Project summary - Research the assigned topic and Mathematician. I was assigned Leonhard Euler and Graph Theory.
MATHEMATICIAN essay
Leonhard Euler was born on the 15th of April in 1707 in Basel Switzerland. Euler had two younger sisters named Anna Maria and Maria Magdalena. Soon after Euler was born, his family moved to Riehen Switzerland. At age thirteen Euler enrolled in college and he had his master in philosophy by the time he was 16. His tutor at the time, Johann Bernoulli recognized his talent in the area of math and Bernoulli convinced Euler’s dad that his son had a great future as a mathematician. John Bernoulli at the time regarded as one of Europe's foremost mathematicians. When Euler was 19 he published a book on sound titled Dissertatio Physica De Sono(Physical Dissertation on Sound. In 1726 Euler unsuccessfully applied for a job as a professor at the University of Basel. On May 17th, 1727 Euler started at a job in Russia working at the Russian Academy of sciences. Euler swiftly moved up in the ranks at the Academy. He was made professor of physics at the Academy in 1731. In 1733 Euler became head of the mathematics department when his friend Daniel Bernoulli left.
On the 7th of January 1734 he married his Wife Katharina Gsell. Leonhard and Katharina had 13 children and of those children only 5 survived childhood. Euler left the academy in 1741 because he was concerned about the turmoil in Russia. He took a teaching job in Berlin. He was working at the Berlin Academy. Euler lived in Berlin for 25 years and he published over 380 articles in his time there. He published his two most famous articles while he was in Berlin. He wrote Introductio in analysin infinitorum (Introduction analysis of the infinite) which is considered to be one of the most influential mathematical texts of all time. In this book he defined a function and presented ways to represent different kinds of functions. Such as; Linear Functions, Square Functions, Cube Function, Reciprocal Function and many others types. Another work that he published was Institutiones calculi differentialis (Foundations of differential calculus). This work was two separate books, one was 9 chapter long and the other was 18 chapters long. This volume was written about the concept of differential calculus. This is a subfield of calculus which is the study of the rate of change in quantities.
Euler was also a tutor for Friederike Charlotte of Brandenburg. He also spent time tutoring the princess of Anhalt-Dessau and Friederike niece. Euler was one of the few dedicated mathematician/Scientist who was able to clearly communicate his ideas on math and science to a lay audience. This was a rare ability for Scientists of his time. In Euler’s later years he had severe eyesight deterioration. In 1766 Euler was rendered almost completely blind due to a cataract in his left eye and in his right eye he had suffered deterioration from a bad flu he suffered nearly 30 years prior.
Leonhard Euler was a pioneering mathematician and physicist. He made important discoveries surrounding Graph theory and field calculus. Graph theory is the study of graphs and how to represent information in the world using numbers and lines. Euler was the only mathematician to ever have two different numbers named after him. He found the number e and figured out how to use this in equations. He used this in e^(i*pi)+1 which is known as Euler Identity. Another equation that Euler designed and proved was the Euler Characteristic. The Euler Characteristic x=V-E+F. This equation defines the topological space of a shape. He revolutionized the terminology used in surrounding graphing and calculus. He introduced the letter e for the base of a natural logarithm, which is also known as Euler's number. Leonhard made important discoveries that revolutionized the way math is done today.
On the 7th of January 1734 he married his Wife Katharina Gsell. Leonhard and Katharina had 13 children and of those children only 5 survived childhood. Euler left the academy in 1741 because he was concerned about the turmoil in Russia. He took a teaching job in Berlin. He was working at the Berlin Academy. Euler lived in Berlin for 25 years and he published over 380 articles in his time there. He published his two most famous articles while he was in Berlin. He wrote Introductio in analysin infinitorum (Introduction analysis of the infinite) which is considered to be one of the most influential mathematical texts of all time. In this book he defined a function and presented ways to represent different kinds of functions. Such as; Linear Functions, Square Functions, Cube Function, Reciprocal Function and many others types. Another work that he published was Institutiones calculi differentialis (Foundations of differential calculus). This work was two separate books, one was 9 chapter long and the other was 18 chapters long. This volume was written about the concept of differential calculus. This is a subfield of calculus which is the study of the rate of change in quantities.
Euler was also a tutor for Friederike Charlotte of Brandenburg. He also spent time tutoring the princess of Anhalt-Dessau and Friederike niece. Euler was one of the few dedicated mathematician/Scientist who was able to clearly communicate his ideas on math and science to a lay audience. This was a rare ability for Scientists of his time. In Euler’s later years he had severe eyesight deterioration. In 1766 Euler was rendered almost completely blind due to a cataract in his left eye and in his right eye he had suffered deterioration from a bad flu he suffered nearly 30 years prior.
Leonhard Euler was a pioneering mathematician and physicist. He made important discoveries surrounding Graph theory and field calculus. Graph theory is the study of graphs and how to represent information in the world using numbers and lines. Euler was the only mathematician to ever have two different numbers named after him. He found the number e and figured out how to use this in equations. He used this in e^(i*pi)+1 which is known as Euler Identity. Another equation that Euler designed and proved was the Euler Characteristic. The Euler Characteristic x=V-E+F. This equation defines the topological space of a shape. He revolutionized the terminology used in surrounding graphing and calculus. He introduced the letter e for the base of a natural logarithm, which is also known as Euler's number. Leonhard made important discoveries that revolutionized the way math is done today.
