## Sunday, November 29, 2015

### PI

As many mathematicians know, Pi is the ratio between a circle’s circumference and diameter. Many of us know that Pi can be approximated at 3.14. However, it is a never ending number. As of 2013, Pi has been calculated to 12.1 trillion digits (The Math Mystery). Even though Pi is such an integral part of many aspects of mathematics, few truly know the early history of Pi. Furthermore, rarely people understand that it comes from nature, and how abundant in nature it really is.
To start off lets first talk about the history of Pi. To this day, we do not know the first to become conscious of this ratio. However, we seem to think human civilizations could have been aware of this as early as 2550 BC. The great Egyptian pyramids were built between 2550 BC and 2500 BC. It turns out that the height and the width come out to approximately 2 times pi. These were measured in cubits, were approximately 18 inches. However, a cubit was measured by the length of a person’s forearm. Therefore, it varied from one person to another. Early texts reveals the Egyptians found an approximation for pi to be 3.16 (Purewal 2013).
One of the biggest contributors to the development of Pi was Archimedes. He is considered to be the first person to calculate an accurate approximation of Pi. He did this by using area of two Polygons. One of these polygons inscribed the circle, while the other was inscribed by the circle. Essentially, he kept increasing the number of sides of the polygons and effectively got closer and closer to the area of the circle. He continued this process until reaching polygons with 96 sides. This process allowed him to get an approximation between 3.1408 and 3.1429 (Smoller 2001).
One of the text that contains an approximation of Pi that is not as frequently talked about is the Bible. The following verse contains an approximation for Pi: “"And he made a molten sea, ten cubits from the one brim to the other: it was round all about, and his height was five cubits: and a line of thirty cubits did compass it round about"(1 Kings 7:23). From this quote, one would think that their approximation was 3:1. However, cubits were the unit of measurement. As mentioned previously, cubits change from person to person, and the approximation could really have been more accurate than it seems (Purewal 2013).  From here, many different mathematicians made contributions for Pi. Now let’s focus on topic that many people are not that familiar with, Pi in nature.
Pi is often thought of just the ratio between circumference and diameter of a circle.  In the Documentary “The Mystery of Math” a professor explains the following example: If a sheet of paper is divided into any amount of sections in which r is the length between lines, and a needle, of length r, is dropped onto the paper, the probability that the needle cuts across a line is 2/Pi. Even though there is no circles or diameters of circles, Pi still comes into play. Experts say that Pi is able to show up in the strangest of places because it is one of only concepts to relate a straight object (diameter) with a rounded one (circle). One of the places that comes into play is the path of rivers. As we know, a river is rarely ever straight, but is not circular either. On average, the actual length of the river over the direct distance from start to finish is about Pi. Similar to rivers, any models that deal with waves has Pi in them. Most commonly, these are light and sound waves. The documentary mentioned earlier also explains that Pi tells us which color should appear in a rainbow, and how middle C should sound like on a piano. On the biology side of things, Pi explains how cells grow into spherical shapes such as oranges or apples (Mystery of Math).
However, Pi is found in many other different aspects of nature. One area that many people do not realize pi is involved in is probability.
Pi is part of a hidden interconnected web that relates many different aspects of our world. It is our job as mathematicians to find these hidden connections, to better understand our world and nature.
References
The Math Mystery: Mathematics in Nature and Universe - Documentary. (2015, June 11). Retrieved
Purewal, S. (2010, March 13). A brief history of pi. Retrieved November 11, 15, from
http://www.pcworld.com/article/191389/a-brief-history-of-pi.html
Smoller, L. (2001, February). The amazing history of pi. Retrieved November 24, 15, from

http://www.ualr.edu/lasmoller/pi.html

### Topology and the Seven Bridges of Konigsberg

By Courtney Correa

Topology is the mathematical study of the properties that are preserved through deformations, twisting, and stretching of objects. (wolfram.com) Tearing of the objects in topology are not allowed, and an example of two shapes that are topologically equivalent are a circle and an ellipse.   Topologists study the shapes and their properties that remain the same when stretched or compressed.  Leonhard Euler is credited with the discovery of topology of networks and in 1735 his work in this field was inspired by the Seven Bridges of Konigsberg problem.  In Konigsberg, which is now modern day Kaliningrad, Russia, a river ran through the city and created a center island.  After the island, the river split into two parts and the people of Konigsberg built seven bridges for people to use to get around the city. (mathforum.org)

Euler presented the question of if it was possible to cross all seven bridges one time, and be able to access each area of land beside each bridge.  He concluded that this is not possible, and he analyzed the geometric position of each bridge and piece of land to prove this.  With a point or node representing each piece of land, and an edge representing each route to a bridge, he counted four nodes and seven edges.  Euler looked at how many vertices each node had and noticed that the nodes could be odd or even.  An odd node has an odd number of vertices and an even node has an even node of vertices.  The problem with the seven bridges was that all of the nodes were odd, and this meant that it did not matter where you started or finished, you would end up getting stuck and not crossing one of the bridges.
When analyzing the Seven Bridges of Konigsberg, Euler created what is known today as a topology of networks.  (mathforum.org) A network represents points and lines that when connected, they create odd or even vertices.  A network can be considered traversable if one can use a pencil to trace the shape without lifting up the pencil and without going over a side more than once.  Euler’s creation of the graph of the seven bridges is not traversable because the four vertices are odd.  Euler’s graphical representation of the Seven Bridges of Konigsberg hinted at the discovery of topology because of the fact that the most important components of the problem were the number of bridges and their endpoints, not their exact positions.  Topology is not concerned with the shape of objects, and the seven bridges problem is a great example of this because we are not focusing on the actual layout, we are focusing on the graph.

