Are there only 5 regular polyhedra ? Why are there only 5 types of regular polyhedra?

There are only five types of regular polyhedra: tetrahedron, octahedron, hexahedron, twelvehedral and icosahedron.

As for the existence of 5 regular polyhedra, that is known for a long time (at the time of Plato in ancient Greece). Graphs and methods for making models can be found in Steinhaus's Kaleidoscope of Mathematics. ①

Proof For a regular polyhedron, it is assumed that all sides of it are regular n-sided and that r edges meet at each apex corner. And so there is:

nf=2e (1)

rv=2e (2)

The right coefficient of 1) is 2 because each side appears in 2 sides, and the right factor 2 of (2) is because each side passes through 2 vertex corners. Substituting (1) and (2) into Euler's equation gives us: or.

Obviously n 3, r 3, because the polygon has at least three sides, and at each apex corner there are also at least three sides. But n 3, and r 3 is not possible, because then there would be , but e 0. So at least one of r and n equals 3.

Let n=3, then , so r=3,4,5, so that e=6,12,30, and f=4,8,20, this gives the tetrahedron, octahedron, and icosahedron.

Let r=3, then , so n=3,4,5, and thus e=6,12,30, and f=4,6,12, this gives the tetrahedron, hexahedron (i.e., cube), and dodecahedron.

Yes, there are only five types: tetrahedron, hexahedron, octahedron, dodecahedron, and icosahedron.

The specific proof can be found here:

There are only five types of tetrahedron, hexahedron, octahedron, dodecahedron, and icosahedron The teacher also said so.

There are only five types of polyhedra that are tetrahedron, hexahedron, octahedron, dodecahedron, and icosahedron.

The so-called regular polyhedron is, of course, to ensure that it is a polyhedron first, and its special feature is that each of its faces is a regular polygon, and the regular polygons of each face are congruent.

That is, cut out the faces of the regular polyhedron and they can coincide exactly. Although the polyhedral family is very large, the members of the polyhedron are very small, only five.

Let the polyhedron have m edges at each vertex, and each face is a regular n-sided shape, and the number of vertices of the polyhedron is v, the number of faces is f, and the number of edges is e. Because the two adjacent faces have a common edge, they are <>

Because two adjacent vertices have a common edge, they <>

And because of the Euler theorem of polyhedra, v+f-e=2 is obtained, which can be obtained from the above three equations.

For the above equation to hold, 2m+2n-mn>0 must be satisfied, i.e. 1 m+1 n>1 2. Because of m 3, so.

So n<6.

When n=3, m<6, so the value that m can take is ;

When n=4, m < 4, so the value that m can take is 3;

When n=5, m < 10 3, so the value that m can take is 3.

When n=3, m=3, v=4, f=4, e=6;When n=3, m=4, v=6, f=8, e=12;When n=3, m=5, v=12, f=20, e=30;When n=4, m=3, v=8, f=6, e=12;When n=5, m=3, v=20, f=12, e=30;So there are only five types of regular polyhedra.

Classical polyhedra.

In the classical sense, a polyhedron (English word from the Greek poly-), which is the root word represents"and more", Edron, from δ representatives"Substrate"，"block", or"faces") is a three-dimensional body consisting of a finite number of polygon faces, each of which is part of a plane, where the faces intersect at edges and each edge is a straight segment.

The edges intersect at the points, which are called vertices. Cubes, pyramids, and prisms are all examples of polyhedra. A polyhedron encloses a bounded volume of three-dimensional space; Sometimes the inner body is also considered part of the polyhedron.

A polyhedron is a three-dimensional counterpart of a polygon. The general term for polygonal, polyhedron, and higher-dimensional counterparts is polysome.

Regular polyhedra The so-called regular polyhedron means that all sides of the polyhedron are congruent regular polygons, and each polyhedral angle is a congruent polyhedral angle. For example, the four faces of a regular tetrahedron (i.e., a regular pyramid) are congruent triangles, with one trihedral angle at each vertex, for a total of three trihedral angles, which can coincide exactly, that is, they are congruent.

The number of species of regular polyhedra is very small. There can be an infinite number of polyhedra, but there are only five types of regular polyhedra: tetrahedron, hexahedron, octahedron, dodecahedron, and icosahedron.

Prove. The number of vertices v, the number of faces f, and the number of edges e

Let each face of the regular polyhedron be a regular n-sided with m-edges at each vertex. The number of edges e should be half of the product of the number of faces f and n (one edge for every two sides), i.e.

nf=2e --

At the same time, e should be half of the product of the vertex number v and m, i.e.

mv=2e --

Obtained by , deposed.

f=2e n, v=2e m, substituting Euler's formula v+f-e=2, there is.

2e/m+2e/n-e=2

After finishing, 1 m+1 n=1 2+1 e

Since e is a positive integer, 1 e > 0. Therefore.

1/m+1/n>1/2 --

It means that m,n cannot be greater than 3 at the same time, otherwise it will not be true. On the other hand, because of the meaning of m and n (the number of edges at a vertex of a regular polyhedron and the number of sides of a polygon), m 3 and n 3. Therefore m and n have at least one equal to 3

When m=3, because 1 n>1 2-1 3=1 6, n is a positive integer, so n can only be 3,4,5

In the same way, n = 3, m can only be 3, 4, 5

So there are several scenarios:

n m type.

3 3 Tetrahedron.

4 3 Regular hexahedron.

3 4 Regular octahedron.

5 3 dodecahedron.

3 5 Regular icosahedron.

Since the above five types of polyhedra can indeed be made geometrically, there can be no other kinds of regular polyhedra.

So there are only 5 types of regular polyhedra.

1. Proof that each face of the regular polyhedron is a regular n edge row, and each vertex is m edges, so the edge number e should be half of the product of f (number of faces) and n, that is, nf=2e (1 formula). At the same time, e should be half of the product of v (number of vertices) and m, i.e. mv = 2e (formula 2).

From formula 1 and formula 2, f=2e n, v=2e m, substituted into Euler's formula v+f-e=2, there are 2e m+2e n-e=2, and 1 m+1 n=1 2+1 e is obtained.

2. Since e is a positive integer, 1 e > 0. Therefore, 1 m+1 n>1 2 (3 formula), 3 formula means that m, n cannot be greater than 3 at the same time, otherwise 3 formula is not valid. On the other hand, because of the meaning of m and n (the number of edges at the vertex of a regular polyhedron and the number of sides of the polygon), m >=3 and n>=3.

Therefore m and n have at least one equal to 3

3. When m=3, because 1 n>1 2-1 3=1 6, n is a positive integer, so n can only be 3,4,5.

4. In the same way, n=3, m can only be 3, 4, 5, so nm type, 33 regular tetrahedron, 43 regular hexahedron, 34 regular octahedron, 53 regular dodecahedron, 35 regular icosahedron, because the above 5 kinds of polyhedra can indeed be made by geometric methods, and there can be no other kinds of regular polyhedra, so there are only 5 kinds of regular polyhedra.