The fascination and beauty of the golden ratio

Slightly edited excerpt from an essay by Dr. Dr. Ruben Stelzner:

The question arises as to why the occurrence of the golden ratio has always been, and still is, associated with beauty. The principle of equality and unity plays a key role here. In the golden ratio, however, the image of perfection no longer arises from the equality of the parts, but from the equality of the proportions. The ratios of the minor parts to the major and the major to the whole are always the same.

The unity of proportions conveys an image of perfection and allows us to perceive the asymmetry of the parts as harmonious. This is evidently connected to a symmetry that integrates the asymmetry: within the golden ratio, symmetry is no longer found in a formal realization (equality of the parts), but rather in a relative one. The proportions of the individual elements are equal. In the golden ratio, the relationship is symmetrical. The symmetry of the parts has been subsumed in favor of the symmetry of the proportions.

The principle of symmetry and asymmetry in the golden ratio

Impressively, however, the symmetry of the parts reappears in the golden ratio upon closer inspection. This becomes easier to grasp visually when the three quantities—minor, major, and whole—are placed side by side ( Fig. 6 ). In this way, an axis of symmetry is also found in the geometry of the golden ratio. A further paradox arises: In the seemingly absolutely asymmetrical relationship of the golden ratio, symmetry is concealed twice: firstly, in the equality of the proportions (minor : major), and secondly, in the equality of the parts (symmetry), as expressed in Figure 6 .

The golden ratio thus impressively combines the principle of symmetry with that of asymmetry. We will repeatedly encounter this "unifying" property of this regularity, also known as the "divine proportion," in the following examples. It will become clear that the golden ratio is perceived as beautiful and harmonious because of its "character of uniting opposites."

If one now mathematically calculates the ratio of the minor to the major and the latter to the whole, the following number results, which is called the constant Phi:

Φ = 1.618033988749894848204586834365638117720309179805762862135...

It states, for example, that the major is 1.61... times larger than the minor, and conversely, that the whole is 1.61... times larger than the major. This endless number after the decimal point belongs to the group of irrational numbers. It is precisely such an irrational numerical ratio that is perceived as particularly harmonious and beautiful. This seems like another contradiction, since isn't it generally the irrational that disturbs people and that they try to eliminate with the help of their rationality?

The Parthenon in Athens

Since time immemorial, we find the golden ratio wherever people have sought to express beauty and where they have attempted to approach the divine ideal. This is generally the realm of art, and in particular its sacred sites, the temples.

The Parthenon in Athens is one of the most famous classical buildings. It is considered the most beautiful and perfect work of ancient Greek architecture. This renowned temple, which crowns the entire Acropolis, was built around 450 BC under Pericles. To this day, it is regarded as a prime example of classical symmetrical architecture.

The precisely symmetrical arrangement of the individual elements is reflected in every detail, from all perspectives. In addition to the symmetry, the proportions of the golden ratio are also incorporated in numerous ways and with astonishing accuracy. The stylized graphics ( Figs. 7 and 8 ) are intended to illustrate this only by way of example.

Figure 7 shows, in a vertical division, the relationship between the temple's substructure and superstructure. The superstructure extends from the pediment to the supporting columns, while the substructure comprises the load-bearing elements, namely the columns and steps. Both parts of the structure are arranged in an impressively precise proportion according to the golden ratio.

Figure 8 shows the ratio of the height (from the base of the staircase to the top of the gable) to the width (width of the structure) of the building. Height and width are related in the same way as minor to major. This two-dimensional representation of the golden ratio is also called a golden rectangle.

Besides the implementation of the golden ratio, numerous other asymmetrical details can be found in the Parthenon. Many stylistic elements of the building deliberately incorporate breaks in symmetry. For example, the columns are not straight but slightly curved inwards. Furthermore, upon closer inspection, they are not positioned at points corresponding to symmetry. Instead, they deviate significantly from these points, though this remains invisible to the untrained eye.

All these details ultimately demonstrate that the Parthenon is a symbiosis of symmetry and asymmetry. Just as symmetry and asymmetry are inherently linked in the golden ratio, these qualities of balance are also evident in every material realization of this principle.

The Parthenon is just one example among many of the application of the golden ratio in famous and large sacred buildings. The realization of human aesthetic perceptions in the form of the golden ratio can be found in numerous other well-known structures, such as the old St. Peter's Basilica in Rome or Cologne Cathedral ( Moessel 1926). Even in the Pyramids of Giza, the proportions of the number Phi are evident with astonishing accuracy ( Hagenmaier 1988). For example, the angle of inclination of the Great Pyramid of Giza ranges from a = 51°50' to a = 51°52'. The cosine of this angle is 0.618. The same value is found in the ratio of the length of the pyramid's side to half the length of its base (356 ÷ 220 cubits). The golden ratios are also found in the most famous of the great stone monuments, Stonehenge, which was built near Salisbury in England approximately 3500 years ago ( Doczi 1996). In art, the proportions of the golden ratio are reflected in the basic structure of numerous well-known paintings ( Doczi 1996), such as Leonardo da Vinci's "The Last Supper", Albrecht Dürer's "Self-Portrait" or Raphael's "The Sistine Madonna".

However, the golden proportions are not merely the product of conscious human creation, as the numerous examples mentioned above might suggest. They appear to be of a more primal nature.

Source:
These observations are an excerpt from an essay by Dr. Dr. Ruben Stelzner