3D space where circumfernce = 4*pi*R

[Spinnor]

Is there a 3D space where the ratio of a circles circumference, C, to its radius, R, is C/R = 4*pi ?

In two dimensions I can imagine a 2D surface that I think would work. In cylindrical coordinates let a surface be defined by

z(r,theta) = A*r*sin(n*theta)

For given integer n we can set A so the ratio of C/R is anything from 2*pi to infinity? If we travel on the surface z(r,theta) once around the origin at constant distance R the distance traveled will be greater then or equal to 2*pi*R?

Can we just specify a space by giving the metric of some space? Let us say say that on some surface,

ds^2 = dr^2 + [2*r*d(theta)]^2,

this gives C/R =4*pi?

Can we do something similar in 3D?

ds^2 = dr^2 + [2*r*d(theta)]^2 + [2*r*sin(theta)d(phi)]^2

Does the above define some space such that C/R =4*pi?


Thanks for any help!

 

[HallsofIvy]

In any Euclidean space, the ratio of a circle's circumference to its radius is 2\pi. In any non-Euclidean space, that ratio is not a constant. So, no, there is no space where the ratio is any constant other than 2\pi.

 

[Spinnor]

In any Euclidean space, the ratio of a circle's circumference to its radius is 2\pi. In any non-Euclidean space, that ratio is not a constant. So, no, there is no space where the ratio is any constant other than 2\pi.

What about the first example I gave, why does that fail my requirements?

Thank!