Beyond the Euler Trail. Mathematics is often thought of as formulas, ratios, and the number Pi. The history of

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1 Patino 1 Prof. Petersen Sierra Patino Math 101 Section April 2016 Beyond the Euler Trail Mathematics is often thought of as formulas, ratios, and the number Pi. The history of math and its roots tend to get ignored and left out when people learn the complex subject. Leonhard Euler was a renowned mathematician. On referring to Euler s work, Keith J. Devlin, a 20th century British mathematician once said, Like a Shakespearean sonnet that captures the very essence of love, or a painting that brings out the beauty of the human form that is far more than just skin deep, Euler's equation reaches down into the very depths of existence. Euler s life might not have been the most extravagant, yet his legacy led to a math concept that is used for modern life. Leonhard Euler was born on April 15 th, 1707 in Basel Switzerland, to his father Paul Euler and mother Marguerite Buck. Euler was raised in a very religious environment, with his father being a pastor, and his mother being the daughter of a pastor. Paul Euler was a friend of Johann Bernoulli, who himself, was a celebrated mathematician at the time. Bernoulli had a prominent influence on the Leonhard Euler, which drove him to his love of mathematics.

2 Patino 2 By the time Euler was a teenager, he had already attended the University of Basel and earned a masters degree. Before the age of twenty, Euler had served in the Navy and was already planning to move to Russia to join the St. Petersburg Academy. Euler had become a professor of physics and later was the head of the mathematics division. He was already well on his way to becoming a cherished mathematician. In 1734, Euler married Katharina Gsell, but just a year later Euler suffered from a fever that was so severe it almost caused him to become blind in one eye. Euler and his wife had thirteen children. Sadly, only five children survived beyond infancy. After being married thirtynine years, Katharina died. Euler later married Salome, Katharina s half sister, in In the mid-1740s, Euler became the appointed the mathematics director of the Berlin Academy of Science and Beaux Arts, which was a relatively new school. Euler took on several management positions and eventually became the head of the school in A year later Euler was given patronage by Catherin II and returned to St. Petersburg Academy. In the early 1770s, Euler had officially lost his sight, after he did not allow appropriate time to recover from an operation. Leonhard Euler eventually died while working in St. Petersburg on September 18, in 1783 from a brain hemorrhage. Euler s legacy will forever live on in the work he accomplished during his life, especially his work in graph theory, which is still used to this day. Leonhard Euler created graph theory in the 1730s when he speculated if it was possible to walk across each of the seven bridges of Königsberg, only once. It was not necessary to start and stop in the same place. For years, the Königsberg bridge problem was considered to be one of the toughest problems to solve. In 1736 Euler revealed the solution. Euler proved that it was impossible for one person to cross each bridge exactly once. To prove it, Euler drew a graph that

3 Patino 3 consisted of vertices, (points that are linked by edges) and edges, (lines that connect vertices). When Euler set out to solve the problem, he started by drawing a graph and eliminating all of the unnecessary details like buildings and rivers. Euler made each vertex a place, each edge a bridge (see figure above with points A-D). He gave each vertex a number to represent how many edges connected to it. This is called a degree. Points A, C, and D all have three degrees each, and point B had a total of five degrees. In order for the graph to be traversable, the graph has to either be an Euler trail, which can only have two or less odd degrees of vertices, or the graph is an Euler circuit, which must have all even degrees of vertices. The graph that is drawn for the Königsberg bridge problem is made up all odd vertices making it not traversable. This solution launched graph theory into mathematics forever. Within graph theory are several types of graphs. An Euler circuit, which I mentioned before, is where the edges can only be traversed once. Vertices may be crossed more than once, and both the end point and the start point are the same. An Euler path is very similar to an Euler circuit, the difference is, the start point and the end point does not have to be the same point. Similarly to Euler circuits, Hamilton circuits start and end at the same point, yet instead of crossing each edge, Hamilton circuits try to

4 Patino 4 touch every vertex. Hamilton paths do the same, but their start and end points don t have to be the same. Graph theory is a very important concept that is still applicable today. It is used in transportation, biology, computer sciences, engineering, security network, and social networking. To show how graph theory is applied to social websites I decided to apply it to my Facebook friends. First I had to decide what it was I am trying to discover with my graph. I opted for the degree of centrality, which measures ones social influence. The more friends that a person had, represents how great of a social influence the person has. If I am a person of great influence then my graph, will look like this: If I am a person of low influence, but high connectivity my end graph will look like this:

5 Patino 5 I created my definitions and rules. Each vertex would represent a friend and each edge would be our friend ship. I also decided that I would base this off of six close friends and would spread out to three mutual friends at the most. To help keep my friends private I decided to go by their first name s initial only. Now it s time draw out my map. I started by picking the six friends I would use. I chose friend from different groups and aspects of my life. Then I compared three of their mutual friends too see if there were any connections. Once I had all of my information, I mapped out all of my connections, and this was my final result:

6 Patino 6 My final result is very similar to what I pictured it would be. While I picked friends that I thought were from different aspects of my life, I knew a lot of my friends shared common interests. I might not be a person of great influence, but I do have high connectivity. Leonhard Euler started his love for math at a young age, when Paul Euler, and Johann Bernoulli taught him the basics. As his love for math grew, so did his family, career and legacy. Euler s life came to a sad end, but he left the world with a new concept that is used even to this day. Many may go their whole lives with out even wondering who it was created the math concepts, but there are those that will keep mathematicians like Euler, from disappearing completely.

7 Patino 7 Works cited Baker, Greg. "Euler and Hamiltonian Paths." Web. 3 Apr "Leonhard Euler Biography." Bio.com. A&E Networks Television. Web. 03 Apr Edwins, Laura. "Leonhard Euler, His Famous Formula, and Why He's so Revered ( Video)." The Christian Science Monitor. The Christian Science Monitor, Web. 01 Apr "Euler - 18th Century Mathematics - The Story of Mathematics." Euler - 18th Century Mathematics - The Story of Mathematics. Web. 02 Apr "Gliffy Editor - Online." Gliffy. Web. 03 Apr "Graph Theory: Part III (Facebook)." Graph Theory: Part III (Facebook). Web. 03 Apr "Graph Theory World of Mathematics." World of Mathematics. Web. 01 Apr