SPECIES

MARCH 2022

ISSUE 12

Math in Nature

Geometrical Shapes Symmetry Fibonacci Sequence Golden Ratio Fractals

WHY DO YOU NEED TO LEARN THE

Lists of evidence that math occurs in nature, particularly in animals and plants

“

to understand

THE universe you must understand the language

in which it's written,

the language of MATHEMATICS

- Galileo Galilei

CONt EnTS

le of b a t

04

08 10

06

04 Geometrical Shapes 06 Symmetry 08 Fibonacci Sequence 10 Golden Ratio 12 Fractals 14 Author's Insight 15 Author's Insight

12

e g ometric h e DEFINITION

Sh ap es

Geometric forms are described as figures that are enclosed by a boundary made up of a specific number of curves, points, and line segments. The practice of geometry had been around for thousands of years in many different civilizations.Every shape has its own name, such as circle, square, triangle, rectangle, and so on. In real life, we are surrounded by different fundamental geometric forms, such as the rectangular shape of the door, or the square size of the window or the circular shape of the pizza and how it forms into individual triangles when cut in a straight line. The word geometry came from the Greek word “geometron” which is the combination of the word “geo” (Earth) and “metron” (measurement). It was derived from a Greek word because the Greeks made the whole concept a more rigorous concept. The Greeks were the first to establish the notion of proofs in a systematic way. A proof is a set of arguments that begin with an initial premise and are intended to demonstrate that a certain statement is true.

Has it ever crossed your mind why most things in nature almost always look like they were aligned and perfectly designed? While sacred geometry isn't always visible with the naked eye, we may readily detect these patterns in animals if we look closely. Take for example the male peacock when attracting females for mating. Male peacocks use their geometric shapes to attract females. Generally, males who have more pleasing geometric shapes in their feathers attract more mates.The average mature peacock has 200 tail feathers, which are lost and regrow every year. Around 170 'eye' feathers and 30 'T' feathers are among the 200 or so feathers. Oscillations are a term used to describe the 'eyes.' The feathers range in length from a few centimetres to nearly 1.5 metres. One might think that geometry is only limited to the aesthetics of animals and plants, but it can also be used practically. Take for example the geometry of flight that eagles and other birds possess. Let’s follow the sequence of this bird taking flight.

When birds take flight, they flap their wings rapidly across a certain angular range, allowing them to acquire height quickly. The huge wings and geometry of an eagle allow it to glide through the air at extremely fast speeds. Our contemporary planes have fixed wings with a geometric design that causes the airflow over them to generate a reduced pressure under the wing, causing the wing to rise upwards. Researchers discovered that birds that are closely related evolutionarily have comparable wing structures, even if they have quite diverse flight patterns, by comparing geometry across species and clades (groups of animals that originated from a common ancestor). Albatrosses, penguins, and loons, for example, all belong to the group Aequornithes and have a wing form that is extremely similar, despite their appearances.

e MM t ry A balanced and proportionate likeness seen in two halves of an item is termed symmetry. One can easily observe if an object has symmetry if it can be divided into two identical pieces. One side of a symmetrical form is the mirror image of the other. The line of symmetry is the imaginary axis or line along which the figure may be folded to generate the symmetrical halves. Symmetry has always been a feature that attracted us because our brain has a sense of familiarity with symmetrical things. This is why it’s much easier to process compared to unsymmetrical things. In addition, it also gives us a sense of order, it’s much more aesthetically pleasing to look at and we almost always tend to correlate it with something positive.

There are two types of symmetry in nature. First is reflective symmetry or bilateral symmetry, this is what we generally think of when we think of a symmetrical object. A great example of reflective symmetry is when a clam gets attached to another clam in nature.

Of course, finding one of these can be deemed hard, however, if you take a closer look these individual clams also are examples of reflective symmetry. Another example is something that all of us had seen at some point in our lives, and those are butterflies. If you cut this beautiful insect in half, you’ll see that both sides have balanced proportions, even the patterns on both wings are perfectly identical. Any straight incision across the central point splits the organism into two parts that are mirror images of each other. This group includes certain floating organisms with radiating portions, as well as some tiny protozoa. All of the animals in this pattern are incredibly little.

And of course, animals are living creatures and they’re not an exception to coming into contact with physical matter (maybe they got hurt and it caused deformation in their symmetry). Some animals have unworldly healing capabilities that make them generate parts of their body. Take this crab for example, you can see that its left legs are much smaller compared to the right ones.

