Is it possible to define a metric where the value of $\pi$ depends on the size or position of the circle measured?

Question

I was thinking about a metric where the further away you are from the origin, the more stretched the distances get, so for an $R^1$ space $d(n,n+1)$ could be less than $d(n+1, n+2)$.

That way the larger the circle centered on the origin for example in $R^2$, the larger $\pi$ gets, since the circumference is out there in stretched space, but the diameter always goes through less stretched places.

But that's just my guess, and I couldn't construct such a metric, I suspect it would violate the triangle inequality so it wouldn't even be a metric.

Is it possible, and if so, what is a good example for such a metric?

Answer 1

Differential geometry of surfaces is a garden of examples in which the circumference $C(r)$ of a circle of radius $r$ (i.e., the arc length of the set of points at distance $r$ from a given point) is not proportional to $r$.

Two of the most widely-known examples are

• The unit sphere, for which $C(r) = 2\pi\sin r$ (note that a circle of radius $\pi$, or any integer multiple of $\pi$, has circumference $0$);
• The hyperbolic plane of curvature $-1$, for which $C(r) = 2\pi\sinh r$ (note that the circumference grows exponentially with $r$).

In each of these examples, the "circumference function" is independent of the center. That is rarely the case: For most surfaces, the circumference-to-ratio function depends on the center of the circle.

As a matter of language, one normally speaks of "$\pi$" only when the ratio of circumference to radius is constant.