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?
On
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.
The straight line and the circle suggest themselves as very basic objects in the study of geometry, so it's natural to see what you can do with a straightedge and a compass. The Greeks quickly worked out how to construct regular polygons with $3$, $4$, $5$, and $6$ sides, so again it must have seemed natural to speculate on what other polygons were constructible. I doubt the problem started out as being important; I think its importance came from its centuries-long resistance to solution.