Pi and Curved Spaces - Is it Constant?

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In summary, pi is a mathematical constant and the ratio of the circumference to diameter in a flat, Cartesian space. However, in a curved space, this ratio can vary depending on the curvature, which means that the value of pi may be more or less than the traditional value. This creates a discrepancy between the universal nature of pi and its measurement depending on the space. Additionally, the definition of a circle may also vary in different geometries, making it difficult to accurately measure pi. However, pi is seen as a constant in mathematics and is not dependent on physical properties of space.
  • #1
DaveC426913
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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?
 
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  • #2
But a mathematician will calculate the ratio of circumference to diameter, and come up with pi, won't he?

Not generally.
 
  • #3
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?
 
  • #4
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?
 
  • #5
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.
 
  • #6
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 [tex]\pi[/tex] 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?
 
  • #7
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.
 
  • #8
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.
 
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  • #9
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:
 
  • #10
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.
 
  • #11
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.
 
  • #12
Pi isn't gotten by experiment, it is a mathematical constant!
 
  • #13
DaveC426913 said:
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, [tex]\frac{c} {d} \ne \pi[/tex] But that is only true of the diameter you are measuring is in the circle's direction of movment.
 
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  • #14
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.
 
  • #15
Agreed :smile:
 

FAQ: Pi and Curved Spaces - Is it Constant?

1. What is Pi and why is it important in mathematics?

Pi, represented by the Greek letter π, is a mathematical constant that represents the ratio of a circle's circumference to its diameter. It is an irrational number, meaning it cannot be expressed as a simple fraction, and has an infinite number of decimal places. Pi is important in mathematics because it is used in countless calculations and formulas in geometry, trigonometry, and other branches of mathematics. It also has many real-world applications, such as in engineering and physics.

2. How is Pi related to curved spaces?

In curved spaces, the concept of a circle changes as the curvature of the space changes. In a flat Euclidean space, the ratio of a circle's circumference to its diameter would always be exactly π. However, in a curved space, this ratio changes depending on the curvature of the space. For example, in a positively curved space, such as the surface of a sphere, the ratio would be greater than π, while in a negatively curved space, such as a saddle shape, the ratio would be less than π. This relationship between Pi and curved spaces is important in understanding the geometry and topology of these spaces.

3. Is Pi constant in all curved spaces?

No, Pi is not constant in all curved spaces. As mentioned before, the value of Pi changes depending on the curvature of the space. In general, the more curved a space is, the larger the value of Pi will be. However, there are some exceptions to this rule, such as in certain non-Euclidean geometries where the value of Pi may be different than expected. Overall, Pi is not a universal constant in all curved spaces, but it still plays a crucial role in understanding the geometry of these spaces.

4. How is Pi calculated in curved spaces?

The calculation of Pi in curved spaces is more complex than in flat spaces. In general, the value of Pi is calculated through the use of mathematical equations and formulas that take into account the curvature of the space. For example, in a positively curved space, the equation for the circumference of a circle would be C = 2πr(cos⁡θ), where r is the radius of the circle and θ is the angle of the curve. In a negatively curved space, the equation would be C = 2πrsinh(θ), where sinh is the hyperbolic sine function. These equations show how the value of Pi changes in different curved spaces.

5. How does the concept of Pi in curved spaces relate to the concept of gravity?

The relationship between Pi and gravity is a complex one. In general relativity, the theory of gravity developed by Albert Einstein, gravity is not a force between masses, but rather a curvature of space and time caused by the presence of massive objects. This means that gravity can be thought of as a curved space, where the value of Pi changes depending on the strength of the gravitational field. This idea is still being explored and studied by scientists, but it shows how the concept of Pi in curved spaces can have important implications in understanding the fundamental forces of the universe.

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