The order and balance found in nature is truly mesmerising. Although studying biology daily and coming across all the phenomena almost every day, I still find myself astonished by the ideas creating the world around us. The complexity of many systems seems to be detached from the natural tendency toward increasing entropy¹, but that only emphasises their wonder.

The brainteaser is whether the solutions observed in nature nowadays come from a simple trial-and-error method (so are contingent²) or follow a known pattern that has already appeared at some point in history (thus are determined³)? Most scientist today agree that both of those forces counteract, driving evolution to the optimum ensuring the survival of the species (e.g. Blount et al., 2018; Wong et al.,2020).

Under such circumstances, some schemes repeating in different evolutionary pathways and organisms become even more fascinating. Especially, regarding the fact that they can also be described by mathematical equations, formulated relatively recently. As Galileo Galilei explained in Il Saggiatore (1623):

[The cosmos / laws of nature] is continually open before our eyes, but it cannot be comprehended until we understand the characters in which it is written, and that is in the language of mathematics. Its characters are triangles, circles and other geometrical figures, without which it is humanly impossible to understand a single word. Without these, we are wandering in vain through a dark labyrinth.

One of the most popular numbers found in many biological patterns is 1.618, also known as The Golden Ratio. The approximate proportion represented by the Greek letter phi (Φ), is a result of the ratio of two segments of a line, divided in a way shown in the Figure 1.

a / b = (a + b) / a = Φ

Fig. 1. The Golden Ratio presented in a graphic from as well as the equation.

It was first introduced by Euclid in Elements around 300 BC, then appeared occasionally in the works of mathematicians throughout the following millennium and was finally calculated explicitly by Michael Maestlin in 1597.

The ratio applies to other mathematical formula too – the Fibonacci sequence. It is a never-ending series of numbers, starting from 0 and 1, and continuing by the rule of addition every two previous numbers:

0 + 1 = 1;

1 + 1 = 2;

1 + 2 = 3;

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, …

By dividing any consecutive numbers, you receive an irrational number that is surprisingly close to the 1.618 of the golden ratios, e.g.:

21 / 13 = 1.615

34 / 21 = 1.619

Those simple formulations turn out to exist not only on paper. In fact, they were only observed by a few very perceptive minds, as they had formed the world around us since the very beginning of life – long before humans explored science, or even before humans themselves existed.

From the nucleus

The golden ratio defines the properties of B-type DNA. Its helical structure makes a complete turn approximately every 10.5 base pairs, measuring 34 ångströms (3.4 nm) in length. The diameter of the helix is about 21 ångströms (2.1 nm), together yielding a ‘wonderful’ proportion close to 1.618 (Fig. 2).

Fig. 2. The properties of the B-type DNA molecule, the original graphics designed by RWhitwam, with own adjustments; under Deed – Attribution-ShareAlike 4.0 International – Creative Commons.

Spiralling arrangements

The growth pattern of plants is strongly conditioned by the Fibonacci sequence (Fig. 3). There is no room for spontaneity – each element spirals according to the ‘golden’ guidelines. Once the flower is pollinated, the rule applies to its seeds too (Fig. 4).

Fig. 3. Just a few examples of spiraling growth paterns observed in botany.

1. A pink rose (Rosa sp.) photographed by Carla Nunziata; under Deed – Attribution-ShareAlike 3.0 Unported – Creative Commons;
2. Echeveria elegans photographed by Diego Delso; under Deed – Attribution-ShareAlike 4.0 International – Creative Commons;
3. Romanesco (Brassica oleracea) photographed by RonPorter; under Deed – CC0 1.0 Universal – Creative Commons.

Fig. 4. Spirals of sunflower seeds, a cap of an acorn, and a pine cone.

1. A sunflower (Helianthus sp.) seeds photographed by Anna Benczur; under Deed – Attribution-ShareAlike 4.0 International – Creative Commons;
2. A cap of an oak’s acorn (Quercus rubrum) photographed by USGS Bee Inventory and Monitoring Lab from Beltsville, Maryland, USA; under Deed – Attribution 2.0 Generic – Creative Commons;
3. A pine (Pinus sp.) cone photographed by böhringer friedrich; under Deed – Attribution 3.0 Austria – Creative Commons.

