Heptadecagon

This mathematical breakthrough provided a method to construct a heptadecagon, a seventeen-sided polygon with equal length sides, using only a compass and straightedge. This was the first instance of a proof of impossibility in the field of geometric construction and solidified the importance of proof in mathematics.
 Carl Friedrich Gauss Germany 1796 Heptadecagon

This mathematical breakthrough provided a method to construct a heptadecagon, a seventeen-sided polygon with equal length sides, using only a compass and straightedge. This was the first instance of a proof of impossibility in the field of geometric construction and solidified the importance of proof in mathematics.