Quest for perfection: those who preceded us

Father of geometry

Writings on the Golden Ratio have their origins in mathematics and geometry. Theano, a Greek philosopher, mathematician, and Pythagorean student was one of the first to write about the divine relationship. Several of her works have been documented but unfortunately none have been preserved.

The next work mentioning the Golden Ratio is by Euclid, a Hellenistic mathematician who was born around the year 300 BC. and was employed in the library of Alexandria.

The handbook Elements of this “Father of Geometry” is one of the most influential works in the history of geometry. In this book Euclid provided proof for the infinity of prime numbers.

Golden cooperation

In 1509, Elements is revived by the Italian mathematician Luca Pacioli. In his book De Divina Proportione he discusses, as the title already gives away, the mathematics of the Golden Ratio. He collaborates with Leonardo da Vinci who makes about sixty illustrations for this book, including the famous Vitruvian man.

The mathematics that Pacioli presents in De Divina Proportione is based on the knowledge that Fibonacci previously shared with the world. The Fibonacci numbers were published around 1200, a sequence where each number is equal to the sum of the previous two: AC / CB = 𝜑. 𝜑, rounded up to three decimal places, is 1.618. This is the Golden Ratio, called the divine ratio by Pacioli.

The clarity with which De Divina Proportione is written in combination with Da Vinci’s illustrations give this book an impact that extends far beyond mathematics.

From bastard to master

Leonardo da Vinci was born in Anchiano on April 15, 1452, as the bastard son of Piero, a notary, and Chataria, a peasant girl. He was mostly raised by his father who soon saw that the boy had a talent for painting. His father introduced him to Andrea del Verrocchio who ran a studio in Florence.

Leonardo was apprenticed to him and became part of the Florence Guild of Artists. This guild controlled all kinds of different media and designed everything related to the city. Here he became a true master of painting and he experimented a lot.

When the fight between the Baroncelli and the Medici families flared up, for a long time there was a war looming between Florence and Naples. The guild had to deal with the defense of Florence and because Leonardo was not allowed to join the army due to his bastard ancestry, he was engaged in designing weapons. This would remain a passion throughout his life that he was only able to truly fulfill for a short time.

The tragedy of the genius

When peace was made and the threat of war faded, Leonardo looked for new challenges. He moved to Milan and tried to pursue his career as a weapons designer. He came up with various inventions such as a tank, a giant crossbow, diving suits and even a robot soldier.

He was far ahead of his time and no invention caught on with his clients. It is the tragedy of the genius: he saw what was possible, but no one could share his horizon with him. It is only hundreds of years later that his ideas can be appreciated and many of his inventions can be found today in everyday objects, such as the ball bearing he first described around the year 1500.

Functioning ball bearings were not first made until much later, around 1740, and it was not until 1907 that the modern ball bearing was used. Leonardo was ahead of his time - many hundreds of years - but it is thanks to his invention of the ball bearing that FritsJurgens’ systems function today as they do, created with ingenious inventions from the past and present.

Founder of symmetry

Leonardo da Vinci studied Vitruvian’s De Architectura for Pacioli, written between 30 and 20 BC. and one of the most important works of construction and architecture. The theories described in this book are at the basis of the works and research of many mathematicians.

Marcus Vitruvius Pollio lived between about 85 and 20 BC and was a Roman soldier, architect and engineer. He wrote De Architectura libri decem: The architecture, in ten volumes. Vitruvius is also called “the first engineer” and his De Architectura is one of the most comprehensive source texts on engineering in Greco-Roman Antiquity.

The work is not only about architecture but also about the symbolic imitation of the order in nature. Vitruvius was - just as Leonardo da Vinci was centuries later - a homo universalis and according to Vitruvius an architect had to have a wide variety of knowledge, such as philosophy, physics, music, medicine, law and astronomy. Books three and four are about public religious buildings and the symmetry that goes with them.

The basis for this symmetry, according to Vitruvius, does not lie in a mirror image but in the correct proportion for both temples and the human body. This is what Leonardo da Vinci bases his drawing of the Homo ad circulum or Vitruvian Man on. Centuries later, it is also what inspired architect Le Corbusier to create The Modulor, a human measure for building design.

Perfection: three principles in balance

Perhaps Vitruvius’ most famous legacy is the three basic principles of good architecture: firmitas: solidity, utilitas: usability, and venustas: beauty. According to him, the three principles should be in balance and should not dominate each other.

The namesake of FritsJurgens, Mr. Frits Jurgens, had a similar philosophy. “A perfect design is innovative, functional, beautiful, and useful.” Based on these values, FritsJurgens still works every day on creating technical door solutions.

According to Vitruvius and technician Frits Jurgens, the beauty of a design, the venustas, is reflected in the degree of functionality of the design. The triad of functionality, beauty and user-friendliness is only possible if all parts have the correct, fitting size in relation to the whole and the other sizes: the eurhythmia. Eurythmia is a Latin word derived directly from the Greek. It means “rhythmic order or movement” or “graceful movement”.

The balance between strength, user-friendliness and beauty gives FritsJurgens pivot hinge systems their strength. Eurythmia: the timeless beauty of hidden perfection and the ultimate door movement.

