Euler's formula of a polygon

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Many formulas have been found in the history of mathematics by Leonhard Euler. They are all called Euler's formulas and are scattered in various mathematical branches.

Euler received a bachelor's degree at the age of 13 and a Master's degree in philosophy from the University of Basel at the age of 15. However, his father wanted him to learn theology. His father was not opposed to reading mathematics until he received a scholarship from the Paris Emy of sciences at the age of 19. Euler was a super-creative mathematician who later moved from Switzerland to Russia and Germany, so all three countries claimed that he was a national scientist.

There are many legends about Euler. For example, Euler's mental calculus is as simple as breathing. On one occasion, his two students added up 17 items of a complex convergence series to 50th digits, with a unit difference between the two. to determine who is right, Euler uses mental arithmetic to perform all operations, finally, the error is found. Euler's article creation speed is very fast. Generally, the previous book has not been printed, and new manuscripts have been written. As a result, his writing order is often the opposite of the Publishing order, making readers very depressed. It is also difficult to collect these large numbers of manuscripts. Switzerland's natural science accounting team draws a complete set of Euler's, which was compiled in nearly 100 s and finally completed in the 1990s S. He did not expect St. Petersburg to suddenly discover a batch of his manuscripts, this complete set has not yet been completed. At the age of 28, Euler was blind, and the other was invisible. It was said that he was overworked and said that he was observing the sun. Despite this, he still completed a lot of papers by mental computing.

Next let's take a look at the most famous and beautiful Euler's formula.

The Euler's formula in topology describes the relationship between a polygon vertex (vertex), an edge, and a face:

V-e + F = x

V indicates the number of vertices in a polygon, e indicates the number of edges in a polygon, F indicates the number of edges in a polygon, and X indicates the Euler Characteristic Number of the polygon ).

X is the topological invariant, which means that no matter how it passes through the topological deformation, it will not change. It is the scope of topology research. The value of X depends on the shape of the geometric object and the orientation of the surface.

Testability-most of the surfaces we encounter in the physical world are oriented. For example, a plane, a sphere, and a ring can be oriented. However, the möbius strip is not targeted, and it seems to have only one "side" in 3D space ". Assuming that an ant crawls along Mobius, It can climb to the other side of the surface without passing through the border.

Genus-the loss of a directed surface is an integer. If any simple closed curve along a geometric surface is cut off, the loss of this curve is 0. If a simple closed curve is not split into two parts after cutting, the loss is 1. After cutting a curve on the surface with a loss of 1, you can find another curve. The loss is 2. And so on.

The Euler's number of closed and oriented surfaces can be calculated by their loss G.

X = 2-2 * g

For example, if the loss of a cube is 0, 8 vertices, 12 edges, and 6 edges, the Euler's formula is

8-12 + 6 = 2

Assume that the loss is 0, the vertices are 4, the edges are 6 GB, And the faces are 4. the Euler's formula is

4-6 + 4 = 2

The strict proof of the first Euler's formula is given by the 20-Year-Old French scientist Augustin Louis Cauchy, which is roughly as follows:

Remove one side from a polygon, and pull the edges of the removed surface to form a flat network of points and curves. Without losing its universality, we can assume that the deformed side is still in a straight line segment. Normal surfaces are no longer normal polygon, even if they are normal at the beginning. However, the number of vertices and edges remains the same as the number of edges in a given polygon .)

Repeat a series of extra transformations of F − e + V that can simplify the network without changing its Euler's number.

1. If there is a polygon with more than three sides, we draw a diagonal line. This adds an edge and a surface. Add the edge until all sides are triangular.
2. Remove all triangles that share the two sides of the network. This reduces a vertex, two sides, and a plane.
3. Remove a triangle with only one side and an external side. This reduces the number of vertices and keeps the number of vertices unchanged.

Repeat steps 2nd and 3rd until there is only one triangle left. For a triangle F = 2 (including the external number), E = 3, V = 3. Therefore, F −e + V = 2. Pass.

Euler's formula of a polygon