As anyone who's actually done geometric construction of n-gons knows, not all construction methods are made equal. Some are very stable (the shape you get is always close to ideal even if you're not absolutely precise, like the construction of a square or octagon), some are very unstable (the notorious construction of the regular 17-gon) and will fail to even produce the correct number of sides if you're not incredibly precise. What's more, different constructions for the same polygon often have differing stability.

What exactly makes a construction more or less stable? The number of steps is obviously important, but it seems more complicated than that. For example, it's fairly easy to construct even large power-of-two-gons (16-gon, 32-gon, etc), accurately because all you need to do is bisect. The pentagon though has as few as nine steps to mark all vertices (maybe less, that's the shortest construction I know), but is fairly difficult to construct accurately (they often come out visibly irregular).

Has anyone studied or written about this? I'm particularly interested in scale-invariant phenomena (physically bigger constructions are obviously easier). There was a reddit thread about this suggesting it's related to numerical analysis, but it seems no real conclusions were reached there.

  • $\begingroup$ Vaguely related: In the old days of drafting, it was generally thought that compasses are more reliable than a straightedge, and compasses with fixed "radius" more reliable than hinged compasses. $\endgroup$ – André Nicolas Jun 7 '16 at 15:40
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    $\begingroup$ I personally have no clue and I think this is a deep question that deserves more attention. Maybe you can post it on MathOverflow to reach the audience with related specialties. $\endgroup$ – Lee David Chung Lin Sep 27 '16 at 8:26
  • $\begingroup$ Which 17-gon construction do you mean? Richmond's (which draws a $\perp$ from $OA$ to find $P_3, P_{14}$ and another to find $P_5, P_{12}$) looks stable to me. And which pentagon? Dixon's looks stable to me, as does Ptolemy's if you draw the 3rd circle to find $P_1, P_4$ then walk around the first circle with radius $P_0P_1$ to find $P_2, P_3$. I see no intersections at very small angles. Any sources of instability I overlooked? $\endgroup$ – Rosie F Feb 26 '19 at 19:54
  • $\begingroup$ Im glad I came across this very old question. Ive been meaning to ask lately what techniques contribute to precision constructions. Whether its using compasses more, or straightedges more, a fair balance of each, or perhaps minimizing the number of shapes drawn/times pen meets paper, or minimizing the number of intersection points one requires, etc. Everything from that to using neusi, sharpening pencils to a pinpoint, etc. Whatever it may be, Id like to know about it. I havent seen any published work on this subject (but I havent really looked that hard). $\endgroup$ – CogitoErgoCogitoSum Jul 18 '20 at 20:22
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    $\begingroup$ @CogitoErgoCogitoSum Unfortunately I still don't have a very good answer to this question (not that I've been actively looking). At this point, I suspect that it might be a good question for someone's thesis project (maybe undergraduate thesis, I'm not exactly sure how hard it would be). If you find any info on it please do share! $\endgroup$ – realityChemist Jul 19 '20 at 2:55

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