Videos:
Websites:

## The documentary starts off by drawing attention to the fibonacci sequence. The Fibonacci sequence is a set of numbers that are produced by adding the two last numbers. A person first start off with 0,1 and then adding both of the numbers up which results in the answer 1. The result is then added to the sequence. So now the sequence is 0,1,1. Then after that you add the last two numbers of the sequence which results in number two. Then result is added to the sequence. The sequence then looks like 0,1,1,2. This pattern continues on and has been seen to appear a lot in our world. For instance, the Fibonacci sequence seems to appear in places such as, the number of pedals in daisy’s, the number of spirals in a sunflower, the bracts of a pinecone, and tree branches. The number of pedals on each daisy’s seems to sound like a random number but this type of flower seems to favor Fibonacci numbers.

So why then does nature seem to favor the Fibonacci sequence? Does nature know how to do mathematical operations? I do not think so. Is the sequence arbitrary sequence? I do not think so either.
Furthermore, in the documentary they mention river sinuosity, which is the curves of the river. The importance of the curves of the is that many mathematicians have theorized that in most rivers the average approximation of a river’s bendiness is pie. They measure its bendiness by  obtaining the length of the river divided by the direct route. When a river is completely straight the river’s sinuosity is 1. To better understand the gist of river sinuosity I watched a video called “Pi me a river” that perfectly describes this theory.

This video explains the dynamic of the theory. River sinuosity is a theory without any concrete applications. But with this in mind, many great mathematicians like Sir Isaac Newton discovered gravity by assuming that an invisible force such as when an apple falls down. He believed that the same force that holds the planets in order is the same invisible force on the moon-gravity. Without having an sort of application Sir Isaac Newton discovered gravity.The sinuosity of a river seems to not have any sort of applications, but is a recurring measurement that appears in rivers. Is this a arbitrary thing? How is it possible that math is an exact approximation to understand how things in nature works? What is the essence of math? What is math’s origin? So this leads me to ask, if math is not arbitrary then is math discovered or invented?

As the documentary progresses, there seems to ask whether math is discovered or invented. If math were to be discovered it would not make sense to state that a person discovered something that has already been there or to explain something that has already been there. For now, let’s say math is invented. Which brings me to my next topic. If math were to be invented, then is math something created by something, someone, or a thing? In the documentary, Max Tegmark is a professor at MIT, who believes that math is like a video game and that the foundation of math built in our brains. A video game is a set of algorithms, properties, and equations of things that are all mathematical which is put in a software that defines the game. Tegmark believes that their are a lot of similarities between this world and video games because math does not only describe reality in some parts but it is the essence of reality. He believes that math can go as far and be our physical reality. Like a digital photo as a person tries to view the picture closer and closer, only pixels are seen. Tegmark believes that our reality is like this and that math is invented by math and that is all too it. But a huge counter argument against this claim is that If math were to be built in our brain then why do we sense that when we have solved a mathematical problem it seems that we found an answer that was not there but it was already there. Discovered instead of invented.
Furthermore, in the documentary a scientist named Liz Duke lemur center in North Carolina does varies experiments with lemurs. She compares the lemurs reaction when choosing between two different quantities of food. She notices that every time the lemurs always pick the amount that has the most. Animals, such as lemurs, have a notion of what math is, without understanding what it is. If math were to be invented from our minds what is its extent? In the documentary a speaker, Mario Livio, states how is it possible that “The product of human thought does so well in defining the universe,” with this in mind is the product of the human thought ineffective-having no significant results? Throughout time many mathematicians have theorized and made equations based on assumptions and experiments that has given them results that have patterns, without explaining or having any applications to back up their points. I watched a video called “The Unreasonable Effectiveness of Mathematics” where Mario Livio describes the strange way in which math seems to work without having any sort of applications, it just works until many years later someone figures out or gives these applications needed.

In brief, we use math everyday, we see math everyday, math is everything, so how could it possibly be ineffective? Math is something significant to us but produces results that describes the world around us. How is it possible that the length of every river to the ocean is approximately pie? Why does nature seem to favor the Fibonacci sequence? Does nature know how to do mathematical operations? How do animals, such as lemurs, have a notion of what math is, without understanding what it is? If math were to be invented from our minds what is its extent? Although people have created this definition of what math is, it does a very good job in explaining the world we live in.  In the documentary “The Great Math Mystery”, explores the question of why does math do a great job in explaining the world we live in today by asking and analyzing math in nature. It seems that the math itself is a mystery that has an answer to everything, but not for itself.