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Old 10-04-2019, 08:48 PM
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Pi is a transcendental number


For an infinitely small circle with an area greater than zero, is pi still defined?
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Old 10-04-2019, 08:58 PM
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For an infinitely small circle with an area greater than zero, is pi still defined?
Your question has a logical contradiction, and if it doesn't, is certainly not answerable in the domain of real numbers.
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Old 10-04-2019, 08:59 PM
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For an infinitely small circle with an area greater than zero
Good luck with that.
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Old 10-04-2019, 09:04 PM
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For a circle with an infinite area, is pi still defined?
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Old 10-04-2019, 09:05 PM
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Assuming that "infinitely small" means "arbitrarily close to zero" - sure. It's pi. The ratio of diameter to circumference is always a constant value, so the limit as diameter-> zero is easy to find
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Old 10-04-2019, 09:23 PM
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Originally Posted by EastUmpqua View Post
For a circle with an infinite area, is pi still defined?
If it has an infinite area, it's not a circle. Same if it has an area of zero.

It sounds to me like you're trying to ask "what is the limit of the ratio of a circle's diameter to its circumference as the area approaches zero?" In that case, the limit is pi.
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Old 10-04-2019, 09:32 PM
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In addition, this has nothing to do with the pi being transcendental. The ratio of the perimeter of a square to one of its sides is 4 no matter how big or small. 4 is an integer. The ratio of the perimeter to the diagonal is 2*sqrt(2) an irrational number but not transcendental.

Last edited by OldGuy; 10-04-2019 at 09:32 PM.
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Old 10-04-2019, 10:26 PM
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Originally Posted by EastUmpqua View Post
For an infinitely small circle with an area greater than zero, is pi still defined?
For a circle with an infinite area, is pi still defined?
The phrasing of these questions makes it sound as if you have the idea that each circle has its own value of pi. I think the mainstream view is that pi is a constant.
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Old 10-04-2019, 10:41 PM
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It sounds to me like you're trying to ask "what is the limit of the ratio of a circle's diameter to its circumference as the area approaches zero?" In that case, the limit is pi.
Is that the same if the area approaches infinity?

Last edited by EastUmpqua; 10-04-2019 at 10:45 PM. Reason: Clarity
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Old 10-04-2019, 10:47 PM
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It sounds to me like you're trying to ask "what is the limit of the ratio of a circle's diameter to its circumference as the area approaches zero?" In that case, the limit is pi.

Is that the same if the area approaches infinity?
Yes. Pi is Pi, by definition; it happens to be the ratio of a circle's diameter to its circumference if that circle is drawn on a flat plane, no matter how large or small, as long as the circle has some nonzero and finite area; if you draw a circle on a sphere or some other non-flat surface, the ratio of its diameter to its circumference will fail to be Pi. Pi doesn't change, you just drew something which isn't a circle on a flat plane.
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Old 10-04-2019, 10:51 PM
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What does any of this have to do with pi being transcendental
?
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Old 10-04-2019, 10:54 PM
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What does any of this have to do with pi being transcendental
?
Absolutely nothing.

Say it again!
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Old 10-04-2019, 10:58 PM
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It sounds to me like you're trying to ask "what is the limit of the ratio of a circle's diameter to its circumference as the area approaches zero?" In that case, the limit is pi.

Is that the same if the area approaches infinity?
I'm not sure that I understand your question. By definition the circumference of a circle is given by C = πD and so the ratio is C/D = π. Are you asking what the limit of a constant is as some other value approaches infinity?
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Old 10-04-2019, 11:06 PM
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What does any of this have to do with pi being transcendental
?
Pretty much answered in #7 -- nothing.
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Old 10-05-2019, 12:30 AM
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Might want to be clear on what sort of limit you are trying to compute. Limits have a particular definition so how you ask the question is pretty important. This is true of most things in math. You might think you are asking one question but a mathematician may parse it differently.

Also have no clue what this has to do with pi being a transcendental number.
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Old 10-05-2019, 02:47 AM
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It may help to learn just what is a transcendental number: "In mathematics, a transcendental number is a real number or complex number that is not an algebraic number—that is, not a root (i.e., solution) of a nonzero polynomial equation with integer coefficients. The best-known transcendental numbers are π and e."

Pi (π) is transcendental and irrational but it is a RATIO of constants, only one of which can be an integer. I doubt that pi floats around much when its circle changes size from infinite (uncountable) to infinitesimal (negligible), not even if warped into extra dimensions. Well, in biblical text, a well's diameter may be 3x its diameter, but that's a miracle. And no, Illinois didn't pass a law that pi=3.0.
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Old 10-05-2019, 06:17 AM
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Absolutely nothing.