The other type of symmetry we commonly find in nature is rotational symmetry. It is when an object (an animal in this case) is rotated on its own axis, but its shape and form remains the same. A perfect example of rotational symmetry is a starfish. The starfish has 72 degrees of rotational symmetry. As a result, the starfish may be rotated 72,144,216,288 and 360 degrees without losing its appearance. The starfish's centre of rotation is its centre.

n a c o i b ci F

SEQUENCE

The Fibonacci sequence is a straightforward but comprehensive pattern. What makes this possible? The Fibonacci number is referred to as a series of numbers wherein each begins with the use of one number, or even zero, succeeded by one and continues in accordance with the rule that every number corresponds to the entire two numbers before it. To demonstrate the Fibonacci sequence, draw a line beginning in the right bottom corner of a golden rectangle within the first square and touch each succeeding numerous squares beyond the corners to form a Fibonacci spiral.

Clearly not. However, there are a few that are rather common and may be found all across the natural world. Consider the Fibonacci sequence, wherein it’s a series of numbers and a related ratio that mirror diverse patterns seen in nature. To be more exact, the Fibonacci number can be found in everything from the smallest to the largest entities in nature. Take, for instance, the whirl of seeds in a pinecone, the curvature of a snail shell, and the swirl of a storm. Is your mind blown? I'm the same way. Keep in mind that nature is all about math. Moreover, it is also known that the Fibonacci numbers are of significance to biologists and physicists since they may be discovered in a wide range of natural objects and occurrences. To be more specific on that thought, the Fibonacci sequence is fundamental in phyllotaxis, wherein it is the analysis of the arrangement of leaves, branches of trees, seeds, or petals in plants with the mission of emphasizing its presence of recurrent patterns.

s u l S i hells t u a N Nautilus shells are amongst the most famous representations of the Fibonacci numbers, wherein they exhibit a proportionate rise of 1.61. To explain in simpler terms why this is an example of the Fibonacci sequence when nautilus shells are cracked open, they reveal a logarithmic spiral made up of enclosed parts known as camerae. With that scenario, every succeeding chamber is the same area as the two cameras that came before it, resulting in the logarithmic spiral. This symmetrical growth takes place since the nautilus evolves at a steady rate throughout its life until it reaches its ultimate size.

spiders

Did you know that spiders inadvertently utilize mathematics to exist in the form of webs? According to the use of math, their webs are susceptible to considerable elasticity and strength in construction. Moreover, the angles are used to produce the characteristic patterns of spider webs. Connecting the spider webs to Fibonacci, the Fibonacci sequence permits the web to be both enhanced and compacted. Also, the Fibonacci sequence is used by nature to support the development and the size of the possible web.

flowers

Every flower is also an example of the Fibonacci sequence. In what way exactly? Seeds form in the focal point of the flower's head and move out. Sunflowers are an excellent example because of their spiraling arrangements. Mostly, each seed head becomes so tightly populated to the point that their numbers might soar to 144 or even higher. In this scenario, the moment these spirals are examined, the number is usually invariably Fibonacci.

e n o r a ti o d l G According to a foreign article, the term "golden ratio" refers to a one-of-a-kind mathematical relationship. For example, if the proportion of the sum of the numbers (a+b) divided by the larger number (a) equals the proportion of the larger number divided by the smaller number (a/b), two numbers are in the golden ratio, which is about 1.618. This is represented by the Greek letter phi. The golden ratio, in essence, matches the concept of the Fibonacci sequence. It is a number pattern formed by combining the current amount in the series with the preceding one, which results in the next number in the pattern. The golden ratio is demonstrated visually as a spiral, often known as the Fibonacci Spiral. It is a logarithmic spiral that increases by the golden ratio component. In particular, it starts with a group of cubes, each proportional to eight successive numbers in the series and covered in the manner shown. Then, from one edge of every cube to the opposite edge, a cyclical curve can be constructed, exposing a spiral and ideally proportional quadrilaterals.

As we all know, the universe is wildly unpredictable. It is, nevertheless, a highly ordered physical environment regulated by mathematical laws. Moreover, the golden ratio is one of the most important representations of these principles. Due to its currency in nature, the golden ratio is also called the "divine proportion." It can be obtained from plants, fruits, seashells, and perhaps even storm cloud patterns. In a more specific example, the quantity of petals on a flower is frequently a Fibonacci number. Also, sunflower and pinecone seeds curl into contrasting spirals of Fibonacci numbers. Even the sides of an unpeeled banana are typically a Fibonacci number, and the number of ridges on a peeled banana is typically a greater Fibonacci number. Nonetheless, the golden ratio may be discovered in geometry as well as in many other areas of our existence, such as biology, architecture, and crafts.

