Leonhard Euler's Polyhedron Formula
Introduction
Leonhard Euler is one of the great mathematicians, who made many remarkable contributions to mathematics. I got to know him when I was learning natural logarithm in math class, which is one of his achievements. He discovered many theorems including polyhedron formula, which states that the number of any polyhedron together with the number of vertices is two more than the number of edges (Kirk, 2007). This formula is widely used in mathematical practice and in real life as well. As polyhedron formula is the basic formula related to figure, I learned it first when I was in elementary school and was keep wondering how this formula is established. In this essay, I will explore Euler’s polyhedron formula by using Cauchy’s proof and
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The value of formula V - E + F does not change until the end of the process, which is still equal to 2. As the formula V - E + F = 2 is valid for the final network, it must also be valid for the polyhedron itself. Therefore, this process proves the Euler’s theorem: V - E + F = 2.
Application
Euler’s polyhedron formula applies to various fields in real life. One of the applications is a soccer ball. A soccer ball, made in 1960, has 32 faces that consist of 12 pentagons and 20 hexagons. As such, soccer ball made of polyhedrons applies to Euler’s formula. Three pieces of pentagon and hexagon meet at each vertex. Then, if their values are applied to the formula, the result becomes:
V - E + F
= 60 – 90 + 32
= 2
Therefore, we can verify that Euler’s polyhedron formula is valid for any polyhedrons which have same connection state with a regular polyhedron or a sphere. Moreover, it is valid not only for a polyhedron that has planar face and straight edges but also for a polyhedron that has curved surfaces and curved edges or sphere.
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Many mathematicians proved the formula in different approaches, and Cauchy is one of them. Cauchy proved the theorem by showing that the value of V - E + F does not change when a polyhedron is transformed into a single triangle. After Euler’s formula was achieved recognition, it has been widely used in various fields in real life such as a soccer ball. As such, his achievement has contributed to development of our lives. When we study the mathematics, it is important not to just memorize the formula but to understand the principle of it. It helps to improve the ability of mathematical thinking. For this reason, exploring Cauchy’s proof was very useful work to broaden my view of mathematics. Through this exploration, I expected that my ability to solve mathematical problems would improve, as I would have a better understanding of
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