Mathematics and its Relation With Nature: Fibonacci, Turing Patterns and Chaos Theory
By: Shamsheer Pal Singh
Tags: Educational, Mathematics, Research, Facts, Nature

Mathematics is the subject which wasn’t discovered but was invented. This broad field is just a creation of the human mind. It acts just like a missing link between the physical world and the theoretical world. Physical world is what we see, observe and feel. That means, we can understand its technicalities just by feeling it or solve its problems by practical means. Whereas the theoretical world is the world of Science and its logic. These stories justify the existent or non-existent things. But, we all realize, those were not enough to feed our hungry minds. To prove them, we needed something more and that was mathematics. This subject comes up with its applications in various fields like Physics, Chemistry, Biology, Commerce, Management, Mental Reasoning etc. One field that always gets neglected is ‘nature’. Mathematics has always been around us; it has always been the secret behind the beauty of various creations of God. Words aren’t the perfect medium to describe the potential of those great minds who discovered sequences and patterns in nature and gave us a new way of understanding called ‘Mathematics’. Well, I am not going to bore you with elementary school facts about discovering various shapes in nature, what we are going see is the application of the subject in true sense.

Fibonacci Sequence

This sequence was found by an Italian Mathematician Leonardo Pisano, called Fibonacci while calculating the growth of the rabbit population. He came up with such a unique and important sequence that literally defined everything about nature and its processes. The sequence followed one simple rule:

(where Fn is nth term of the series)

We can obviously start plotting the values by putting the values of n as 0, 1, 2 up to infinity. The special thing about this series is that it is capable to generate all those numbers that govern the laws of nature. In easy words, there exists no such group in nature which has number of elements other than the ones generated by the Fibonacci sequence. For instance, number of seeds in sunflower follow the same spiral pattern as generated by the Fibonacci series for different values of n. Not only sunflowers but many other plants follow the Fibonacci sequence during their growth. While calculating the growth of the rabbit population, Pisano found that the growth also takes place in the Fibonacci sequence of 0, 1, 1, 2, 3, 5, 8, 13…… Isn’t this amazing how nature multiplies itself in same pattern and that pattern is governed by laws of Mathematics? Moreover, the theory to prove these laws is so strong that we could confidently generalize the whole concept for every living being on the earth.

Fun Fact : Golden Ratio is another theory in mathematics that makes its connection well with nature. It is the ratio of two consecutive Fibonacci numbers and has an approximate value of 1.618. It is known as the ratio of beauty as it is believed as all beautiful things in nature have some of their parts present in this ratio, a perfect example of which is human face.

Flowers and branches: Some plants express the Fibonacci sequence in their growth points, the places where tree branches form or split. One trunk grows until it produces a branch, resulting in two growth points. The main trunk then produces another branch, resulting in three growth points. Then the trunk and the first branch produce two more growth points, bringing the total to five. This pattern continues, following the Fibonacci numbers. — Robert Lamb, How Stuff Works

Fibonacci Spiral

Fibonacci Spiral in Hurricane | Source: Xah Math

Symmetry

A symmetric structure is one which can be divided in two similar equal in proportionate halves. Keeping this definition in mind, we can derive many results and conclude that we are surrounded by symmetrical objects.

There are two main types of symmetry:

Reflective Symmetry: Where one half of the object reflects the other half i.e. it is mirror image of the other half. Butterfly is the best natural example for this type.

Rotational Symmetry: Where object can be rotated about its centre to get number of positions that match the initial position. In the field of mathematics, circle is a common geometric shape that possess such symmetry. Many species of flowers can be categorized into this kind.

The human body is full of symmetrical objects. Our ears, eyes, nose, lips which comprise the face are great examples of reflective symmetry. They all give rise to a symmetrical human face. Psychologically, humans are attracted to symmetrical faces. Yes, the definition of beauty lies solely in yet another concept. Symmetric things are usually more attractive. Biologists believe that people with symmetry are generally more fit. Many microorganisms in Protozoa kingdom also possess a wide range of symmetry.

Symmetric Pattern in Onion

Fractals

Mathematically, Fractals are subsets of Euclidean figure having statistical figures same as the main figure. In layman terms, they can be explained as patterns that exist inside a solid geometrical figure or are a part of it and have similar characteristics. These patterns re-occur at smaller scales. Generally, we can notice fractals in nature when we go deep into the structure and formation of an object. Structure of snowflakes is such an example. The best and natural example of fractals in nature are trees. It’s amazing that these divine creations can not only be explained based on Biology but also Mathematics. Trees originate from roots, these roots (especially fibrous roots) are the first example of the fractal as they form a structure that has smaller parts like the bigger ones, that grow into more such smaller parts. Trees have branches that are another example of fractals as they replicate themselves into similar network-like structures. The leaves on these roots contain veins (the thin small lines on the leaf) that originate from midrib and form a network of veins replicating the parent vein and resulting in numerous such structures. Yet another example of fractals. Walking more into the lap of the mother nature, we can find rivers. These rivers form deltas which are self-branched patterns that resemble fractals too.

