Nerd-Science:

Mathematics Rocks the Universe

Published in

The Startup

9 min readMar 30, 2020

“Music is the pleasure the human mind experiences from counting without being aware that it is counting”- Gottfried Wilhelm von Leibniz

Before becoming a nerd obsessed with science, I was a metalhead teenager. I thought only rock, metal and cross over deserved to be called music and other genres were simply unworthy. Fortunately, now I’m more open-minded and my music taste is more complex than before. Anyway, during that period I discovered some of my favourite groups and among them, Tool captured my interest in a special way. This band is famous for its technical ability and its powerful and evocative lyrics. One song that exemplifies these rare skills is Lateralus, the title track of their third studio album. The reason for its success is the instrumental compartment that blends with astounding lyrics and visionary inspiration to produce a milestone able to trigger the imagination and inspire the listener. Tool placed spiritual and mystical references into a song designed on a specific mathematical concept, the Fibonacci Sequence. This unusual choice created a strongly evocative track which matches the language of the universe itself. The term Lateralus is a linguistic mix of Vastus lateralis muscle and lateral thinking. In fact, it proposes to use this ponder method to embrace the randomness and feel the rhythm of the universe. So, without further ado, let’s disobey Keenan by overthinking and overanalysing Lateralus.

As I already mentioned, the structure of Tool song is carefully planned to follow the Fibonacci Sequence, a series in which each number is the sum of the two preceding ones. The overall rhythm follows the sequence with keen precision, in particular, the time signature of the drum is 9/8, 8/8, 7/8. Also, the lyrics are based on the Fibonacci Sequence both in the content and in the syllable breakdown of each verse.

Fibonacci Numbers: 0; 1; 1; 2; 3; 5; 8; 13; 21; 34; 55; 89; 144; 233; 377; 610; … They are all obteined by sum up the preciding number with the following one. For example: 0+1=1; 1+1=2; 1+2=3; 2+3=5; 8+5=13; …
The syllabic analysis of the first verse reveals the following pattern:[1] black [2] then [2] white are [3] all I see [5] in my infancy [8] red and yellow then came to be

Now it’s time to go deep into this mathematical concept, its history and its implications. The sequence was introduced in Europe by Leonardo of Pisa (also known as Fibonacci), an Italian mathematician which lived from 1170 to 1250 circa. He spent part of his life in the Arabian city of Béjaïa in which he learned to use the Hindu-Arabic numerical system, the types of numbers that we currently use. This discovery convinces him to write a book to popularize them in Europe, where at the time the Roman Numbers were still in use. So, in 1202 he publishes Liber Abaci (literally Book of Abacus) in which he proposed a series of numerical problems and showed their resolution using the Arabic system. Many of the problems proposed were not invented by him, he simply put together all the knowledge acquired in Arabia. The Fibonacci Sequence appears in this book as the solution to a problem but, probably, the Italian mathematician was not aware of the potential of these numbers and their relationship with other mathematical concepts. The Sequence remained largely unnoticed until Johannes Kepler studied it further. He understood that the division between a Fibonacci number and the preceding one produce an approximation of ɸ, the irrational number called Golden Ratio. Two quantities are in Golden Ration when their fraction equals the sum of the two divided by the larger one.

(1+√5)/2=1,6180339887…=ɸ

Probably Ancient Greeks were the first to study ɸ, they called it Extreme and Mean Ratio due to its appearance in geometry. However, it is also possible that the Indian or Egyptian population discovered this number even before. In this case, the term discovery is not casual, humans used ɸ in geometry, music and architecture on multiple occasion, but its most prominent presence is in nature. The Golden Ratio can be found in anatomy, botany, zoology, biomechanics, climatology, chemistry, physics and even in astronomy.

One of the most common geometrical representation of ɸ is the Golden Spiral. An approximation of this figure can be constructed by drawing a series of squares with side lengths that are successive Fibonacci Numbers and then tracing circular arcs that join the opposite corners of them. The image produces in this way is a Fibonacci Spiral, the same figure that Tool cites several times in the lyrics of Lateralus.

“We’ll ride the spiral to the end and may just go where no one’s been”

Another property of the Golden Ratio is self-similarity, the property by which the whole object has the same shape of the parts that compose it. The first time that ɸ appears in a book was as the solution of a geometrical problem: the division of a segment so that the greater and the lesser part are in the same proportion as the entire segment to the greater part. This is a problem related to self-similarity present in Euclid’s Elements and reflects the fractal properties of the Golden Ratio. Nevertheless, ɸ is one of the simplest examples of fractals and it can be written as an infinite series of fractions (or square roots).

Lateralus reflects the fractal properties of the Extreme and Mean Ratio thanks to its construction following the Fibonacci Sequence. This particular design confers to the song the sensation to be able to continue indefinitely, similarly to the virtual images produced with fractal geometry that seems to be zoomed endlessly. References to the fractals can be found also in the philosophy of the track which is based on the realization to be part of the universe but not the entirety of it. Which means that humans cannot have total control but, because the universe is self-similar, everyone could feel its rhythms and be in harmony with their surroundings.

