Topology

Topology is the study of the properties of a shape which do not change under various contortions and deformations such as bending or stretching, but not ripping or attaching. Simply, topology can be used to describe an object by the number or "holes" it has. Which means that any object that has one hole can be bended or stretched into any other object with one hole, any object that has two holes can be bended or stretched into any other object with two holes, and so on. For example, topologically, A donut (or torus) and a coffee cup are the same object. Therefore, a donut and a coffee cup can be described as homeomorphic, which is a term derived from the Greek words ὅμοιος (homoios), meaning same and μορφή (morphē), meaning shape. These commonalities between homeomorphic objects reveal greater mathematical relationships between the number of holes an object has and its mathematical identity.

These commonalities between homeomorphic objects reveal greater mathematical relationships between the number of holes an object has and its mathematical identity.

Although the concept of topology and its more specific subtopic of homeomorphism is a bit complex, it is incredibly relevant to mathematics and physics when more deeply studied, and has lead to a greater understanding of mathematical objects and physical principles, and even lead to some incredible work which could be applied to data transmission in the field of topological photonics.

Something that we can take away from this baffling field of study is that sometimes by stepping back and evaluating something simply, we can find greater meaning in it than from up close.

Often times in our busy lives, we approach situations with a multitude of thoughts swirling in our heads, and ultimately end up over analyzing and overthinking each intricacy. With the basic principles behind topology, we can see that sometimes an epiphany can surface when we take one step back and approach the situation from a greater distance as to see it fully rather than trying to understand it up close.  

The Fibonacci Sequence

This simple, yet intriguing system of numbers first originated in the Hindu-Arabic arithmetic system, and reached the western world by 1202 in the book Liber Abaci by Italian mathematician, Fibonacci (also known as Leonardo of Pisa and Leonardo Pisano). Fibonacci first encountered these numbers while studying in North Africa, but the first time he actually noticed the sequence was in a mathematical problem about rabbit breeding.

As we begin to look at our universe through a Fibonacci lense, we can see Fibonacci’s numbers blazing from sources all around… not just when it comes to rabbit love. Fibonacci’s sequence can be spotted in the spiral shapes of galaxies, hurricanes, shells, pinecones, and even flowers. Out of the Fibonacci sequence, we get a beautiful spiral formation. By connecting a series of quarter-circles fitted squares with dimensions of each Fibonacci number as so, we form a particularly captivating image.

More interestingly, as we take the ratio of two successive numbers in the sequence, we find that the very closely approximate the Golden Ratio, commonly notates with the Greek symbol phi, or 1.618034. This ratio is also incredibly common in both nature and artwork. For example, creations such as the violin and famous works of art such as the Mona Lisa correspond distinctly to the golden ratio.

The Fibonacci Sequence teaches us that our universe holds an intrinsic and powerful underlying relationship between some of the tiniest elements like flowers, and the grandest elements like galaxies. The Fibonacci Sequence also reveals to us the subconscious mathematical awareness that humans, to such an extent that it reveals itself in our artistic creations. From the Fibonacci Sequence we learn that humanity and nature are more connected than we may know, and we can also see how incredible it is that such a simple mathematical concept seems to govern our world so exquisitely whether we are aware of it or not.

The Möbius Loop

Any argument involving möbius loops only has one side… and it’s that they are absolutely awesome.

For those who don’t know, möbius loops are a phenomenon of the topological realm, and what is so fascinating about them is that they only have one side (and one edge).

If you would like to have a go at creating your very own mathematical wonder, simply cut out a strip of paper, twist one end π radians (or 180 degrees), and tape it together. The true simplicity of this surface is perhaps what makes it most wondrous.

Now, let's open up the treasure chest that we call a möbius loop. Mathematicians refer to the möbius loop as a non-orientable surface, which means that it has no defined "back" or "front," and any attempt to assert so would end in contradiction. This makes the möbius loop a quite unique entity indeed, and baffling enough, it is the only one of its kind that we can experience in three dimensions.

...But if you're curious, the four dimensional analogue of a möbius loop is something that we can represent fairly well in our three dimensional space, and is called a Klein bottle, which you can create by attaching together two möbius loops at the edge. Although this object intersects itself in our three dimensional version, in four dimensions it has one side just as the möbius loop. (If you can conjure up an image of the five dimensional analogue, please let me know!)

But beyond the mathematics, möbius loops represent that with each journey we take, each time we travel along that loop, we gain a new perspective and a greater appreciation for something we may have initially made out to be simple. The fact that a mundane loop of paper can transform into a brilliant phenomenon of the mathematical world truly reveals that there is infinite hidden beauty to be found within everything; if we are just willing enough to travel along that loop and explore the point opposite to where we started, we will always return to our starting point with a more well-rounded understanding and appreciation for what we may have thought we knew before.

Möbius loops teach us to constantly continue to explore and reevaluate, to deepen our understanding of the world, and to never let our appreciation for its wonder cease to grow.

Sources:

https://d7hftxdivxxvm.cloudfront.net/?resize_to=fit&width=350&height=527&quality=95&src=https%3A%2F%2Fd32dm0rphc51dk.cloudfront.net%2FpaXcYnaRTdj-GijxCGRf4A%2Flarge.jpg

http://leseloop.at/wp-content/uploads/roth_grundriss_inside.png

https://www.math.hmc.edu/~gu/curves_and_surfaces/surfaces/moebius.html