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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?

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    $\begingroup$ +1 Nice one! This never occurred to me, "we're gonna construct everything using only compass and straightedge, let's go," that's a weird motivation. $\endgroup$ – Lord Soth Apr 13 '13 at 3:57
  • $\begingroup$ I wouldn't say particularly that the Greeks gave these constructions "so much importance", but I just think it was a natural problem to consider in plane geometry. If what you're after is some sort of practical application of these constructions I am not aware of any. The Greeks were probably considering such objects in the vein of "pure mathematics". It's kind of like asking why Fermat's last theorem was so important. $\endgroup$ – dezign Apr 13 '13 at 3:57
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    $\begingroup$ If one can speculate, the regular pentagon and its close relative the pentagram were of symbolic importance to the Pythagoreans. The division of a line segment in mean and extreme ratio (golden ratio) can be done by straightedge and compass, giving a construction of the regular pentagon. And in mathematics, once a problem becomes attractive, there is an urge to generalize. Also, the importance to Greek mathematicians of the construction of regular polygons is perhaps overstated. They decidedly did not restrict constructions to straightedge and compass. $\endgroup$ – André Nicolas Apr 13 '13 at 4:29
  • $\begingroup$ Related: math.stackexchange.com/questions/22686/… $\endgroup$ – I. J. Kennedy Dec 16 '13 at 20:33
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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.

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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.

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