Construction

I have personally always been fascinated with the idea of construction in the context of geometry. For those of you who aren’t familiar with it, the method of construction allows one only to use a straight edge and a compass, both without any markings. With these two tools, and quite a bit of smarts, ancient mathematicians were able to do such things as construct the perpendicular bisector of a line segment or the angle bisector of an arbitrary angle. Using the method of construction, ancient mathematicians figured out how to create regular polygons with 3, 4, and 5 sides. Because they were also able to construct the midpoint of a line segment, they were able to double the number of sides in their polygons, giving them 3x, 4x, and 5x sided polygons where x=2^k for some positive integer k. However, they were never able to generalize their results to construct a polygon with n sides for any integer n. In 1796, when he was 19 years old, Carl Friedrich Gauss came up with a proof that a heptadecagon, or a 17-sided regular polygon, could be created using construction. A method of construction for a heptadecagon wasn’t discovered until 1825 by Johannes Erchinger. Though Gauss himself never constructed any 17-gons, he was able to generalize his proof a bit, giving the sufficient conditions for a n-sided polygon to have the possibility of begin constructed. That condition is that n is a distinct Format prime multiplied by 2^k, where k is any non-negative integer. Gauss was unable to prove that this condition was also necessary, though a hand was lent by Pierre Wantzel. Hence the name Gauss-Wantzel Theorem.