3D space where circumfernce = 4*pi*R

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In summary, the conversation discusses the possibility of a 3D space where the ratio of a circle's circumference to its radius is a constant other than 2π. The conversation concludes that in any Euclidean space, the ratio is 2π and in non-Euclidean spaces, it is not a constant. The first example given does not meet the requirements for a constant ratio.
  • #1
Spinnor
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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!
 
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  • #2
In any Euclidean space, the ratio of a circle's circumference to its radius is [itex]2\pi[/itex]. 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 [itex]2\pi[/itex].
 
  • #3
HallsofIvy said:
In any Euclidean space, the ratio of a circle's circumference to its radius is [itex]2\pi[/itex]. 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 [itex]2\pi[/itex].

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

Thank!
 

1. What is the formula for calculating the circumference of a 3D space where circumference = 4*pi*R?

The formula for calculating the circumference of a 3D space where circumference = 4*pi*R is C = 4*pi*R, where C is the circumference and R is the radius.

2. What does the value of 4*pi*R represent in this formula?

The value of 4*pi*R represents the total distance around the 3D space, also known as the circumference, in terms of the radius (R).

3. Can this formula be used for any 3D shape or only for circles?

This formula can only be used for 3D shapes that have a circular cross-section, such as spheres, cylinders, and cones. It cannot be used for other 3D shapes such as cubes or pyramids.

4. How is this formula related to the formula for calculating the circumference of a circle?

This formula is directly related to the formula for calculating the circumference of a circle, which is C = 2*pi*r, where r is the radius. In this formula, the radius is multiplied by 2*pi, while in the 3D space formula, the radius is multiplied by 4*pi.

5. Can this formula be used to find the volume of a 3D space?

No, this formula can only be used to find the circumference of a 3D space. To find the volume of a 3D space, a different formula, such as V = (4/3)*pi*r^3, would need to be used.

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