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What are the three major drawing problems in history?
Plane geometric drawing is limited to rulers and compasses. The so-called ruler here refers to a ruler that can only draw straight lines without scale. Of course, many kinds of figures can be made with rulers and compasses, but some figures, such as regular heptagon and regular nonagon, can't be made. Some problems seem simple, but they are really difficult to solve. The most famous of these problems are the so-called three major problems.

The three main geometric problems are:

1. Turn a circle into a square-find a square with an area equal to a known circle;

2. Divide any corner into three equal parts;

3. Double Cube-Find a cube, and make its volume twice that of the known cube.

Both circles and squares are common geometric figures, but how to make a square with the same area as the known circle? If the radius of a circle is known as 1 and its area is π( 1)2=π, then the problem of turning a circle into a square is equivalent to finding a square with an area of π, that is, making a line segment with a ruler (or a line segment with a length of π 1/2).

The second of the three major problems is the problem of bisecting an angle. For some angles, such as 90. 、 180。 It is not difficult to divide into three parts, but can all corners be divided into three parts? Like 60. If you can divide it into three parts, you can get 20 parts. Angle, then the regular 18 polygon and the regular nonagon can also be made (Note: each side of the regular octagon is connected into a circle with a circumferential angle of 360. / 18=20。 )。 In fact, the problem of angle trisection is caused by the problem of finding regular polygons.

The third problem is cubes. Eratoseni (276 BC ~ 65438 BC+095 BC) once described a myth that a prophet had to double the size of the cube altar when he got the Oracle. Some people advocate doubling the length of each side, but we all know that this is wrong, because the size has been eight times the original.

These problems have puzzled the mathematician 1000 years, but in fact, none of these three problems can be solved by a ruler and compass through limited steps.

After Descartes founded analytic geometry in 1637, many geometric problems can be transformed into algebraic problems to study. In 1837, Wantzel gave a proof that it is impossible to draw any angle and cube with a ruler. In 1882, Lin Deman also proved the transcendence of π (that is, π is not the root of any integer coefficient multiple), and the impossibility of changing the square of a circle is established.