The quantity of the flower's petals follows the Fibonacci sequence every time. The lily, which has three petals, buttercups, which have five petals, and the daisy, which has thirty-four petals, are its prominent examples. Specifically, phi arises in petals as a result of Darwinian mechanisms selecting the optimal packing arrangement; per petal is positioned at 0.618034 per rotation (from a 360° circle), providing for the highest potential sunlight exposure and other elements.

When it comes to honeybees, they also emulate the Fibonacci sequence in a variety of unique ways. By splitting the number of females in a colony by the proportion of men, the most profound example is discovered. Take note that females often outpace males. The result is almost always somewhat nearer to 1.618. Furthermore, the honeybee family tree maintains a well-known arrangement. Males have a single parent, while females have two. As a result, men have, in particular, 2, 3, 5, and 8 grandparents, great-grandparents, great-great-grandparents, and great-great-grandparents, correspondingly, in their family lineage. Females have 2, 3, 5, 8, 13, and so on, continuing the very same sequence. And, as previously stated, bee physiology fits the Golden Curve quite well.

How frequently does each of us search for the patterns of the trees, considering that we are somehow familiar with them in our everyday lives? Of course, most of us haven't really tried searching, haven't we? Nevertheless, the Fibonacci number starts with the emergence of the trunk and next are the spirals directed outwardly as the tree enlarges in size and height as well. Additionally, we also observe the golden ratio in each of their branches, wherein as they started with one trunk that then separates into two parts, then one of the current tree branches sprouts into two again, and so on and so forth.

A fractal is a mathematical set that has an infinitely repeating pattern. The word is most commonly used to describe patterns and other forms of mathematical art. Basically, you start with a Form with Pattern A and repeat Pattern A off the shape, increasing the pattern's overall complexity while simultaneously increasing the number of repeats with each iteration. To make it easier for you to understand, take this for an example: Assume you're on a beach. Find a stick and make a large triangle in the sand with it. Draw another triangle, one-third the size, on each side. Do the same thing with each of the smaller triangles. Simply keep drawing triangles until your stick no longer works. That's when we relinquish control to computers. Even if things are too little for us to see, they have a knack for figuring out what they look like. Do you want to see what it's like below? Then just use your mouse wheel to scroll around and have a look. You'll notice that no matter how far out or near you look, everything appears the same all the way down.

You might think to yourself, “what’s the purpose of fractals in nature?” Well, if you take a close look, you’ll see that fractals in nature demonstrate how "basic" nature's rules are. Fractals are formed by repeatedly repeating the same process, which is how nature prefers to form. Though we can’t deny it, fractals are indeed very satisfying to look at, but their purpose doesn’t stop there. Fractals are actually found naturally, brain cells and broccoli, that's quite remarkable in themselves.

The animal respiratory system is an example of a Fractal that starts with a single trunk (similar to a tree) that branches out and grows into a much finer grained network of cavities. You can see the trachea branching out into the primary left and right bronchus and then to the secondary bronchi, to the tertiary bronchi to the bronchioles and lastly to the alveoli, where the body would distribute the oxygen to different organs via the capillaries. Another example would be the famous Romanesco broccoli. Natural fractals are formed by the spirals of florets on a Romanesco broccoli. A fractal, to put it as simply as possible, is a pattern that seems almost the same at all scales. So, if you zoomed in more and closer on a head of Romanesco broccoli with your camera, each frame would seem just like the one before it, despite the fact that you were focussing on progressively little pieces. Plants in the natural world can occasionally resemble these self-repeating patterns. For example, many ferns have leaves that display fractal architecture. Spiral nautilus shells, pinecones, and ice crystals all feature self-repeating patterns in connection to this.

“Why do you need to learn the connection of mathematics with our nature?”

n s i i g s h ' r t o h t Au

We need to learn the connection of Mathematics to nature because it helps us interpret nature so well. There’s so much we don’t know about nature and mathematics helps us understand the world we revolve in. However, don’t be mistaken, because mathematics wasn’t made to represent nature. Ultimately, it's just a language we made up to look for and explain patterns in nature. Nature has always existed; math just provides us with the skills to discuss it and make predictions based on it. Now to answer the question in a more elaborated matter, let’s first talk about the importance of it in more superficial reasoning.

