The concept of impossible constructions in mathematics arouses a unique interest from mathematicians who wish to find answers that no one has found before them. For the Greeks, some impossible constructions had not proven impossible at the time, but simply unfeasible. For them there was excitement at the idea of ​​being the first to do it, excitement that lay in the discovery. There are some impossible constructions in Greek mathematics that will be examined in this chapter. They all share the same constructability criteria: they must be built using exclusively compass and straightedge, and were defined as the three "classical problems of antiquity". It was believed that the need to use only compass and ruler came from Plato himself. 1The Greeks were able to do many things with just a compass and straightedge (although these were not their only tools, the Greeks in fact had access to a wide variety of tools as they were a fairly modern society). For example, they found the means to construct parallel lines, bisect angles, construct various polygons, and construct squares of equal or double the area of ​​a given polygon. However, three constructions that they could not achieve with these two tools alone were the trisection of the angle, the doubling of the cube, and the squaring of the circle. The first impossible construction to examine is the trisection of an angle. Its purpose, dividing an arbitrary angle into three equal angles, could have proved useful for a variety of fields. However, mathematicians repeatedly failed to find a solution using only compass and ruler. People began to think about it around the 5th century BC in Greece, at the time of Plato. T...... half of the card...... the impossible construction is doubling the cube. In this scenario you are given a cube and the goal is to build a larger cube with twice the volume of the first cube. If you were to assign the length x and the volume v to the length of one side of the first cube, then x3 = v. If the first cube, for example, had a volume of 1, then the second cube would have a volume of 2. Its length would then be the cube root of 2 and, as demonstrated by Galois theory, any root of a polynomial third degree is not a constructible number.Works Cited1. Anglin, W.S., Mathematics: A Concise History and Philosophy, Springer-Verlag, New York, 1994, pp. 75-802. Angular trisection. Available: http://en.wikipedia.org/wiki/Angle_trisection. Last accessed 16 September 20133. Squaring the circle. Available: http://en.wikipedia.org/wiki/Squaring_the_circle. Last access September 13th 2013