Say it again!
Pi, Huh!

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Old 10-05-2019, 07:59 AM
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For any circle in flat space with non-zero finite size, the ratio of the circumference to the diameter is the same, and is equal to a single value somewhere in between 3.1415926535897932384 and 3.1415926535897932385. We can then take the limit of this as the diameter approaches infinity, or zero, or anything else, because a constant is the easiest thing in the world to take a limit of. And if you don't understand what I'm talking about with limits, then you need to take a calculus course before you're even qualified to ask the questions you're asking.

Oh, and don't get too hung up on that qualification of "in a flat space". The real world isn't a flat space, and that's completely irrelevant to this question, because the real world doesn't contain any perfect circles, either. The only perfect circles are the abstractions we talk about in math, and we can set them in whatever kind of space we want, and the space most often wanted (because it's the simplest one) is a perfectly flat space.
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Old 10-05-2019, 08:30 AM
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For any circle in flat space with non-zero finite size [...]

Oh, and don't get too hung up on that qualification of "in a flat space". The real world isn't a flat space[...]
I get more hung up on the idea that Pi is the ratio of two dimensions of a circle. I like the second sentence of Wikipedia's article on Pi:
"Originally defined as the ratio of a circle's circumference to its diameter, it now has various equivalent definitions and appears in many formulas in all areas of mathematics and physics."

For one thing, circles don't have Pi as the exact ratio of their circumference to diameter if there is any mass in the neighborhood, which Chronos may have been alluding to. This might appear a fastidious nitpick, but hey, we're talking about a number that has been calculated to "many trillions of digits" (same article).

IMHO the definition in terms of the ratio of two dimensions sounds almost like an experimental result.
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Old 10-05-2019, 08:51 AM
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And no, Illinois didn't pass a law that pi=3.0.
It was Indiana, and the actual story is still dumb.



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For one thing, circles don't have Pi as the exact ratio of their circumference to diameter if there is any mass in the neighborhood, which Chronos may have been alluding to.
Not quite. Chronos was referring to two different things (1) the fact that mathematical abstraction doesn't rely on the real world and (2) to the extent we know anything, it's that the real world doesn't conform to Euclidean geometry (the sort of geometry we teach the kids). Hyperbolic geometry fits better, but at relatively small scales, things act much like they do in classic geometry.

A circle is a circle. The phrase 'mass in the neighborhood' does not compute - there isn't such a thing as 'mass' in geometry.

That leads to the question of how much the real world can be accurately modeled by classic geometry, but that's an entirely different discussion.
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Old 10-05-2019, 08:54 AM
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... circles don't have Pi as the exact ratio of their circumference to diameter if there is any mass in the neighborhood, which Chronos may have been alluding to.
As Chronos notes, circles are an abstract concept, unaffected by mass. In the real world we find at best only rough approximations to them.


ETA: What G.A. said.

Last edited by Xema; 10-05-2019 at 08:55 AM.
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Old 10-05-2019, 09:14 AM
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It was Indiana the actual story is still dumb.
Dumb indeed. It seems to be a tale of an awkward attempt to designate a rational substitute for an irrational number.

Which is curious, because useful rational approximations have been around since antiquity. Everyone knows of 22/7, accurate to about 1 part in 2500. Much better is 355/113, accurate to better than 1 part in 11 million.
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Old 10-05-2019, 10:38 AM
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One certainly can, if one desires, speak of circles in spaces with curvature, such as one in the vicinity of a mass and affected according to Einstein's equation. And if one speaks of such circles, then the ratio of that circle's circumference to its diameter will not be equal to pi. But one seldom has need to speak of such.
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Old 10-05-2019, 10:45 AM
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I got curious and went off looking around for more on the subject, and I turned up the following absolute gem:
"I fundamentally believe that it's related to the curvature of space-time itself. If space-time had a different curvature, I believe that would have a different value. For our 3-dimensional space/time pairing, I suspect that 3.00000000000. . . would be the value if the cosmological constant were exactly one."
[https://www.physicsforums.com/thread...ce-time.9869/]
This is such a breathtaking stake in the sand of physics forums that I don't think I can contribute a single thing to make it glisten any brighter.

