Geometry for Elementary School/Some impossible constructions
In the previous chapters, we discussed several construction procedures. In this chapter, we will number some problems for which there is no construction using only ruler and compass.
The problems were introduced by the Greek and since then mathematicians tried to find constructions for them. Only in 1882, it was proven that there is no construction for the problems.
Note that the problems have no construction when we restrict ourself to constructions using ruler and compass. The problems can be solved when allowing the use of other tools or operations, for example, if we use Origami.
The mathematics involved in proving that the constructions are impossible are too advanced for this book. Therefore, we only name the problems and give reference to the proof of their impossibility at the further reading section.
Squaring the circle
The problem is to find a construction procedure that in a finite number of steps, to make a square with the same area as a given circle.
Doubling the cube
To "double the cube" means to be given a cube of some side length s and volume V, and to construct a new cube, larger than the first, with volume 2V and therefore side length ³√2s.
Trisecting the angle
The problem is to find a construction procedure that in a finite number of steps, constructs an angle that is one-third of a given arbitrary angle.
Proving that the constructions are impossible involves mathematics that is not in the scope of this book.
The interested reader can use these links to learn why the constructions are impossible.
The Four Problems of Antiquity have no solution since any solution would involve constructing a number that is not a constructible number. The numbers that should have been constructed in the problems are defined by these cubic equations.
It is recommended to read the references in this order:
- Four Problems of Antiquity http://www.cut-the-knot.org/arithmetic/antiquity.shtml
- Constructible numbers http://www.cut-the-knot.org/arithmetic/rational.shtml
- Cubic equations http://www.cut-the-knot.org/arithmetic/cubic.shtml