The ancient Greeks were fascinated by exploring the constructions that they could make with only a compass and straightedge. To make their constructions, they agreed that the straightedge could not have any markings that could be used to make measurements. Furthermore, the compass had to be assumed to collapse after constructing any circle, so that it could not be used to mark off identical distances by setting it using two points and then repositioning it.
Even with these strict rules, the Greeks were masters of constructions. One of the easiest constructions is to form the perpendicular bisector of a line segment. They also were able to inscribe regular polygons of many different side lengths inside a circle. The first several of these side lengths are 3, 4, 5, 6, 8, 10, 12, 16, and 20. German mathematician Karl Friedrich Gauss (1777 - 1855), a great prodigy at an extremely young age, was able to determine mathematically which regular polygons were possible to construct and which were not. He discovered a class of regular polygons that were possible to construct, of which the Greeks had been unaware. The smallest of these, the heptadecagon, has 17 sides. Gauss must have thought this discovery very important, as he asked for a heptadecagon to be placed on his gravestone.
The ancient Greeks were puzzled, however, about the possibility of the three constructions listed below, with which they were having great difficulty.
- Trisecting an angle, or dividing any given angle into three equal angles.
- Squaring a circle, or constructing a square equal in area to the area of any given circle.
- Doubling a cube, or given the side of a cube, constructing the side of a cube that has twice the volume.
Mathematicians tried for many centuries to find solutions for these three constructions. It took the development of "abstract algebra" in the nineteenth century to prove that these constructions were impossible. The reason for the impossibility is that these constructions require either constructing a cube root or constructing . Cube roots and are examples of non-Euclidean numbers, which are not possible to find using only a compass, a straightedge, and the given rules. Gauss realized that the first two constructions were impossible. Doubling a cube was shown to be an impossibility by Ferdinand Lindemann in 1882.