Last edited by Napier; 10-05-2019 at 10:46 AM.
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Old 10-05-2019, 11:34 AM
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Originally Posted by Napier View Post
For one thing, circles don't have Pi as the exact ratio of their circumference to diameter if there is any mass in the neighborhood, which Chronos may have been alluding to. This might appear a fastidious nitpick, but hey, we're talking about a number that has been calculated to "many trillions of digits" (same article).
Pi is the mathematical definition of the ratio. It is related to the physical result of measuring a specific circle, but measurements belong in the realm of physics, not math. Pi is always exactly pi on a theoretical two-dimensional plane. The measured ratio of a diameter of a circle to its perimeter in our three-dimensional world is never exactly pi. Neither finding contradicts or nitpicks the other.
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Old 10-05-2019, 12:27 PM
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Originally Posted by Qadgop the Mercotan View Post
Pi, Huh!
What is it good for?
Absolutely nothin!

Pi ain't nothin but a Dedekind partition,
Friend to no one but the mathematician.
Oh, pi has shattered many a Rationalist's dreams,
Made him confused, bitter and mean.
Life is much too short and precious
To spend calculating digits these days.
Pi can't give a belief that all numbers have a finite representation,
It can only take it away.
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Old 10-05-2019, 12:55 PM
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I got curious and went off looking around for more on the subject, and I turned up the following absolute gem:
"I fundamentally believe that it's related to the curvature of space-time itself. If space-time had a different curvature, I believe that would have a different value. For our 3-dimensional space/time pairing, I suspect that 3.00000000000. . . would be the value if the cosmological constant were exactly one."
[https://www.physicsforums.com/thread...ce-time.9869/]
This is such a breathtaking stake in the sand of physics forums that I don't think I can contribute a single thing to make it glisten any brighter.
Note that quote is from fringe theories, as stated by the author:


Quote:
Disclaimers--Please Read: The above "title," -pH D," is not an academic title, but a joke (the academic title would be Ph.D. or PhD). Such as a joke may be. Also, the concepts proposed in this paper are strictly my own conjectures, and are not, so far as I know, theories held by any legitimate physicist of the creditable physics community. They are offered purely as wild speculation, food for thought, and absolutely should not be represented as fact and may be utterly diametrically opposed to reality, although I do include factual information (look it up if you don't know!). Finally, the contents of this are copyright April, 2000 by John A. Knouse, and may not be used or represented without attribution nor for commercial purposes without my explicit consent.


Circles are not rotationally invariant in 4D spacetime so that whole question really has to go back to your choices of projection etc....

In Riemannian geometry, the diameter of a circle is pi while it would be 2 in the Euclidean space.

It also would depend on how you define a circle. If you define it as a line of constant curvature with no endpoints, as several mathematical fields do, and have infinite radius you have a line. If you choose to Lie transformations can turn circles to lines or points, and points are merely zero radius circles.

Note how the answer the OP can change depending on if you choose the Projective, Mobius, Riemannian, high school, or .... geometry?

Even if you move to the extremes of hyperbolic geometry circumference is still "2*pi* sinh(r)" and even pi is greater than 3.14 it doesn't matter. If spacetime is so curved as to limit your ability to simplify to the euclidean case two observers will simply not agree on anything but the spacetime interval between any two events . As spacetime is the ONLY invariant one events are not local, and because it is a scalar yet still a vector field there is another fun place you can find pi hiding in with a "infinitely small circle with an area greater than zero" even if that fact is masked through the tensor calculus na diff geometry conventions.

The question is if it is convenience or convention that cause SO(1,n), SO(n), and SU(2) to be used in relativity and quantum mechanics. I personally don't know if that is answered or answerable but the value of pi itself is far less important than the properties of Lie groups when working with infinitesimals and the convenience of using Euclid's third postulate and Euler's Identity to solve very hard problems.

Anyway I can't show the math on this here but the more mainstream view is that under strong curvature it is most likely that pi > 3.14 in higher dimensions.

Last edited by rat avatar; 10-05-2019 at 12:57 PM.
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Old 10-05-2019, 01:09 PM
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If you use extended reals with infinitesimals, then the OP's question makes sense, but the ratio is still pi.
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Old 10-05-2019, 02:25 PM
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. . . If you define a circle as a line of constant curvature . . .
Are circles and straight lines the only curves that have constant curvature? That seems true for Euclidean geometry (although I would have no idea how to prove it) but wonder if there might be some 'novel' constructions that could also satisfy the condition.

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Old 10-05-2019, 02:47 PM
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Are circles and straight lines the only curves that have constant curvature? That seems true for Euclidean geometry (although I would have no idea how to prove it) but wonder if there might be some 'novel' constructions that could also satisfy the condition.
Don't helices have constant curvature?
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Old 10-05-2019, 02:57 PM
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Are circles and straight lines the only curves that have constant curvature? That seems true for Euclidean geometry (although I would have no idea how to prove it) but wonder if there might be some 'novel' constructions that could also satisfy the condition.
Circles and lines are the only smooth curves of constant curvature in a plane in Euclidean geometry.

