Studies of Proportions

The first studies about harmonious proportions raise old Greece. At that time, Phidias, one of the main ones responsible for the acropolis of Athena, and Ictinus, the architect of Partenon, were the first ones to demonstrate the potential of the design based on the line divisionin proportional segments, what later would come to be known as golden section.

One of the first records of this subject is in the book De Divina Proportione, of Fra Luca Pacioli in 1509. The proportion that him called of "divine" derived from the division of a line in such a way that the reason of the entire line with the largest segment corresponded to the reason of the larger segment in relation to the smallest. When expanded, that proportion became the Fibonacci series. Fibonacci was an Italian mathematician - Pisa - of the century X. the series that takes his/her name is formed by numbers that correspond to the sum of the previous two (0,1, 1, 2, 3, 5, 8, 13, n...).

The Golden Section

This proportionality system is based on the regular pentagon and his correlate pentagram. For the designers prone to the mathematics, the golden reason is represented by the Greek letter Φ (phi), in honor to the Greek sculptor Phidias, and it corresponds to the irrational number 1,61803398.

The image above illustrates the proportional relationships in the regular pantagon and its respectives pentagram. Adapted of HURLBURT, 1978.

The square

The square was also studied in order to meet harmonious proportions, a lot of combinations were based on the simple division by the square parts, that is a natural division of the golden rectangle. In Japan, the square has a fundamental role in the traditional architecture and decoration, visible above all in the production and asymmetrical arrangement of tatamis (rugs of aprox 0,91 X 1,83m). Besides, one of the dimensional patterns more used now for paper format (DIN) is used of the rectangle based on the square "root of 2" (1,4142).

The image above shows the proportionality used in tatames and the construction of the proportional rectangle with the diagonal of the saquare. Adapted of HURLBURT, 1978.

A confusion between the two presented proportional systems has appeared in Paris, when a cubist group used that kind of rectangle in an exhibition called La Section d'Or.

Dynamic symmetry

In the XX century several personages used the golden section as a design element. One of them was Jay Hambidge, whose book Elements of the Dynamic Symmetry, published in 1920, still preserves a great amount of followers. The author was probably the first one to relate the golden proportion with the logarithmic spiral.

The image above presents 8 golden rectangles in a Fibonacci series, positioned to create the logarithmic spiral. Source: HURLBURT, 1978.

The author also points that when the diagonal of the rectangle intercepts the side of the original square, it creates a harmonic subdivision. He took borrowed the expression "dynamic symmetry" of Plato, and one of their sources of the I century, Vitrúvio.

The Modulator (Modulor)

The first grid system clearly identifiable as aiming the design was created by Le Corbusier. Originally his proposal was applied to architecture, but he quickly conceived applications in the graphic area.

The book cover of Le Corbusier was designed using the concept of the book. Source: HURLBURT, 1978.

The proposal of Le Corbusier was important as it related the golden proportion to the human body. That elaborated system produced a series of countless mathematical proportions, used above all in the architecture. Besides, it influenced typographic designers of Germany and Switzerland to create more modern grids guided to the design.