As I already mentioned the Golden Ratio appears extensively in nature from human heartbeat to quantum mechanics. However, why this number is so present? Probably the secret of its success is the relationship with fractals, Fibonacci Sequence and logarithmic spirals. All these properties confer to ɸ its charm and allow it to be present on countless occasions. Moreover, it is also related to symmetry, rhythm and geometry of the pentagon so it appears extensively also in music, art and architecture. Often, it was placed there on purpose, as in the case of Lateralus, but sometimes it appears randomly, only because the Golden proportions and rhythms are pleasant for humans. The Extreme and Mean Ratio appears to be hidden also in the trajectory of the flight of peregrine falcon (Falco peregrinus) pursuing its prey and in the human gait. Researches demonstrated that a trajectory that approximates the Golden Spiral is the best to allow peregrine falcon to maintain the eye contact of its prey. Other researches, instead, showed a proportion of ɸ between stance and swing phase of healthy humans, this was not observed in people affected by Parkinson disease. The reasons why so many organisms evolved structures or movement pathways that are related to the Golden Ratio are still unknown. Probably, this is due to biomechanical reasons and optimization of space. Indeed, all the things in the universe follow the laws of physics, it doesn’t matter if they are living or non-living. The Golden Ratio is a reminder that even the most complex life-forms evolved following the same rules as molecules and galaxies. This brings to my mind a quote from Full Metal Alchemist: “One is All, All is One” that strongly connects with the concept of embracing the universe present in Lateralus.

One of the best examples of Fibonacci Numbers in nature is phyllotaxis, a branch of botany that studies the arrangement of leaves around the stem. The majority of plants have leaves positioned in an alternative pattern, following a sort of spiral when observed from above. The rotational angle from one leaf to the next can be represented as a fraction of the full rotation around the stem. This ratio is often composed of Fibonacci Numbers both at numerator and denominator. Moreover, it’s also possible to represent the same angle in degree, in this case, we obtain 137.51°, also known as the Golden Angle.

Rotational angles in various plants: 1/3 in beech and hazel; 2/5 in oak and apricot; 3/8 in sunflowers, poplar, and pear; 5/13 in willow and almond.
Golden angle: 360° x (2-ɸ)=137.51°

Anyway, this rule does not apply to all plants, some show rotational angle fractions that are not composed of Fibonacci Numbers. Sometimes, there are multiples of Fibonacci Numerals while in other cases it’s possible to recognize Lucas Numbers. This Series is strongly related to the Fibonacci Sequence because it follows the same rules. Each number of the Lucas Series is the sum of the two preceding ones but with a different starting point from Fibonacci Sequence. Even the relationship with the Golden Ratio is equal: the ratio of a Lucas Number and the preceding one approaches ɸ.

Lucas series: 2; 1; 3; 4; 7; 11; 18; 29; 47; 76; 123; …

However, these observations trigger another question: why phyllotaxis is so tightly linked to the Golden Ratio? Also in this case, the answer is related to physics, mathematics, evolution and physiology of plants. Arranging leaves in Golden Angles is a good solution to maximize sunlight exposition and minimize hindrance. Similarly, the Golden proportions optimize space in flowers and strobila in order to harbour as many seeds as possible. Probably, these arrangements were positively selected during evolution and so, they became prominent patterns in the existing flora. The optimization of space is a property related to the irrationality of the Extreme and Mean Ratio. In fact, ɸ is defined as the most irrational number, it can be approximated very well neither by whole numbers nor by any fraction of them. I have to admit that a title like this fits especially well with a progressive-metal band like Tool. Moreover, the irrationality allows packing more objects in the same amount of space than it would be allowed by an arrangement based on fractions of whole numbers. However, there is more, other researches also related phyllotaxis to auxin hormone properties, repulsion of particles and distribution of the newly formed buds. The research in this field is not over yet so we expect to find new reasons for these arrangements.

Golden Ratio and Fibonacci Sequence are also linked to many misconceptions due to their peculiar charm and their high presence in the natural world. However, they are not the only numbers to show these features. Mathematics is the basic language to represents nature and ɸ is only a beautiful example of this concept. There are many erroneous identifications of the Extreme and Mean Ratio and Nautilus pompilius is related to one of them. Even though the shell of this animal describes a logarithmic spiral, its shape does not coincide with the Golden Spiral. It should be, also, noticed that it is neither an approximation of this geometrical figure nor its ratio range approaches ɸ. In other words, the relationship between Golden Ratio and nautilus shell simply doesn’t exist, this is only a misinterpretation. Anyway, the charm of Nautilus pompilius and its ability to control buoyancy using the chambered shell cannot be undermined by this.

The last example underlines the relationship between the Golden Ratio and myth. In fact, ɸ was able to fascinate not only scientists but also philosophers, artists and even occultists of all time. Unfortunately, this led to many myths related to magic, alchemy and pseudoscience. These last interpretations are perfectly in line with Lateralus which mixes mathematics with spiritual and even occultism references. The song also mentions the steps of the Magnus Opus to obtain the Philosopher Stone, a clear allusion to alchemy. These references complete the symbolism of the Fibonacci Sequence and Golden Ratio. Tool was able to represent these numbers in every aspect from science to mythology. This confers to Lateralus a charm able to remain in the mind of every listener and reveal a world in which the scientific method fuses with mysticism and spirituality.

References:

Literature:

Adam J.A., 2011, Mathematics in Nature: Modeling Patterns in the Natural World, Chapter 10
Iosa M., 2018, Il Numero Meraviglioso: la Sezione Aurea
Okabe T., 2015, Biophysical optimality of the golden angle in phyllotaxis, in Scientific Reports 5, 15358.
Tucker V.A., Tucker A.E., Akers K., Enderson J.H., 2000, Curved flight paths and sideways vision in peregrine falcons (Falco peregrinus), in Journal of Experimental Biology

Websites:

Credits:

Matteo Bondini for the beautiful drawing of Nautilus pompilius

Nerd-Science:

Mathematics Rocks the Universe

Written by Andrea Vissani