I am not sure what happens with higher dimensions or with unit circles etc...so someone else will have to answer.

Obviously a sphere would be smooth curves of constant curvature in 3 dimensions as would a plane
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Old 10-05-2019, 02:58 PM
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Don't helices have constant curvature?
Indeed. I meant in the plane.
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Old 10-05-2019, 03:09 PM
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Don't helices have constant curvature?
Sure, but you can specify an arbitrary torsion and get different curves. Anyway, this is in three dimensions, not the Euclidean plane. To make things interesting, if you wanted you could certainly consider curves in the three-dimensional sphere and other Riemannian manifolds; that makes perfect sense.
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Old 10-05-2019, 03:15 PM
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ETA to define a circle, AFAIK it is the set of all points in the plane at a fixed distance from some specified point. That is, you have the centre and the radius.
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Old 10-05-2019, 07:46 PM
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It was Indiana, and the actual story is still dumb.
See, I was right. Illinois didn't.

And yes, I knew it was Indiana. That was deliberate. I possess little shame, just enough to turn me from posting snide, fringe-y jokes about circles. [bad taste omitted]
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Old 10-05-2019, 08:57 PM
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Would it be threadshitting to say that the question is not well-formed?
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  #37  
Old 10-05-2019, 08:58 PM
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ETA to define a circle, AFAIK it is the set of all points in the plane at a fixed distance from some specified point. That is, you have the centre and the radius.
Though you should specify that you're using Euclidean distance - with a different metric you'd get a different shape and possibly a different number of points. With a taxicab metric the set of all points at a fixed distance from a specific point would look square, and if the space was discrete, you would have only a finite number of points on that square (possibly zero).
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Old 10-05-2019, 09:16 PM
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Quoth rat avatar:

In Riemannian geometry, the diameter of a circle is pi while it would be 2 in the Euclidean space.
What the heck are you talking about here? I can't figure out any way to make this sentence make sense.
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Old 10-05-2019, 09:33 PM
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Though you should specify that you're using Euclidean distance - with a different metric you'd get a different shape and possibly a different number of points. With a taxicab metric the set of all points at a fixed distance from a specific point would look square, and if the space was discrete, you would have only a finite number of points on that square (possibly zero).
If you allow non-Euclidean geometry, then, sure, e.g., following the Bertrand–Diguet–Puiseux theorem, consider a point on a smooth surface and a small geodesic disc centered on it; the Gaussian curvature at that point is equal to the limit, as the radius decreases to 0, of (3/πr3)(2πr - the circumference). Note that this does not yield a "different value of π" for small circles in the limit.

ETA I agree circles may be natural geometric objects of interest not only in Euclidean space

Last edited by DPRK; 10-05-2019 at 09:36 PM.
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Old 10-05-2019, 09:35 PM
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If you allow non-Euclidean geometry, then, sure, e.g., following the Bertrand–Diguet–Puiseux theorem, consider a point on a smooth surface and a small geodesic disc centered on it; the Gaussian curvature at that point is equal to the limit, as the radius decreases to 0, of (3/πr3)(2πr - the circumference). Note that this does not yield a "different value of π" for small circles in the limit.
Agreed.
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Old 10-06-2019, 12:25 PM
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What the heck are you talking about here? I can't figure out any way to make this sentence make sense.
It could of been worded better....but I am assuming you didn't catch I was referencing the unit sphere. I deleted Cp and other math notation to avoid losing viewers.

The generalized diameter is the greatest distance between any two points on the boundary of a closed figure.

In Euclidean space, the shortest distance between two antipodal points on the unit sphere is 2; as expected.

If one naively copies that generalized definition of diameter onto Riemannian geometry:

The shortest distance between two points on the surface of a sphere, as measured along the surface of the sphere, an orthodrome or a segment of a great circle. The shortest distance between two antipodal points on the unit sphere is π (Pi).

As this example only depended on Differential Geometry I expect most math fans probably picked up on my admitted opaque context. Define your terms and check your assumptions when switching domains is the point.

Edit to add for other responses: Note due to p = 2 being the only Lp group with SO(3) our value of pi 3.14.... is the lowest value it has as p > 2 loses rotational invariance.

