From a purely mathematical point of view, it has been proved that it is impossible to divide any angle equally; But from a philosophical point of view, any angle may be divided into any equal parts. Because mathematics is only a tool for human beings to describe the world, the whole mathematical system is based on several basic assumptions. For example, do decimal numbers exist in the objective world? No. Many mathematical problems may be caused by several basic assumptions of the foundation of the mathematical building, or they may be very simple problems in the mathematical system composed of binary, ternary and multivariate, or there are no such problems at all.

How to prove that it is impossible to draw any angle with a ruler?

By reducing to absurdity:

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Given an arbitrary angle ∠ α,

Firstly, Cos(A) is generated,

Suppose we can divide ∠a into three parts at this time,

Then we can do cos(A/3),

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According to the cos triple angle formula, we can get:

4*cos^3(A/3) - 3*cos(A/3) = cos(A)

Let cos(A/3) = x, and we will get the ternary linear equation:

4x^3 - 3x - cos(A) = 0

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Different values of cos(A) lead to different solutions of the above equation;

However, for most ∠A,

The solution of equation 4x 3-3x-cos (a) = 0 will be in the form of [cube root].

That is, cos(A/3) will be in the form of [cube root].

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However, from an arithmetic point of view, ruler and gauge drawing can only do five operations:

Add, subtract, multiply, divide and square.

Only these five operations can't get the form of [cube root] in any case.

So the ruler can't draw the amount of [cube root];

So cos(A/3) can't be made;

So ∠A can't be divided into three equal parts.

(This is the general idea of proof. If you want to prove it rigorously, you have to write too much. There's no need here. After all, it's ok to get to know the idea. )