If part of a circle is a straight line of 1 meter, what is the minimum circumference of this circle?

Answer

Dear Wim,

A line segment of 1 meter (or whatever length) cannot be part of a circle: it is curved everywhere. It is true that the curvature decreases as the circle is larger. Consider, for example, a large circle on the Earth’s surface: if you look at only one meter from the circumference, the curvature is imperceptibly small. That’s the physical answer: above a certain size, the curvature becomes imperceptibly small. (How big depends on the measurement method.)

While I’m not familiar with the artwork in question, I suspect its creator wants us to conduct a thought experiment. As I wrote, the curvature decreases as the radius increases. This suggests a limit process: in the limit where the radius is infinite, the curvature is zero. However, this leads to a paradox: if the circumference is not curved, how can it be a circle? And if the artist shows us a piece of the circumference, where is the center of the corresponding circle?

This paradox can be disentangled by noting that “the limit of properties of finite circles” does not necessarily correspond to “the properties of an infinite circle”. Although “infinite circle” is correct Dutch, in mathematics it does not correspond to something that is unambiguously defined (at least not in ordinary geometry, as it is taught in school). Hence the paradox.

Now I am very curious about the artist and the work in question: will you let me know in a comment?

Regards,
Prof. dr. Sylvia