Last edited by rat avatar; 10-06-2019 at 12:28 PM.
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Old 10-06-2019, 02:28 PM
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Why are you measuring across the sphere in Euclidean geometry, but along the sphere in Riemannian geometry? You can measure in either way in either geometry. And the Riemannian result approaches the Euclidean one, in the limit where the size of the sphere is much less than the radius of curvature.
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Old 10-06-2019, 04:18 PM
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Why are you measuring across the sphere in Euclidean geometry, but along the sphere in Riemannian geometry? You can measure in either way in either geometry. And the Riemannian result approaches the Euclidean one, in the limit where the size of the sphere is much less than the radius of curvature.

Because in that case the chord metric is not an intrinsic metric or length metric. In this case the induced intrinsic metric uses the great-circle distance. Sure you could respond with the chordal distance when someone asks you how far it is from NYC to London but until someone develops low cost robotic tunneling machines it doesn't do them very good to travel between the two. Chordal distance or Euclidean Distance is little value to travel on that curved surface either.

Note the generalized definition I linked to.

Quote:
"greatest distance between any two points on the boundary of a closed figure"
While the Euclidean metric defines the distance between two points as the length of the straight line segment connecting them. That Euclidean metric is not assumed in non-Euclidean geometry.

Obviously math proofs are not offered in plain English so you can find some wording mistake to pick at, but why do you assume that a geometry that is not Euclidean will follow Euclidean rules? Heck even the taxicab metric in Euclidean space would be an example were you may use different definitions of "distance"...which is what a metric is.

Last edited by rat avatar; 10-06-2019 at 04:21 PM.
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Old 10-06-2019, 05:13 PM
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I'm assuming that by "Riemannian geometry" you mean "Riemannian geometry", and that by "pi" you mean "pi", and so on. The problem is not that I'm not understanding what you're saying in words; the problem is that what you are saying in words is just plain wrong.
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Old 10-06-2019, 05:27 PM
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I guess he is saying that if you stand somewhere on a positively curved surface, measure out a small circle, and divide the circumference by the diameter, you will obtain a number ever so slightly less than "pi". Which is true...
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Old 10-06-2019, 06:59 PM
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I'm assuming that by "Riemannian geometry" you mean "Riemannian geometry", and that by "pi" you mean "pi", and so on. The problem is not that I'm not understanding what you're saying in words; the problem is that what you are saying in words is just plain wrong.
This is General Questions, how about offering a cite vs simply offering a retort due to personal incredulity?

Non-Euclidean spaces have Circles have a perimeter/diameter ratio that is different from pi. I chose a special case of a Riemannian circle

They probably pulled the idea from Springer's intro to smooth manifolds but they agree with the jist of my original claim.

Quote:
The diameter of the Riemannian circle is π, in contrast with the usual value of 2 for the Euclidean diameter of the unit circle.
I get that Riemannian geometry is still mostly on dead trees so feel free to cite books if you wish.

The fact that the perimeter/diameter ratio == pi in the Euclidean case is a special case for Euclidean circles.

Last edited by rat avatar; 10-06-2019 at 07:01 PM.
  #47  
Old 10-06-2019, 07:12 PM
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On a curved surface, you would not expect the circumference divided by the diameter to be constant for all circles, big and small and with different centers, nor the angles of a triangle to add up to pi, etc. To recover a constant you would have to take the limit, which is basically the Euclidean case.
  #48  
Old 10-07-2019, 12:50 PM
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Quote:
Originally Posted by rat avatar View Post
This is General Questions, how about offering a cite vs simply offering a retort due to personal incredulity?

Non-Euclidean spaces have Circles have a perimeter/diameter ratio that is different from pi. I chose a special case of a Riemannian circle

They probably pulled the idea from Springer's intro to smooth manifolds but they agree with the jist of my original claim.



I get that Riemannian geometry is still mostly on dead trees so feel free to cite books if you wish.

The fact that the perimeter/diameter ratio == pi in the Euclidean case is a special case for Euclidean circles.
I think the problem is terminology. Riemannian geometry describes a class of different geometries, where the metric's behaviour is restricted in a certain way. Euclidean geometry and spherical geometry are both examples of Riemnnian geometries. Also when you say circle, what you mean specifically a circle with circumference 2pi*r.
  #49  
Old 10-09-2019, 10:30 AM
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Quote:
Originally Posted by Chronos View Post
I'm assuming that by "Riemannian geometry" you mean "Riemannian geometry", and that by "pi" you mean "pi", and so on. The problem is not that I'm not understanding what you're saying in words; the problem is that what you are saying in words is just plain wrong.
What can you do when a ≠ a?
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