Beauty of Numbers & Patterns — A World Shaped By Math
“Mathematics, rightly viewed, possesses not only truth, but also the supreme beauty - a beauty cold and austere”
Mathematics is a tool. Play with it any way you want and see if you can make something. Don’t worry if you break the tool, we’ll rebuild it, together.
Today, we’ll be talking about the essence of mathematics and how it shapes the world around us. The intention behind this post is to show the beauty of math to people, how it governs nature without most of us even noticing it.
Well, we want to talk about so much but we have to keep it compact, to give a brief idea over a lot of things. Hope you enjoy the read.
One of the things about Mathematics that we love the most is it’s uncanny ability to reveal hidden beautiful patterns in our everyday life, the nature around us. These patterns can be sequential, spatial, temporal, and even linguistic. There are connections between things that don’t seem connected, but can be observed with the intellect of math. One beautiful example is — fireflies flashing in unison and a pattern that can be solved mathematically. You may ask, what’s the point of that practically? Well, engineers at Cornell Institute used the above work to make little electronic clocks get in sync, which is a very important aspect in the distributive computing and internet itself.
In short, we can say mathematics is the science of patterns.
Talking more about patterns, lets have a glimpse of “Chaos Theory” (we’ll be going into deep in later post), which is a hot topic among many mathematicians. ‘Chaos’ is an interdisciplinary theory stating that within the apparent randomness of complex systems, there are underlying patterns, constant feedback loops, repetition, self-similarity, fractals and self-organisation.
One of the simplest example to understand is the ‘Butterfly Effect’ that describes how a small change in one state of a deterministic nonlinear system can result in large differences in a later state, e.g. a butterfly flapping its wings in Brazil can cause a tornado in Texas. Such kind of phenomena are often described by fractal mathematics. Chaos theory and chaotic models have applications in many areas including geology, economics, biology, meteorology etc, and can help demystify the huge dynamic complex systems.
What is a fractal? — A fractal is a never ending pattern. They are the images of dynamic systems — the pictures of ‘Chaos’. Geometrically, they exist in between our familiar dimensions, nature is full of fractals, for instance: trees, mountains, seashells, clouds, ferns, even human body ! These things look very complex and non-mathematical. Now, think what it took to produce what you see. You’ll realise it takes endless repetition and that gives rise to one of the defining characteristics of a fractal, a self similarity. Fractals have vast applications in astronomy, fluid mechanics, image compression etc as they hold the key to describe the real world better than traditional science.
Let’s have a closer look at some of the real world fractal examples around us.
- Fern — As you look deeper and deeper, you see a never ending repetitive pattern.
- Koch Snowflake — A beautiful example of a fractal with infinite perimeter but finite area. The idea is, make an equilateral triangle. Now make another equilateral triangle above the previous one, but in opposite direction. You’ll see small equilateral triangles on the boundary. Keep doing the same for them, and keep doing, keep doing…
When you keep doing it, soon after some depth, you’ll start seeing the resemblance of pattern with a snowflake.
Fractal Antenna — Above example of ‘koch snowflake’ shows a fractal of perimeter increasing infinitely while it’s area can be bounded. Using such property, fractal antenna was invented, using a self-similar deign to maximise the length of material that can receive much weaker signals and transmit signals over long distance without compromising the area and volume taken by the antenna due to it’s fractal nature. This is very compact and have useful applications in cellular telephone and microwave communications.
So, you see, there are examples around us shaped by mathematics with hidden patterns, without us even knowing about it. That’s the beauty of math.
Now let’s have a look at one of the famous mathematical number sequence, the ‘Fibonacci Sequence’. The Fibonacci sequence is a recursive sequence, generated by adding the two previous numbers, the first two numbers of the sequence being 0 and 1.
So, Fibonacci sequence is 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89 …
An interesting fact is that the number of petals on a flower always turns out to be a fibonacci number. Statistically, this sequence appears a lot in botany. Another example is if you look at the bottom of pine cone, and count clockwise and anti-clockwise number of spirals, they turn out to be adjacent fibonacci numbers (see image below).
Let’s have a look at a property of fibonacci numbers. I’m gonna write continuous sums of squared fibonacci numbers.
Squared Fibonacci Sequence: 0, 1, 1, 4, 9, 25, 64, …
Continuous sums:
0 = 0 x 1
0 + 1 = 1 x 1
0 + 1 + 1 = 1 x 2
0 + 1 + 1 + 4 = 2 x 3
0 + 1 + 1 + 4 + 9 = 3 x 5
0 + 1 + 1 + 4 + 9 + 25 = 5 x 8 … and so on. (You see every time product of the sum is two consecutive fibonacci numbers)
Well, there’s a mathematical explanation for the pattern we see above. Suppose you have squares of sides representing fibonacci numbers, and assemble them in the way shown below. The above pattern is nothing but area of the rectangle formed by joining the squares (continued fibonacci squares sum).
Fibonacci spiral recurs throughout the nature — in the seed heads of sunflower, the petals of a rose, the eye of the hurricane, the curve of a wave, even the spiral of galaxies!
It seems that when we keep comparing ratios of two consecutive fibonacci numbers, as we move further in the sequence, the ratio approaches a value of 1.618034… which is called Φ or better, the golden ratio. This ratio has a beauty of special kind and is important to us. Why? — The golden ratio appears everywhere — DNA, human body, eye of hurricane etc — it appears in various structures of nature. This occurrence of Φ in various aspects of nature, gives rise to the question that ‘Was our universe intelligently designed, or is it just a cosmic coincidence?’
Now, we would like to end this article and we hope that we have inculcated a feeling of love for math in the readers. Next time, you step outside your home, I’m sure most of you would give a shot to decode the math and find patterns in your surroundings 😄. Good day fellow readers and math-a-holics!
