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I'm interested in finding a word (if it exists) for describing how regular polygons are less regular than circles. Even though they are regular, it seems apparent that there is some characteristic that separates them from circles.

Basically, the points that make up the perimeter of a circle can be described by a single function instead of a piecewise function and all of these points are equal in all respects, whereas in $n$-gons, they become less "equal" with decreasing $n$.

Specifically, I want to describe the difference described above between a hexagram and a circle in one word (or maybe two). Right now, what I wrote reads "irregular" shape (referring to the hexagram), but that's just not right – a regular hexagram is not an irregular shape.

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    $\begingroup$ The symmetry group of a regular polygon is finite, that of the circle is infinite $\endgroup$ Sep 20, 2015 at 17:29

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Similar to what Hagen said in a comment, the circle has infinitely many lines of reflection symmetry, while the regular polygon only has finitely many. Also, any rotation of the circle about the centre preserves it, whereas only finitely many rotations (modulo a full turn) of the regular polygon preserves it.

On the other hand, if you are talking about a parametrization of the shapes, then the circle has a smooth (differentiable) parametrization while the regular polygon does not. Another possible characterization is that there is no point equidistant from all points on the regular polygon, unlike for a circle.

Ultimately though, your question is too vague, but I can understand that perhaps you don't know the terminology to make it precise. Hope that one of the above is what you're looking for. If not, try to explain more clearly what you are looking for.

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  • $\begingroup$ I know the question is not very precise, I wrote based on intuition instead of actual terminology – I'm looking for the term after all! The smooth parametrization (which I was talking about) and the equidistant points are interesting… The reason I'm asking for this is that I have a (physical) system of points that each repel each other at long ranges only, leading to differing total forces depending on the shape they're arranged in. Then I actually just wanted a word to describe the geometric reason for that concisely, but the different perspectives you and Hagen gave are interesting too. $\endgroup$ Sep 21, 2015 at 3:11
  • $\begingroup$ @MrArsGravis: I guessed so that's why I attempted an answer. But I don't know what could be different between a high-sided polygon and a circle in physics, so I'm afraid I can't quite help still. $\endgroup$
    – user21820
    Sep 21, 2015 at 3:18
  • $\begingroup$ Well, for a high-sided polygon the effects would be negligible. It's mainly the concave regions in a hexagram, pentagram etc. that matter because the points that lie at the "spikes" are separated without any other points in between them (in a circle, there are no such empty regions at its perimeter). $\endgroup$ Sep 21, 2015 at 3:26
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    $\begingroup$ @MrArsGravis: Well what I just described does not distinguish between smooth curves, since it only quantifies the 'bentness of corners'. Yours can be made rigorous by just having a constant speed parametrization and taking the standard deviation in distance from centre, and might be what you're looking for since it quantifies deviation from the circle. $\endgroup$
    – user21820
    Sep 21, 2015 at 3:46
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    $\begingroup$ Looking back at this with four more years of experience doing mathsy stuff, both the smoothness (i.e. $\int \left| \frac{\partial \vec{v}(t)}{\partial t} \right| \mathrm{d}t$ for deformations of $S^1$ = in 2 dimensions) and standard deviation characterizations you offer are the obvious and natural answers to this question. There is no word for this AFAIK. Smoothness or circularity, resp., perhaps. Props for mentioning that the parametrization must be constant speed, which should be especially important for strong deformations! $\endgroup$ Dec 7, 2019 at 3:52

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