Swirling traps

The fractal compositions of botanical world encouraged people to broaden their perspective and look for the ‘golden’ patterns everywhere. As a result, many spirals were blindly attributed to the Fibonacci sequence, despite having little or nothing in common with it. The most prominent example is the Nautilus shell (Fig. 5). After a detailed examination of the shells’ profiles, researchers found that their growth was not dependent on the 1.618 ratio (Bartlett et al., 2019).

Fig. 5. The sagittal section of the shell of the cephalopod genus Nautilus, photographed by Sérgio Valle Duarte; under Deed – Attribution 3.0 Unported – Creative Commons.

The natural human tendency to discern patterns is a blessing as well as a curse. Although it may give rise to even revolutionary theorems, it can also completely mislead the assessments and cause fallacy.

It is believed that the architecture of the Great Pyramids and the Parthenon follow the Golden Proportion too (Fig. 6), although being built centuries before the term was introduced. While some of their ratios do, in fact, correspond the Φ, many studies have shown that these assumptions are exaggerated, with inaccurate measurements distorting the truth. Despite scientific evidence, the myth continues to be perpetuated.

Fig. 6. Singular proportions found in architecture, such as the Great Pyramids or the Parthenon, unintentionally follow the Golden Ratio rules. Graphics designed by Hasanisawi; under public domain.

At the turn of the 15th and 16th centuries, the ratio was proposed as a compositional standard in the arts by Luca Pacioli in his book De Divina Proportione. It is now found in many famous paintings, drawn on top of the Girl with a Pearl Earring or the Mona Lisa (Fig. 7). However, it should be noted that there is no evidence from artists themselves supporting the use of the golden ratio in the creation of any of the paintings mentioned below. The inclination to proportion and balance seems to be deeply rooted in humans’ perception, making them see it where it has never even been. As landscapes and portraits certainly benefit from harmony, such a perspective is far from realistic, existing on canvas solely for the purpose of visual attraction.

Science is based on observations but cannot rely on presumptions – it needs examination. Therefore, it is still a matter of discussion, to what extend is the world around us ruled by randomness. However, whether contingent or determined, it surely is remarkable and astounding.

Fig. 7. Windmills at Wijk bij by Jacob van Ruisdael, the Girl With a Pearl Earring by Johannes Vermeer and the Mona Lisa by Leonardo da Vinci with the Fibonacci spiral. Although composiotions of paintings seem to follow the Golden Proportions, they were never used to create those arts. The affinity with the Fibonacci spiral may merely suggest that the painters were visionaries of their time, highlighting the remarkable quality of their paintings; all under public domain, Creative Commons.

1. Entropy is a term derived from thermodynamics, stating that all isolated systems drive to chaos and disorder. ↩︎
2. Processes known as the motors of evolution, such as genetic drift or mutations, are regarded as stochastic, random, and therefore unpredictable. ↩︎
3. That notion is often explained by convergent evolution. The most common example would be the evolution of wings in many drastically different organisms (such as birds, mammals and insects), as a result of adaptation to the surroundings. ↩︎

References

• Blount, Z. D., Lenski, R. E., & Losos, J. B. (2018). Contingency and determinism in evolution: Replaying life’s tape. Science, 362(6415), eaam5979.
• Wong, T. W. (2020). Sources of evolutionary contingency: chance variation and genetic drift. Biology & Philosophy, 35(4), 36.
• Galileo, G. (1990). Il saggiatore [The assayer]. Discoveries and opinions of Galileo. New York: Anchor Books. (Originally published in 1623.).
• Shea, W. R., & Davie, M. (Eds.). (2012). Selected Writings. Oxford University Press.
• Bartlett, C. (2019). Nautilus spirals and the meta-golden ratio chi. Nexus Network Journal, 21(3), 641-656.
• Livio, M. (2008). The golden ratio: The story of phi, the world’s most astonishing number. Crown.
• Naini, F. B. (2024). The golden ratio—dispelling the myth. Maxillofacial Plastic and Reconstructive Surgery, 46(1), 2.