What makes a geometric construction more or less stable?

Question

I'm not entirely sure if this is "research level" math or not, but I asked on Math.SE and it was suggested I try asking here, so hopefully it's of interest to this community. (Original question on M.SE; I've done some rewording but this is essentially the same question.)

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My question relates to the physical construction of geometric figures on a plane (so just normal, everyday Euclidean constructions). Some constructions are very "stable," here meaning that even if you're not being terribly careful you usually end up with something quite close to the ideal result (eg. constructing an equilateral triangle). Others are quite "unstable," such that you need to be very careful or your result could end up quite dramatically wrong (eg. constructing a 17-gon). Importantly, different constructions of the same figure seem to have differing stabilities.

What makes one of these constructions more or less stable? Number of steps is obviously important, but I think there's more to it. For example, I can construct even quite large (power of 2)-gons accurately because all I have to do is bisection, whereas constructing a pentagon takes relatively few steps but is liable to come out visibly uneven unless I'm being quite careful.

Likewise, scale is obviously a factor (a square inscribed in a 2ft diameter circle is much easier to construct with accuracy than the same figure in a 2cm diameter circle), but I'm more interested in arbitrary scale phenomena.

Has anyone written about or done research into this? I dug up a reddit thread on this topic which suggested that it's related to numerical analysis, but they didn't reach any real conclusions. A Math.SE user suggested I post this here to reach out to people with specialties in that field. If someone can answer this, feel free to post the same answer on the Math.SE question.

Answer 1

In the case of regular polygons inscribed in a circle, you get addition of errors if you start with a chord that's supposed to be one side of the polygon and copy it all the way around in one direction. You may be better off proceeding equally in both directions until it is time to close the polygon, or perhaps even constructing a chord that subtends two sides and getting the intermediate vertex using a perpendicular bisector.

I invite readers to do some experiments with the regular pentagon. We use Ptolemy's pricedure as our point of departure, which I describe below:

1. Construct a pair of perpendicular lines, which intersect at O.

2. Construct a circle centered at O, passing through points A,B,C,D on the perpendicular lines in rotational order.

3. Bisect line segment OA to generate its midpoint M.

4. Draw an arc centered on M passing through B, which intersects line segment OC at E.

5. Set the compass radius to natch OE, whose chord length is correct to intercept an arc of 72°, and mark off successive points at this distance along the circle starting at B, generatong points F,G,H,I.

6. Connect B,F,G,H,I in rotatational order using the straightedge to draw the pentagon.

Experiment 1: In the Step 5, mark off two points starting from B going clockwise amd two going counterclockwise. Compare your result with marking all the points offing just one direction.

Experiment 2: Instead of placing E in Step 4 on line segment OC, extend OA and place E on this extension, not OC. Then continue as below:

5. Draw the arc centered at B passing through E, whose radius is the chord length that intercepts a 144° arc. This intersects the circle at points F and G.

6. Perpendicularly bisect BF to identify point H on the circle, on the minor arc from B to F.

7. Draw the arc centered on B passing through H which also intersects the I intially constructed circle at I.

8. Go back to Step 6 of the original construction.

Experiment 1 explores multiplying a slightly off-ideal chord by a smaller number, while Experiment 2 involves dividing instead of multiplying the chord. Readers are invited to comment on the relative accuracy if pentagons obtained by these methods.