Pi and Curved Spaces - Is it Constant?

[DaveC426913]

pi is the ratio of circumference to diameter.That is true in a flat, Cartesian space.

In a curved space, a physicist measuring the ratio of the circumference of a circle to its diameter will come up with a different value, which will be more or less than pi depending on which way his space is curved.

But a mathematician will calculate the ratio of circumference to diameter, and come up with pi, won't he?

So, pi is universal? But our measurement of it depends on our space?

 

[Hurkyl]

But a mathematician will calculate the ratio of circumference to diameter, and come up with pi, won't he?

Not generally.

 

[mathman]

To Dave. What is it that bothers you? You made two statements that are true. (1)Pi is the ratio (in a flat plane) of the circumference to the diameter of a circle. (2)In a curved space this ratio can be more or less than pi depending on the curvature. What is the problem?

 

[Icebreaker]

Then "pi" on Earth is literally different from "pi" in free space? And if I really measured it with instruments sensitive enough, you will actually see the difference?

 

[whozum]

no, it depends on what kind of geometry system you are using. if you drew a circle on a horse's saddle and measured the ratio of the diameter to the circumference you wouldn't get pi.

C = pi*d only on a flat plane.

 

[Jameson]

Right, the "circle" has to be a locus of points equidistant from the center... and this in many cases just cannot work in real world applications. But \pi can be found in real world measurements.

In a curved space, a physicist measuring the ratio of the circumference of a circle to its diameter will come up with a different value, which will be more or less than pi depending on which way his space is curved.

Is this really the definition of a circle though? Technically a circle is a two dimensional shape, but does this even fall under a 3-D representation of a circle?

 

[DaveC426913]

What I mean is that pi the calculation is not dependent on geometry. (Many formulae for calc'ing pi are found in the Wikipedia entry).

So, theoretically, a physicist could tell the curvature of the space he exists in by (very!) carefully measuring the diameter of a circle and comparing it to the circumference. If his space is actually curved, he will get a value != pi.

 

[DaveC426913]

This all comes from reading Sagan's book 'Contact' (decades ago). If you recall the ending, our heroine discovered a message embedded in the value for pi that was clearly contrived by an intelligent entity.

I'm trying to figure out what thing exactly was manipulated in order for this to happen.

If pi were different depending on the geometry of space, then our I.E. could simply have subtlely bent space so as to make the correct numbers show up. But our heroine was not measuring pi, she was calculating it. So what was the thing that was manipulated by I.E.?



Not that all of this is about a piece of fiction, I'm after a more philosophicxal question about what does it mean to the universe to have a fixed ratio between a diameter and a circumference? Again - not as a measurement, but as a calculation.

 

[eNathan]

I never thought about this. According to the theory of relativity, pi would be different when measuring circles when v is near c. I can calculate it but what's the point... :wounder:

 

[matt grime]

Pi is fixed as the ratio between the diameter of a circle and its circumference *in the euclidean sense*. Ok, pi doesn't vary, but the idea of what a "circle" is does in different geometries depending on the metric. Pi is more properly a constant of integration. It can also be calculated using series to some degree of precision given enough time.

 

[DaveC426913]

I dunno.
Your ruler will compress just as the circle you're measuring will.

Hmm. Which means it's the same in curved space. which means that, even in curved space, you ought to get a measurment = pi.

 

[matt grime]

Pi isn't gotten by experiment, it is a mathematical constant!

 

[eNathan]

I dunno.
Your ruler will compress just as the circle you're measuring will.

Hmm. Which means it's the same in curved space. which means that, even in curved space, you ought to get a measurment = pi.

Yes, relavent to yourself pi would be the same because length does not contract for you own reference frame.

But if you measure a circle which is in movment, \frac{c} {d} \ne \pi But that is only true of the diameter you are measuring is in the circle's direction of movment.

 

[HallsofIvy]

1. pi is a mathematical constant- that's true but ignores the original definition of pi. IF you take pi to be that number, then it, like anything else in mathematics is independent of what physical properties space has.

2. IF by "pi", you mean the ratio of the circumference of any circle to its diameter, then pi is only defined in Euclidean space. In non-Euclidean space, circles of different diameter will not have the same ratio of circumference to diameter.

 

[eNathan]

Agreed