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This article is to share with you the example analysis of the application of abstract mathematical theorems in python. The editor thinks it is very practical, so I share it with you to learn. I hope you can get something after reading this article.

This paper introduces a theorem in group theory, which has many names, as follows:

Burnside counting theorem, Cauchy-Frobenius Lemma, orbit counting theorem

The description of this theorem is abstract, as follows:

Given group G, set X, and G acts on X, and the definition is:

Number of orbits acting on it =

To prove the theorem, the following is an application to illustrate the meaning of the theorem:

Given a cube, and given three different colors, the surface of the cube can be colored with only one color, and ask how many different coloring methods there are. (if the cube is the same after rotation, the two coloring methods are regarded as the same coloring method.

This problem can be calculated by listing all the coloring methods one by one, but a simpler algorithm can be obtained by using the orbit counting theorem:

The natural rotation of a cube is seen as a group G, and the coloring arrangement of six faces is regarded as a set X

The number of orbits of action, that is, the number of equivalence classes X is divided under the action of group G, each equivalence class is a set of elements that can remain the same through the action of group G (cube rotation).

Then the different coloring methods to be solved are actually the number of orbits of this effect:

The rotation of a cube can be divided into five categories:

1, do not move and rotate 1

2There are three axes of symmetry passing through the center of the face, with 2 rotations of +-90 degrees along any one of them, for a total of 6 rotations.

3, 3 axes of symmetry passing through the center of the face, rotated 180 degrees along any one of them, for a total of 3 rotations

4, 6 central symmetrical axes passing through the edge, rotated 180 degrees along any one of them, for a total of 6 rotations

5, 4 cross-vertex symmetry axes, each with +-120 degrees rotation, a total of 8 rotations

There are 24 rotations.

Then

Because of these five types, the rotation g of the same type corresponds to

Is the same, only need to calculate the g corresponding to any one of each class

, of which

According to the definition, it is the same number of shading under rotation.

1, without rotation, obviously every coloring method remains the same, with a total of 3 ^ 6

2. Rotate 90 degrees, requiring that the four faces around the axis have the same color, and the other two faces are random, then there are 3 ^ 3 kinds.

3, rotate 180 degrees, the four faces around the axis are required to be the same, and the other two are random, a total of 3 ^ 4

4, rotate 180 degrees, the requirements are the same, a total of 3 ^ 3 possibilities

5The three faces are the same as 1 group, a total of 2 groups, a total of 3 ^ 2 kinds of coloring possibilities

Therefore, according to the orbit counting theorem

That is, there are 57 coloring methods with different rotation.

The above is an example analysis of the application of abstract mathematical theorems in python. Xiaobian believes that there are some knowledge points that we may see or use in our daily work. I hope you can learn more from this article. For more details, please follow the industry information channel.

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