First, we all know that students usually have a tougher time with mathematics compared to other subjects. The most common answer to this is because students always had a hard time correlating it with their surroundings, they have a hard time using other mathematical concepts that aren’t addition, substruction, division, etc., in real-life scenarios so they have the tendency to not develop any sort of mutual understanding towards the subject. But by simply correlating Mathematics in nature, students can begin to appreciate the subject because they can finally understand where and how those patterns and designs came to be. Second, ordinary people are generally more emotionally linked to images of trees, grass, and animals because they are immersed in the real world and have direct contact with them. As a result, when the physical world's mathematics is given as symbols, the ordinary individual is completely lost. However, when mathematics is given aesthetically in nature, the typical individual is more connected and hence may more easily identify beauty. Take for example the geometry of flight. We all know that birds fly, and we know that the shape of their wings plays an important role in that, but if weren’t for math we would never have known the concept of “wing loading” (ratio of weight to wing area that determines how fast a bird or plane must fly to retain lift: wing loading = weight/wing area (kilograms per square meter). Well, is understanding mathematics in nature just a way for us to better appreciate math in the real world? Not really. Because math also helps us appreciate nature as well. Just like what Dr. Thomas Britz said, “Maths is not only seen as beautiful—beauty is also mathematical.”

h g i t s n i s ' r o h t u A

It is important that we, as individuals, understand the relationship between mathematics and nature. Why is this so? Because mathematics is a science of structure and order. Numbers, shapes, algorithms, chance, and change are its realms. Mathematics, being a discipline of abstract objects, focuses on logical reasoning rather than observation as its standard of evidence, but it still uses simulation, observation, and experimentation to uncover the truth.

Mathematics exposes unseen patterns that enable us to comprehend our surroundings. However, we must first acknowledge that mathematics does not exist in nature; rather, mathematics was invented by people to represent entities and phenomena that we experience in our surroundings. To elaborate, consider the relationship between nature and mathematics; nature is that portion of experience that contains uniformity and control. In contrast, mathematics is an accurate language for describing order. As a result, we can all claim, as Galileo did, that math is the language of nature. We can identify the melodious language of mathematics simply by looking at our surroundings. For example, when we look closely at the structures and interactions in nature, we can see beauty and perfection in various geometric patterns. The petals of flowers show perfect harmony and symmetry, as do others in nature, such as the shell of a snail and even our galaxy. The aforementioned symmetry is known as the divine proportion or golden ratio. It appears as a recurrent number in certain natural phenomena. Moreover, this quantity in nature can be seen in the most unforeseen scenes, such as the multiplication of certain particular plants or the snail's spiral. Nevertheless, every spiral pattern is abundant in nature. To add information regarding snail shells, they can be seen in the fog's turbulence from the chimney or in the arrangement of galaxies' matter, and for all we know, this occurrence is self-sufficient of the material engaged in the process. Overall, there is, indeed, a correlation between mathematics and nature. The potency of mathematics lies in its abstract notions, which are not visible to the naked eye but are used to categorize real-world entities.

REFERENCES:

Albright, R. (n.d.). Spider Web. Prezi.Com. https://prezi.com/eb1endffzrjo/spider-web/ Contributor, T. (2016, October 1). Fibonacci sequence. WhatIs.Com. https://whatis.techtarget.com/definition/Fibonacci-sequence How Many Times Have You Spotted Fibonacci In Nature? Here Are 7 Examples For You. . . (2022, January 17). The Stemettes Zine. https://stemettes.org/zine/articles/fibonacci-in-nature/ Mathnasium of Pflugerville. (2013, February 20). Mathnasium. https://www.mathnasium.com/examples-of-the-golden-ratio-in-nature National Geographic Society. (2012, November 16). The Golden Ratio. https://www.nationalgeographic.org/media/golden-ratio/ CK-12 Foundation. (2019, November 20). Rotation Symmetry. https://www.ck12.org/geometry/Rotation-Symmetry/lesson/Rotation-SymmetryGEOM/ Roos, D. (2021, June 16). Why Do We Get So Much Pleasure From Symmetry? HowStuffWorks. https://science.howstuffworks.com/why-do-get-somuch-pleasure-from-symmetry.htm#:%7E:text=The%20simple%20explanation%20for%20our,Nature%20is%20full%20of%20symmetry Norman, G. (n.d.). Understanding the World Through Math. Asia Society. https://asiasociety.org/education/understanding-world-through-math J. (2020, September 24). What is a Fractal? - The Ultimate Guide to Understanding Fractals. Iternal Technologies. https://iternal.us/what-is-afractal/ Science Learning Hub. (2011, September 11). How birds fly. https://www.sciencelearn.org.nz/resources/303-how-birds-fly Morales, O. M. (2021, February 10). Math Behind the Beauty of Romanesco Broccoli Explained. Science Times. https://www.sciencetimes.com/articles/29586/20210210/math-behind-beauty-romanesco-broccoli-explained.htm Study Pug. (n.d.). How Geometry Has Shaped the World Around Us. StudyPug. https://www.studypug.com/blog/how-geometry-has-shaped-theworld-around-us/