Euler formula
A famous mathematician, Swiss, spent most of his time in Russia and France. Master's degree at the age of 65,438+07. He began to study mathematics in his early years with the appreciation of the mathematical genius Benuri, and studied mathematics after graduation. He is the most prolific writer in the history of mathematics. He published more than 700 papers before his death, leaving more than 100 to be published after his death. His works involve almost all branches of mathematics. He used f(x) first.
polyhedron
Definition of polyhedron
Geometry surrounded by several planar polygons.
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(2)
(3)
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Related concepts of polyhedron
Faces of polyhedron
Sharp ridge
pinnacle
convex polyhedron
If any face of a polyhedron extends to a plane, if all other faces are on the same side of this plane, such a polyhedron is called a convex polyhedron.
Classification of polyhedron
tetrahedron
Pentpolyhedron
Hexagonal polyhedron, etc
polyhedron
regular polyhedron
Each face is a regular polygon with the same number of sides, and a convex polyhedron with the same number of sides at each vertex is called a regular polyhedron.
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(2)
(3)
regular tetrahedron
regular hexahedron
regular octahedron
pyritohedron
Regular icosahedron
polyhedron
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Simple polyhedron
The surface can become a spherical polyhedron after continuous deformation.
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discuss
Question 1: (1) Count the vertices V, faces F and edges E of the following four polyhedrons and fill in the table.
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(2)
(3)
Graphic number
Vertex number v
Face number f
Edge number e
( 1)
(2)
(3)
(4)
Law:
V+F-E=2
four
six
four
eight
six
12
six
eight
12
20
12
30
(Euler formula)
(4)
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( 5 )
Question 1: (2) Count the vertices V, faces F and edges E of the following polyhedron and fill in the table.
five
eight
five
seven
eight
12
Graphic number
Vertex number v
Face number f
Edge number e
(5)
(6)
V+F-E=2
(Euler formula)
Simple polyhedron
discuss
Question 2: How to prove Euler formula?
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B
C
D
E
A 1
B 1
C 1
D 1
E 1
A
B
C
D
E
A 1
B 1
C 1
D 1
E 1
discuss
Thinking 1: If the number of faces of a polyhedron is f, the number of vertices is v, and the number of edges is e, then the number of polygons, vertices and edges of a planar figure are respectively
Thinking 2: Let f faces of a polyhedron be N 1, N2, ..., NF polygons. What is the sum of the internal angles of each surface?
(n 1-2) 1800+(N2-2) 1800++(nF-2) 1800 =(n 1+N2 ++ nF-2F) 1800
Thinking 3: What is the relationship between N 1+N2++NF and the number of edges e of polyhedron?
n 1+n2+ +nF =2E
f，V，e。
Question 2: How to prove Euler formula?
discuss
A
B
C
D
E
A 1
B 1
C 1
D 1
E 1
A
B
C
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E
A 1
B 1
C 1
D 1
E 1
The sum of the interior angles of a polygon = (e-f) 3600.
Thinking 4: Let the largest polygon in a plane figure (namely polygon ABCDE) be an M polygon, then what is the sum of its internal angles with all the polygons in it?
2(m-2) 1800+(V-m)3600 =(V-2)3600
∴(E-F) 3600= (V-2) 3600
Question 2: How to prove Euler formula?
discuss
A
B
C
D
E
A 1
B 1
C 1
D 1
E 1
A
B
C
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E
A 1
B 1
C 1
D 1
E 1
V+F-E=2
Euler formula
Question 3: Application of Euler Formula
11The Nobel Prize in Chemistry in 1996 was awarded to three scientists who made great contributions to the discovery of C60. C60 is a molecule composed of 60 C atoms, and its structure is a simple polyhedron. This polyhedron has 60 vertices, from which three sides are drawn, and the shape of each face is pentagon or hexagon. Calculate the number of pentagons and hexagons in C60 molecules.
Solution: Let C60 molecule have X and Y pentagonal and hexagonal surfaces respectively.
The problem means that the number of vertices V=60, the number of faces X+Y, and the number of edges E= (3×60).
According to Euler formula, we can get 60+(x+y)-(3×60)=2.
On the other hand, the number of sides can also be expressed by the number of sides of a polygon, that is
(5x+6y)= (3×60)
X = 12 and Y = 20 can be solved from the above two equations.
A: C60 molecules have 12 and 20 pentagonal and hexagonal faces.
Example 2, is there a simple polyhedron with 7 sides?
Solution: suppose there is a simple polyhedron with the number of sides E=7.
According to Euler formula, V+F=E+2=9.
Because the number of vertices V≥4 and the number of faces F≥4 of a polyhedron, there are only two cases:
V=4, F=5 or V=5, F=4.
However, a polyhedron with four vertices has only four faces and a tetrahedron has only four vertices. Therefore, it is assumed that there is no simple polyhedron with 7 edges.