The fundamental relationship between trigonometric functions and complex exponential functions is defined by Euler’s formula. Euler’s theorem, also known as Euler’s equation, is a fundamental equation in mathematics and engineering with many applications. Euler’s formula is straightforward but crucial in geometrical mathematics. It is concerned with the Polyhedron shapes. This Euler Characteristic can aid in the categorization of the shapes. Let us research Euler’s Formula
What do you mean by Euler’s formula?
Let’s learn a little more about the polyhedron shape before we get into Euler’s formula. A polyhedron is a solid structure with flat faces that are only bordered by straight lines on its surface. Each face in it is, in reality, a polygon. A polygon is a closed-form made up of points connected by straight lines in the flat 2-dimensional plane.
Euler’s Formula and its History:
Leonhard Euler (1707–1783) was a Swiss mathematician who was regarded as one of history’s greatest and most prolific mathematicians. He was blind for most of his career, but he nevertheless managed to write one paper a week with the aid of scribes. Euler’s Polyhedral Formula is a well-known formula that he devised. The rest of Euler’s formula deals with complex numbers.
Euler’s formula states for polyhedron that that certain rules are to be followed:
F + V – E = 2
Where,
- F denotes the number of faces
- V denotes the number of vertices or corners
- E includes the number of edges
What is a Polyhedron?
A polyhedron is a three-dimensional solid formed by joining polygons together. The word ‘polyhedron’ is derived from two Greek words: poly, which means many, and hedron, which means surface. Polyhedrons are classified according to the number of faces they have.
Parts of a Polyhedron:
Every polyhedron has three essential parts:
- Face: the flat surface that makes up a polyhedron is called the face. These faces are regular polygons in nature.
- Edge: the regions where the two flat surfaces meet to form a line segment are named the edges.
- Vertex: It is the point of intersection of the edges of the given polyhedron. A vertex is also named the corner of a given polyhedron. The plural of vertex is known as the vertex.
Euler’s Formula for Various Shapes
Euler’s Formula is only valid for a polyhedron that follows a set of laws. The form must not have any holes and must not intersect itself, according to the law. It also can’t be made up of two parts glued together, such as two cubes joined at one vertex. This formula will work for any polyhedron if all of these laws are followed correctly. As a result, this formula will work for the majority of common polyhedra.
In reality, there are a variety of shapes that give different answers to the sum FE. The Euler Characteristic X is a name given to the response to the sum FE. FE=X is a common abbreviation for this. Some forms, such as the “Double Torus” surface, may also have a negative Euler Characteristic. As a result, the value of abstract figures can become very difficult.
This formula can be used in the Graph theory as well:
- To prove a given graph as a planar graph, this formula is applicable and very beneficial.
- This formula is very useful to prove the connectivity of a graph in geometry.
- To find out the minimum colors required to color a given map, with the distinct color of adjoining regions, this formula can be used.
A widely used example to prove Euler’s theory is given below:
For the tetrahedron shape prove the Euler’s Formula:
Solution:
In a given tetrahedron,
- F (Number of faces) include = 4
- V (Number of Vertices) include = 4
- E (Number of Edges) include = 6
Thus,
F + V – E = 4 + 4 – 6 = 2. Hence it is proved.
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