(1) Background: Behind Euler's formula is a new geometry, which only studies the relative order of the positions of various parts of a graph, regardless of the size of the graph. This is the "geometry on rubber film" (position geometry) founded by Leibniz and Euler. Now this subject has developed into an important branch of mathematics-topology.

(2) History: One of the most interesting theorems about convex polyhedron is Euler formula "V-E+F=2", which was actually discovered by Descartes around 1635. Euler independently discovered this formula in 1750 and published it in 1752. Because Descartes' research was not discovered until 1860, this theorem is called Euler formula instead of Descartes' formula.

Euler, born in the Swiss city of Basel, went to university of basel to study at the age of 13, and was carefully guided by the most famous mathematician at that time (John johann bernoulli, 1667- 1748).

Euler has made many achievements in mathematics, and the solution of the famous problem of the Seven Bridges in Konigsberg initiated the research of graph theory. Euler also found that no matter what shape of convex polyhedron, there is always a relationship between the number of vertices V, the number of edges E and the number of faces F, that is, V-E+F=2. V-E+F, which is called Euler characteristic, has become the basic concept of topology. Mathematical formulas and theorems named after Euler can be found everywhere in mathematics books. At the same time, he has made brilliant achievements in physics, astronomy, architecture, music and philosophy. Euler also created many mathematical symbols, such as π( 1736), i( 1777), e( 1748), sin and cos( 1748), tg( 1753).

1733, at the age of 26, Euler became a professor of mathematics at the Academy of Sciences in Petersburg. 1735, Euler solved an astronomical problem (calculating the orbit of a comet), which took several famous mathematicians several months to solve, but Euler used his own invented method and completed it in three days. However, overwork made him suffer from eye diseases and unfortunately lost his right eye.

Euler's life is a life of struggle for the development of mathematics. His outstanding wisdom, tenacious perseverance, tireless spirit of struggle and noble scientific ethics are always worth learning.

There are four Euler formulas.

(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 and e be the number of edges and 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.