Fractal dimension is the ratio providing the statistical index for the complexity of pattern. It compares how the detail in a pattern is changing with respect to the scale it is being measured on.

One characteristic feature of fractals is that the Fractal dimension of the subsets (branches of parent figure) strictly increases the fractal dimension of main figure.

Repeating Identical (to parent leaf) Patterns in Leaves

Ideal Structure of Snowflake | Source: Cosmos Magazine

Pattern Formation

This is an application where mathematics combines with biology. This relation was explained by ‘Father of Modern Computer Science’, Alan Turing. He introduced a term known as Mathematical Models. While this is an extremely deep topic, this article is going to focus on the theory and how mathematics contributes to it. Pattern formation (also known as morphogenesis) is a self-occurring phenomenon in which two stabilizing processes give rise to instabilities that produce spatial patterns. These patterns are a part of nature. We see them everywhere. In animals as stripes, dots, in natural elements like rocks, wood and leaves as different colours. These patterns can differ within the same species. In 1952, Turing expressed mathematical models that defined this formation in a set of coupled reaction-diffusion equations which describe the ways in which cells differentiate in a concentration-dependent manner in response to a chemical pre-pattern. In short, those models gave us an insight on behaviour these patterns by defining some equations which tell us how the cell is going to differentiate when it undergoes concentration dependent chemical processes. Afterall, cell differentiation in response to some chemicals is the main cause of pattern formation in nature. So, it’s evident that every design or art of nature we see is coded in mathematics which I believe is a mind-blowing fact.

Turing’s work on pattern formation has triggered new research programs in both mathematics and biology. In the mathematical domain, the focus fell on exploring the rich variety of behaviours of the system of nonlinear parabolic equations, while theoretical and experimental work in biology targeted the discovery and detailed analysis of the structure and function of morphogens. — Maria Serban, University of Copenhagen

Wave Pattern in Desert

Turing Models Depicting Pattern Formation

Chaos Theory

As the name suggests, this phenomenon accounts for the natural caused my elements of nature. These chaotic movements can range from atomic levels to trajectory of a group of asteroids approaching Earth. While this chaos happens to be completely different from the one, we get to see on a Monday morning, mathematics helps us understand more about this theory as it simply converts the particle behaviour to numbers and equations. With the help of these equations, we can easily predict the future of chaos or find out what happened before the observation period was started. There is a great significance of mathematics in this field since its discovery, it has been applied in numerous fields like weather detection, determining behaviour of atomic constituent molecules, prediction of asteroids and even defining various processes in the Universe.

Now, having the required intel, one must remember that a chaos here is defined as apparent random states of disorder of dynamical systems. These states importantly depend on one factor known as Initial conditions. Mathematically, these conditions are nothing, but the parameters required to define the motion of certain particles. It suggests and proves that with the help of one simple differential equation of three variables — x, y, z (denoting the 3 planes of motion) with respect to time (t) can be enough to understand the chaos theory. One common example of this is that it is used by the meteorological department to analyse the chaotic wind pattern based on its initial conditions and hence predict the weather. This clearly tells us how maths is helpful in defining nature, proving its laws and has the power to even predict the future of some processes that we say are governed by the forces of nature. It’s funny how big phenomena can be converted to some variables and numbers which affects our daily life almost completely.

Fun fact: This theory lays down the foundation of the famous ‘Butterfly Effect’.

In fact, the mere act of opening the box will determine the state of the cat, although in this case there were three determinate states the cat could be in: these being Alive, Dead, and Bloody Furious. — Terry Pratchett, Lords and Ladies

Chaos Patterns in a 2-D Plane

There is no doubt that we, our life and nature are science, also there is no doubt that Mathematics is the language made to understand Science. It truly surrounds us and helps us derive a meaning out of a strange natural phenomenon. It’s really astounding how these strange things can be converted into numbers and these numbers can successfully predict the fate of that natural element. Next time you get bored in a math class, try going deeper into the concept afterwards. There is surely a mind-blowing simple application hidden behind that concept that directly or indirectly affects your life.