva Fibonacci's biography accessed on 7 August 2015
http://icp.ge.ch/dip/maths/IMG/pdf/act_cult_biofib.pdf

A fairly comprehensive mathematics resource:
Jean-Christophe MICHEL The Golden Ratio accessed August 7, 2015
http://www.gecif.net/articles/mathematiques/nombre_d_or/

A very lively and clear video
Mickäel Launay MicMaths: the golden ratio accessed August 7, 2015
https://youtu.be/DxmFbdp7v9Q

The golden ratio in nature:

• The golden ratio in nature accessed on August 7, 2015
  http://leventtourne.free.fr/livreouvert/NombreOr/introduction.html
  
• A proposed activity for 12-year-old students - mathematics and earth sciences and life.
  
• The Golden Number in Nature - accessed August 7, 2015
  http://maupassant.clg.ac-amiens.fr/gdmv2/projets/co-intervention/5c/382-co-inter-maths-svt
  
  on the same theme - for students of about 15 years
• Jérôme Ygé: The content of the various sessions devoted to the golden ratio, accessed on 7 August 2015
  http://yge.fr/atelier/descriptif.html
  

Architecture, Golden Number and Modulor

• College Golfe des Pictons - Louvre Pyramid and Golden Number accessed August 7, 2015
  http://golfedespictons.vendee.e-lyco.fr/espace-pedagogique-d-etablissement/matieres-enseignees/mathematiques/pyramide-du-louvre-et-nombre-d-or--490.htm
  
• Golden Number and Architecture, Reality or Coincidence - accessed August 7, 2015
  http://www.scoop.it/t/nombre-d-or-et-architecture
  
• Le Corbusier Foundation: the Achievements - accessed August 7, 2015
  http://www.fondationlecorbusier.fr/corbuweb/morpheus.aspx?sysId=64&sysLanguage=en-fr&itemPos=1&sysParentId=64&clearQuery=1
  

Site of four students from ECE, an engineering school located in Paris:
The Modulor - accessed on August 7, 2015
http://www.lenombredor.free.fr/modulor.htm

Télérama La cité radieuse - accessed August 7, 2015
http://www.telerama.fr/scenes/voyage-avec-le-corbusier-1re-etape-dans-la-cite-radieuse-suivre-la-course-du-soleil,125256.php

Le Monde Le Corbusier fasciste? accessed on 7 August 2015
http://www.lemonde.fr/idees/article/2015/06/22/le-corbusier-fasciste_4659424_3232.html

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