V+F-E=2

This formula is called Euler formula. This formula describes the unique laws of the number of vertices, faces and edges of a simple polyhedron.

Euler, a Swiss mathematician, went to university of basel to study at the age of 13, and was carefully guided by Bernoulli, a famous mathematician. Euler is the most prolific outstanding mathematician in the history of science. He started publishing papers at the age of 19 until he was 76. In his tireless life, * * * wrote 886 books and papers, of which more than 700 papers were written before his death. In order to organize his works, the Petersburg Academy of Sciences spent 47 years.

It is no accident that Euler's works are surprisingly numerous. His tenacious perseverance and tireless academic spirit enable him to work in any harsh environment: he often kneels with his children in his arms to finish his papers. Even during the 17 years after his blindness, he did not stop studying mathematics, and dictated several books and more than 400 papers. He died while writing the calculation essentials for calculating Uranus' orbit. Euler will always be our respected teacher.

Euler's research works involve almost all branches of mathematics, including physical mechanics, astronomy, ballistics, navigation, architecture and music! There are many formulas, theorems, solutions, functions, equations and constants named after Euler. The mathematics textbook written by Euler was always regarded as a standard course at that time. 1Gauss, a great mathematician in the 9th century, once said that "studying Euler's works is always the best way to understand mathematics". Euler was also the inventor of mathematical symbols. Many mathematical symbols he created, such as π, that is, sin, cos, tg, ∑, f (x), are still in use today.

Euler not only solved the problem of calculating the trajectory of comets, but also solved the problem of moon deviation, which was a headache for Newton. The perfect solution of the famous "Seven Bridges in Konigsberg" opened the research of "Graph Theory". Euler found that no matter what shape the convex polyhedron is, there is always a relationship among the number of vertices V, the number of edges E and the number of faces F. V+F-E=2, which is the so-called Euler formula. V+F-E, that is, Euler characteristics, has become the basic concept of "topology". So what is "topology"? How did Euler discover this relationship? How did he study it? Today, let's follow the footsteps of Euler and explore this formula with reverence and appreciation. ......

The Significance of euler theorem

(1) Mathematical Law: The formula describes the unique law among the number of vertices, faces and edges of a simple polyhedron.

(2) Innovation of ideas and methods: In the process of theorem discovery and proof, it is conceptually assumed that its surface is a rubber film, which can be stretched at will; Methods Cut off the bottom surface and turn it into a plane figure (three-dimensional figure → plan).

(3) Introduction of topology: From three-dimensional graph to open graph, the shape, length, distance and area of each face have changed, while the number of vertices, faces and edges remain unchanged.

Theorem leads us into a new field of geometry: topology. We use a material (such as rubber wave), which can be deformed at will, but it can't be torn or stuck. Topology is to study the unchangeable properties of graphics in this deformation process.

(4) Propose a polyhedron classification method:

In Euler's formula, f (p)=V+F-E is called Euler's characteristic. Euler theorem told us that simple polyhedron f (p)=2.

In addition to simple polyhedrons, there are non-simple polyhedrons. For example, dig a hole in a cuboid and connect the corresponding vertices at the bottom to get a polyhedron. Its surface cannot be transformed into a sphere by continuous deformation, but it can be transformed into a torus. Its Euler characteristic f (p)= 16+ 16-32=0, that is, the Euler characteristic of polyhedron with holes is 0.

(5) euler theorem can be used to solve some practical problems, such as: Why are there only five regular polyhedrons? What's the relationship between football and C60? Is there a regular polyhedron with 7 sides? wait for

Euler theorem's proof

Method 1: (Using the Geometry Sketchpad)

Gradually reduce the number of edges of polyhedron and analyze v+f-e.

Firstly, the simple tetrahedron ABCD is taken as an example to analyze the proof method.

When a face is removed, it becomes a plane figure, and the number of tetrahedral vertices e, the number of edges v and the number of remaining faces F 1 remain unchanged after deformation. Therefore, in order to study the relationship between V, E and F, we only need to remove one surface and turn it into a plane figure, and prove that V+F 1-E= 1.

(1) If one edge is removed and one face is reduced, V+F 1-E remains unchanged. Remove all faces in turn and become a "tree".

(2) Every time an edge is removed from the remaining tree, a vertex is reduced, and V+F 1-E remains unchanged until only one edge remains.

In the above process, V+F 1-E remains unchanged, and V+F 1-E= 1, so a removed surface is added, and V+F-E =2.

For any simple polyhedron, this method has only one line segment left. Therefore, this formula is correct for any simple polyhedron.

Method 2: Calculate the sum of internal angles of polyhedron.

Let the number of vertices v, faces f and edges e of a polyhedron. Cut off a face to make it a plane figure (open graph), and find the sum of all angles in the face ∑ α.

On the one hand, the sum of internal angles is obtained by using all the faces in the original image.

There are f faces, the number of sides of each face is n 1, n2, …, nF, and the sum of the internal angles of each face is:

∑ α = [(N 1-2) 180 degrees +(N2-2) 180 degrees+...+(NF-2)180 degrees].

= (n1+N2+…+nf-2f)180 degrees.

= (2e-2f) 180 degrees = (e-f) 360 degrees (1