TITLE: Importance of construction of polygons
QUESTION [9 upvotes]: Why did the ancient Greeks give so much importance to the construction of regular polygons with $n$-sides using only ruler and compass and tried to study for what $n$ was such a construction possible? Until Gauss-Wantzel, this was a famous open problem in Euclidean geometry. 
Can anyone throw any light on its importance?

REPLY [3 votes]: It's worth mentioning that many of the great Greek geometers, including Archimedes of Syracuse and Apollonius of Perga, used the neusis ruler unapologetically.  With this marked ruler, they were able to trisect arbitrary angles and, equivalently, solve cubic equations.  This did allow for the construction of the regular heptagon, for example.