PI

By Robin Prokaski

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

November 24, 2015, from https://www.youtube.com/watch?v=8gd-gUlBv_s

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:

https://www.youtube.com/watch?v=2iovbcPwAro

https://www.youtube.com/watch?v=Kw6g31HFMDA

Websites:

http://mathforum.org/isaac/problems/bridges1.html

http://mathworld.wolfram.com/Topology.html