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Today it is well known that a regular heptagon cannot be constructed with straightedge and compass, since it would require to solve an equation of third degree which is not possible with the standard Euclidean tools.
However, a marked ruler or a compass which can carry a distance to a different point would allow us to solve such an equation; and Greeks exploited such tools to trisect an angle. Is there any reference of such an use to the problem of constructing an heptagon?
We have Archimedes's construction of regular heptagon due to Arab transcription by Thabit Ibn Qurra, see Mendell's translation. Aaboe gives modernized exposition and commentary in Episodes from the Early History of Mathematics, MathWorld's Heptagon has a shorter version along with modern variants.
The construction uses neusis of sorts, i.e. marked straightedge as in the trisection of the angle, but the condition for marking the point is unusual, it requires two areas to become equal (in typical neusis one verges a line until it is tangent to something or some segments are in a given ratio). Neither Archimedes nor Thabit indicate how it is supposed to be met. According to Aaboe, this type of neusis is unique in all of extant Greek corpus, but he points out that one could determine when the areas are equal by intersecting two conic sections. The remaining straightedge and compass construction is itself much more intricate and relies on a "miraculous" coincidence at a key point.
For a broader perspective see Hogendijk's Greek and Arabic Constructions of the Regular Heptagon.
