Marseilles

France’s maritime gateway to the Mediterranean Sea and North Africa, Marseilles is the country’s largest seaport and its oldest and second largest city.

Attractively situated on the Gulf of Lyons just east of the Rhône delta, it is a bustling commercial and industrial centre, home to over 1 million inhabitants, including a growing proportion of people of African origin who have added vigour and variety to its lifestyle. Because of its history (from around 600 B.C. when the Greeks founded a city here) and its size, Marseilles provides the visitor with much to explore. Its crowning glory is the cathedral of Notre-Dame-de-la-Garde (left).


















Unité d'Habitation / Marseilles Block

South of the centre of the city on the Boulevard Michelet (the N559) is one of the most influential buildings of the twentieth century. It is an apartment block, known as the Unité d’Habitation, built in 1947 and designed by the great Swiss architect Charles-Edouard Jeanneret, known commonly as Le Corbusier. (He was also a painter, sculptor and city planner.)

The building’s architectural significance lies in its use of raw concrete, its style of construction, and its inclusion of communal facilities (e.g. a shopping precinct and a rooftop kindergarten), features which were widely emulated, though without the same devotion to harmonious design, for high rise apartment blocks all over the world.

Similar but less successful apartment blocks designed by Le Corbusier may be found in Berlin and Nantes. The mathematical significance of the Unité lies in the architect’s deliberate and overwhelming use of the Golden Section and Fibonacci Series in the design, using a system which he termed ‘The Modulor’.

Further reading ...

Le Corbusier (1954), The Modulor, Faber and Faber.
Huntley, H.E. (1970), The Divine Proportion, Dover.

























Golden section and Fibonacci sequence (I)

To understand Le Corbusier’s Modulor, we need to begin with two of its elements: the golden section (or golden ratio) , and the Fibonacci sequence. The golden section may be defined conveniently in terms of the golden rectangle, which has the property that the ratio of the length of the smaller side to the greater is equal to the ratio of the length of the greater side to the sum of the lengths of the two sides. The sides of the golden rectangle are in the ratio :1, where is calculated by:
           1/ = /(1 + ), so that 1 + = 2 and 2 – = 1.

Solving this quadratic equation gives = (1 + 5)/2 = 1.618... The Ancient Greeks, who first defined this ratio, and Le Corbusier (among many others) believed that the golden rectangle has proportions which are most appealing to the eye.

Leonardo of Pisa or Fibonacci (son of Bonaccio), was the most eminent mathematician of the Middle Ages. In his Fibonacci sequence each term is the sum of the previous two terms. The most basic and famous sequence is:
                          1, 1, 2, 3, 5, 8, 13, 21, 35, ...  
However, a general Fibonacci sequence, beginning with any two numbers is:
              u0, u1, (u0 + u1), ( u0 + 2u1), (2u0 + 3u1), (3u0 + 5u1), ...

The coefficients of u0 and u1 are terms of the basic Fibonacci sequence.    





















Golden section and Fibonacci sequence (II)

Now consider a sequence of rectangles beginning with a golden rectangle (1) of unit width. Along the length of this rectangle, add a square (2) to produce a new rectangle of length 1 + and width . This has the dimensions of a golden rectangle. Can you prove this? Along the length of this second rectangle, add a square (3) to produce yet another golden rectangle, then a third square (4) along its length to produce another golden rectangle and so on.

Prove that the lengths of this sequence of rectangles form a Fibonacci sequence. Furthermore, show that the terms of this sequence form a geometric progression with first term 1 and common ratio .

That is, show that the sequence 1, , 2, 3, 4, ... is both ‘Fibonacci’ and geometric.

This is the essential property of the Modulor, as we shall see.




















The Modulor (I)

Le Corbusier wanted to design mass housing for the post-World War II reconstruction which was modularised, relatively cheap and yet inhabitable. To achieve this, he argued, the proportions needed to be based on the proportions of the human body so that people would feel ‘at home’, and the measurements compatible with each other to facilitate the modular construction. He appealed to an idea of the Ancient Greeks (and Egyptians), that the navel divides the upright body in the ratio :1. He added the notion that the tips of the fingers of the upstretched arm are at twice the height of the navel. A diagram (above) using these proportions became the trademark of his ‘Modulor’.

Le Corbusier sought a way to construct a double square with an inscribed golden rectangle using straightedge and compass.

  Investigate  Show that the method of construction of the golden rectangle is accurate (left figure, top right), but that the construction of the double square (below) is not.

























The Modulor (II)

Having demonstrated the constructibility of the proportions for his modular grid, Le Corbusier looked for a standard height upon which to base the Modulor. He settled on 183 cm, the height of a ‘six-foot detective’, giving 113 cm as the navel height (the dimension of the unit square), 226 cm as the height of the fingertips of the upraised arm, and 86 as the height where the hand rests. He then defined a

    Red Series (Rn) : 5, 11, 16, 27, 43, 70, 113, 183, 296, 479, 775, 1254, ...
and a
   Blue Series (Bn) : 1, 6, 7, 13, 20, 33, 53, 86, 140, 226, 366, 592, ...

– Fibonacci sequences containing the key dimensions 113 and 183, 86 and 226. Allowing for rounding off, the system worked well in metric and imperial units.

Using the values in the two Series, Le Corbusier was easily able to demonstrate that any square or rectangular region whose dimensions corresponded to those values could be dissected in seemingly limitless numbers of ways into smaller regions whose dimensions also took values from the Series. Here was the proportional method which would allow modularisation of components for building yet provide needed variety. He was proud to boast that in his Unité d’Habitation he only needed 15 standard measurements for the whole project, including dimensions of windows, doors, furniture etc. The Modulor system was successfully used in a whole range of other design projects, including factories, sculpture, typography and offices.