Graph Theory proofs and problems
Eulerian path is a line that touches all ends of graph only once. The S.B.o.K. is an example of a failed Eulerian path. A vertex is set at the end of each edge. There are different types of vertices; isolated vertex, leaf vertex, pendant vertex, source vertex, and sink vertex.
Euler found out that there is no way to solve this problem. You can not cross all the bridges in a single path without having to re cross somewhere. Even though this problem couldn’t specifically be solved, the math involved in finding that out was the basis for much of graph theory. Euler looked at how many spots on land needed to be visited. The bridges were just paths between these points. He was then able to reduce this to.
Then Euler tested if he could trace the graph without having to retrace any of the edge. Euler figured out that the number of lines entering a vertex must be even in order to be able to cross the bridges without repeating.
There are 6 vertices and 3 edges coming into each of these vertices. You know that there will be no way to solve this without crossing. You know this because there are an odd number of edges entering each vertex.
This is a successful example of Vertex coloring. The original problem that needed to be solving with Color Theorem was with mapping. When mapping you cannot have two of the same colors directly next to each other.
- Seven Bridges of Konigsberg:
Euler found out that there is no way to solve this problem. You can not cross all the bridges in a single path without having to re cross somewhere. Even though this problem couldn’t specifically be solved, the math involved in finding that out was the basis for much of graph theory. Euler looked at how many spots on land needed to be visited. The bridges were just paths between these points. He was then able to reduce this to.
Then Euler tested if he could trace the graph without having to retrace any of the edge. Euler figured out that the number of lines entering a vertex must be even in order to be able to cross the bridges without repeating.
- Utility Problem:
There are 6 vertices and 3 edges coming into each of these vertices. You know that there will be no way to solve this without crossing. You know this because there are an odd number of edges entering each vertex.
- Graph Coloring:
This is a successful example of Vertex coloring. The original problem that needed to be solving with Color Theorem was with mapping. When mapping you cannot have two of the same colors directly next to each other.
- With a hexagon the max number of colors that you would need would be two because the you can alternate with each color.
- When any shape has an odd number of vertices then you need three colors.
- Hamiltonian Path
- Platonic solids
project reflection
Project Reflection
Write/type a reflection about this project. HONESTY.
A couple things about math: Yes there are times when math is not applicable to a real world situation. But it teaches you skills that go way beyond solving equations and such. Math teaches you to think critically, use logic and to solve problems.
Things to touch on:
i. Did you enjoy researching your topic?
I did not really enjoy researching graph theory but I did enjoy researching Leonhard Euler. It was interesting seeing how Euler’s discoveries play it to the origination of Graph Theory.
ii. Did you enjoy researching your chosen mathematician?
I researched Leonhard Euler and it didn’t interest me until I started realizing how his discoveries tied back to Graph Theory.
iii. Would you have preferred if your mathematician and math topic were directly connected? For example, Isaac Newton is an inventor/discoverer of calculus.
The fact that my mathematician and topic tied together made it easier to learn about both. Alone they were not as intriguing but tying them together made it more interesting.
iv. Would you have liked to work in partners?
Usually I enjoy working in partners much more but this project’s layout seemed to better fit working without a partner. If the layout were changed a little I could see enjoying working with a partner more.
v. Did you have enough time in class to work?
More class time would have been helpful but it was still doable with the amount of time allowed.
vi. Was I available enough to help?
I wish that I could have gotten more help from you but I know that trying to help twenty students all at once is hard enough. I felt like you did really well helping as many people as you can.
Write/type a reflection about this project. HONESTY.
A couple things about math: Yes there are times when math is not applicable to a real world situation. But it teaches you skills that go way beyond solving equations and such. Math teaches you to think critically, use logic and to solve problems.
Things to touch on:
i. Did you enjoy researching your topic?
I did not really enjoy researching graph theory but I did enjoy researching Leonhard Euler. It was interesting seeing how Euler’s discoveries play it to the origination of Graph Theory.
ii. Did you enjoy researching your chosen mathematician?
I researched Leonhard Euler and it didn’t interest me until I started realizing how his discoveries tied back to Graph Theory.
iii. Would you have preferred if your mathematician and math topic were directly connected? For example, Isaac Newton is an inventor/discoverer of calculus.
The fact that my mathematician and topic tied together made it easier to learn about both. Alone they were not as intriguing but tying them together made it more interesting.
iv. Would you have liked to work in partners?
Usually I enjoy working in partners much more but this project’s layout seemed to better fit working without a partner. If the layout were changed a little I could see enjoying working with a partner more.
v. Did you have enough time in class to work?
More class time would have been helpful but it was still doable with the amount of time allowed.
vi. Was I available enough to help?
I wish that I could have gotten more help from you but I know that trying to help twenty students all at once is hard enough. I felt like you did really well helping as many people as you can.