I have a problem I can't figure out

[CTho9305]

Ha, I bet you came expecting a tech support question .

Consider the fact that e ^ pi * i = -1. Here I take e to be defined by the value you reach with continuous compounding and also defined as the number for which f(x)=e^x is its own derivative.

I take pi as the ratio of the diameter of a circle to the circumference, when the center lies in the same plane as the circle (as in, you dont raise the center point making a cone). Pi can be calculated on a computer to billions of digits using some simple geometric rules.

Now picture curved space - use a cloth with a marble sitting on it as an example. If you draw a circle on this cloth (say, with the center where the marble is) and then measure the radius and circumference along the cloth you will come up with a larger radius than expected. By this reasoning, the curvature of space affects the value of pi.

Following from this, one can reach a couple conclusions:
[*]e depends on the curvature of space
[*]we are in perfectly flat space or space that is conveniently curved precisely to the amount required for the equation to hold

There is no reason at all for the value of e to change.

The obvious answer is that pi won't actually change, but I don't see why it wouldn't. If we measured pi, would it match the calculated pi exactly to the first few billion digits? I have a guess, but my guess doesn't make much sense to me so I won't state it for fear of looking like an even bigger idiot

 

[bizmark]

If I were to venture a guess, I would say that your definition of pi is wrong, and that the *real* definition of pi is actually given by the equation, and that that value simply happens to match up with what we find in circles in our space, which is sufficiently flat that we don't notice any discrepancies.

 

[Armitage]

I would say that your definition of pi is only valid for non-curved space.
Taken another way ... measuring along the cloth is not valid. You have to measure the diameter of the circle in the plane of the circle.

 

[Cattlegod]

Originally posted by: ergeorge
I would say that your definition of pi is only valid for non-curved space.
Taken another way ... measuring along the cloth is not valid. You have to measure the diameter of the circle in the plane of the circle.

his analogy of measuring the cloth in the surface of the cloth signifies mass curving space, so it would be in the plane even though it is cone shaped.

assuming what you have said is true, the reason we do not see a difference is because the actual curving of space may be very small. if the curving is very small, then a large object like the earth would be virtually unmeasureable. this is similar to an algorithm that estimates an integral, you sample area so small over a curved line that it appears to be a flat straight line over those samples, giving you a very accurate estimation.

the reason i say that space isn't curved is because i *think* light travels exactly along the curves of space, a huge object like a star far far away does not even bend light all that much when it passes by. a much smaller object like the earth would bend it even less.

 

[sao123]

The real problem is this...
your equations hold true in planar geometry only. once you leave the plane (flat surface) and go into curve space, the equation changes.
C = 2 * Pi * R
Area = Pi * R * R

Pi is still constant.

C = 2 * Pi * R * Z
Area = Pi * R * R * Z

Where Z is some derivative of the equation of the definition of your new curved space. z = f'(x).
In planar geometry, the Derivitive of a plane surface is 1, since f(x) = linear, so Z = f'(x) = 1.
(1 stands for linear space).

 

[CTho9305]

Originally posted by: sao123
The real problem is this...
your equations hold true in planar geometry only. once you leave the plane (flat surface) and go into curve space, the equation changes.
C = 2 * Pi * R
Area = Pi * R * R

Pi is still constant.

C = 2 * Pi * R * Z
Area = Pi * R * R * Z

Where Z is some derivative of the equation of the definition of your new curved space. z = f'(x).
In planar geometry, the Derivitive of a plane surface is 1, since f(x) = linear, so Z = f'(x) = 1.
(1 stands for linear space).

ah, didn't know that

 

[earthman]

Your result will be consistent at any given point in space, to observable limits. What the results are in some other area of space is irrelevant, since they are never directly comparable. To travel there would take time, which would negate the original result since the universe is not static. Such is my understanding, in any case. Anyway, why does pi have to be consistent everywhere...perhaps it isn't. I remember reading a story about aliens that teleported themselves across the universe to near earth and were confused to discover that pi, which used to be exactly 3 in their original frame of reference, seemed to be some new irresolvable value.