There is a relationship between the number of vertices v, the number of faces f and the number of edges e of a simple polyhedron.

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.

Understand Euler

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, σ, 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 solve some practical problems.

For example, 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.

Remove a face to make it a plane figure. After deformation, the number of vertices v, edges v and remaining faces f 1 of tetrahedron remain unchanged. 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 figure), and find the sum σ α of all angles in the face.

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:

σα=[(n 1-2) 1800+(N2-2) 1800+……+(nf-2) 1800]

=(n 1+N2+…+nf-2f) 1800

=(2e-2f) 1800 =(e-f)3600( 1)

On the other hand, the sum of interior angles is obtained by using vertices in an open graph.

Let a cutting surface be an N polygon, and the sum of its internal angles is (n-2 n-2) 1800, then among all V vertices, there are N vertices on the edge and v-n vertices in the middle. The sum of the internal angles of the v-n vertices in the middle is (V-N) 3600, and the sum of the internal angles of the N vertices on the side is (N-2 n-2) 1800.

So, the sum of the internal angles of each face of a polyhedron:

σα=(v-n)3600+(n-2) 1800+(n-2) 1800

=(v-2) 3600。 (2)

from( 1)(2):(e-f)3600 =(v-2)3600。

So v+f-e=2.

Euler theorem's application method

(1) score:

a^r/(a-b)(a-c)+b^r/(b-c)(b-a)+c^r/(c-a)(c-b)

When r=0, 1, the value of the formula is 0.

When r=2, the value is 1.

When r=3, the value is a+b+CB+C.

(2) Complex number

From e I θ = cos θ+isinθ, we get:

sinθ=(e^iθ-e^-iθ)/2i

cosθ=(e^iθ+e^-iθ)/2

(3) Triangle

Let r be the radius of the circumscribed circle of the triangle, r be the radius of the inscribed circle, and d be the distance from the outer center to the inner center, then:

d^2=r^2-2rr

(4) polyhedron

Let v be the number of vertices, e be the number of edges and f be the number of faces, then

v-e+f=2-2p

For example, p is an Euler feature.

A polyhedron with p=0 is called a zero-class polyhedron.

A polyhedron with p= 1 is called the first polyhedron.

(5) Polygon

Let the number of vertices of a two-dimensional geometric figure be v, the number of divided regions be ar, and the number of strokes be b, then there are:

v+ar-b= 1

(For example, a graph consisting of a rectangle and two diagonals, v = 5, ar = 4, b = 8)

(6) euler theorem

In the same triangle, its circumscribed circle, center of gravity, center of nine o'clock and vertical center line.

In fact, there are many Euler formulas, and the above are just a few commonly used ones.

Euler theorem is used to calculate the number of pentagons and hexagons in football.

Q: The football surface is made of pentagonal and hexagonal leather. A * * *, how many such pentagons and hexagons are there?

A: Football is a polyhedron, which satisfies the Euler formula F-E+V = 2, where F, E, v E and V respectively represent the number of faces, edges and vertices.

Suppose that there are regular pentagons (black leather) and hexagons (white leather) with X and Y on the football surface, then

Number of faces f = x+y

The number of sides e = (5x+6y)/2 (each side consists of a black leather and a white leather * * *).

The number of vertices v = (5x+6y)/3 (each vertex is used by three skins * * *).

According to Euler formula, x+y-(5x+6y)/2+(5x+6y)/3 = 2 and x = 12 are obtained.

So * * * has 12 pieces of black leather.

So the black leather * * * has 12× 5 = 60 sides, all sewn together with the white leather.

In the case of white leather, three of the six sides of each white leather are sewn together with the side of the black leather, and the other three sides are sewn together with the sides of other white leather, so half of all the sides of the white leather are sewn together with the black leather.

Then a piece of white leather should have 60× 2 = 120 sides, 120 ÷ 6 = 20.

So there are 20 pieces of white skin in the * * * dynamic. Euler's rotation theorem points out that if a rigid body is displaced by at least one fixed point in three-dimensional space, the displacement must be equal to the rotation around the fixed axis containing the fixed point. This theorem is named after the Swiss mathematician leonhard euler. In mathematical terms, the relationship between two coordinate systems with arbitrary origin in three-dimensional space is rotation around a fixed axis containing the origin. This also means that the product of two rotation matrices is still a rotation matrix. The rotation matrix that is not identity matrix must have an eigenvalue of a real number, and this eigenvalue is 1. The eigenvector corresponding to this eigenvalue is collinear with the fixed axis around which the rotation revolves [1]. Directory [hidden]1apply1.1rotation generator 1.2 quaternion 2 reference 3 reference [Edit] Apply [Edit] rotation generator main items: rotation matrix, rotation group If we set the unit vector as a fixed axis, and assume that we rotate around this fixed axis with a tiny angle value δ θ. To the first power approximation, the rotation matrix can be expressed as follows: one angular rotation around a fixed axis can be regarded as several continuous small rotations around the same fixed axis; The angle of each small rotation is 0, which is a big number. In this way, the rotation around the fixed axis angle can be expressed as follows: We can see the basic explanation of Euler's rotation theorem: all rotations can be expressed in this form. The product is the generator of this rotation. It is usually a simple method to analyze with a generator instead of using the whole rotation matrix. The knowledge of generator meta-analysis is generally regarded as rotation group's Lie algebra. [Edit] Quaternion According to Euler's rotation theorem, the relative orientation of any two coordinate systems can be set by a set of four numbers; Among them, three numbers are direction cosine, which is used to set the feature vector (fixed axis); The fourth number is the angle value of rotation around the fixed axis. Such a set of four numbers is called quaternion. Quaternions as described above do not involve complex numbers. If quaternions are used to describe two consecutive rotations, they must be calculated in complex form using irreducible numbers derived from William Tam Hamilton. In aviation application, the method of calculating rotation with quaternion has replaced the direction cosine method. This is because they can reduce the required work and minimize rounding errors. In addition, in computer graphics, it is very valuable to simply perform spherical linear interpolation between quaternions.