# In what way is a regular polygon less regular than a circle?

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.

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

• 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! Dec 7, 2019 at 3:52