Mathematical efficiency

The Golden Ratio can also be found in nature. The intrinsic beauty of the ratio is reflected in various natural creations such as the proportions between branches on trees, the fruit of a pineapple or the propagation of flower bulbs - 1,618 times as many bulbs every year.

We take as an example the petals of a flower. The golden angle is approximately 137.5° and divides a circle according to the Golden Ratio. If each petal forms a golden angle in relation to its predecessor, the disc is filled most efficiently, and light falls optimally on each petal. If the petals were to grow at an angle of 120°, for example, a layer of petals would form at three points that grow in exactly the same place and therefore do not catch sunlight.

The path of least resistance

This origin in nature could explain the way in which the Golden Ratio makes whatever is designed with it so perfect to the human eye. Because things that have an unconsciously recognizable relationship are easier for the brain to process than if it does not conform to the Golden Ratio.

This makes the universal aesthetics of the divine relationship plausible and useful in all kinds of media, from painting and architecture to advertising and everyday products.

Human abstraction

Pacioli was not the last to take the inventions of Vitruvius as the basis for much of his work. Between 1942 and 1955, the Swiss-French architect Le Corbusier developed an architectural measurement system based on the theory of the Vitruvian man and the Fibonacci series and called it The Modulor.

Le Corbusier’s goal was to create a mathematical approach to the human scale that allows buildings to be designed based on human dimensions. This abstraction of the human body is exactly what Vitruvius used when he created the Vitruvian Man.

Le Corbusier’s The Modulor

The Modulor consists of two series, the red series and the blue series. For the red series, Le Corbusier takes a measure of 183 cm - a measure he takes for the length of the human body - as a starting point and divides it repeatedly by 𝜑. The blue series follows the same principle but with a size of 226 cm. According to Le Corbusier, this is the human measure with an outstretched arm. This corresponds to a doubling of the navel height of 113 cm, which also occurred in the red series.

The red series: 183, 113, 70, 43, 27 …
The blue series: 226, 140, 86, 54 …

Le Corbusier describes his series based on Fibonacci’s rows as follows: “… rhythms that are apparent to the eye in their interrelationships. And these rhythms are the basis of human activities. They resound in man through an organic inevitability, the same refined inevitability that tracing the Golden Ratio causes in children, old men, savages and scholars.”

Geometric masterpiece

The most famous example of a building based on The Modulor by Le Corbusier is the Unité d’Habition. This housing unit or vertical city complies in various ways with the series drawn up by Le Corbusier.

The dimensions of the individual apartments, the loose elements of the façades and different types of superstructure on the roof can all be traced back to The Modulor. According to some studies on the Unité, the three largest dimensions - the length, width and height of the building - are also based on The Modulor.

The illusion of the Golden Ratio

Many architects and artists are said to have consciously applied the Golden Ratio in their works. One of those artists is Piet Mondriaan. His works seem measured, thought out.

At first glance, a conscious connection with the Golden Ratio is obvious. However, Mondriaan only worked for a short period with measured modules of 16 by 16, each of which had exactly the shape of the painting - apart from this period, Mondriaan rejected any mathematical calculation for his art: “Coincidence must be as distant as calculation.”

While experimenting, Mondriaan’s works changed over time, but the spontaneity and experimental character of the paintings remain a constant. He rejects the application of the Golden Ratio of one of his most loyal students, Marlow Moss, and sticks to his own intuitive source of the creative process.

Charles Bouleau investigated three works by Mondriaan for the appearance of the Golden Ratio. In Tableau I, Two-Line Composition and Broadway Boogie Woogie, Bouleau discovers an underlying grid that satisfies the values of the Golden Ratio.

Traces of overpainting can be found in many of Piet Mondriaan’s works. He restlessly experimented with the boxes and lines on the canvas, until a satisfying composition was created. So, despite rejecting geometrical theories in art, it may be that, through a constant search for perfection, Mondriaan has created an equivalent of the Golden Ratio in some of his works.

Casual beauty?

The same may be true for the designers of well-known architectural feats such as the Pyramid of Kukulkán, the Great Mosque of Kairouan, the Stupa of Borobudur, the Pyramid of Cheops, the Parthenon, Notre Dame and the Taj Mahal.

For some buildings it is doubted whether the creators could already have had knowledge of the Golden Ratio, while other buildings sizes deviate, in yet others only part of the building meets the Golden Ratio. What can generally be said is that each of these designs has a strong aesthetic appeal to many people. Perhaps the designers’ intention was not to apply the Golden Ratio - whether or not because of the lack of knowledge of it - but to strive for perfection.

Just as with works by Piet Mondriaan, it may be that perfection in these buildings was pursued in such a way that it makes sense that the divine proportions would present themselves in the final result.

Designed, painted, created: absolute perfection

The pursuit of absolute perfection, whether or not by applying the Golden Ratio, is a goal in itself regardless of the medium.

By ‘repainting’ what is being created, over and over again, this divine relationship can eventually arise: a building that at first glance immediately seems logical, a painting that demands your full attention before you even have entered the room properly, a movement so smooth and fluid that it becomes completely self-evident - Le Corbusier designed it, Mondriaan painted it and FritsJurgens strives for it every day. Absolute perfection is the goal.