Is a Perfect Circle Possible Given the Limitations of Pi and Space?

[Josh S Thompson]

Since pi is irrational does that mean that a perfect circle could never be produced?
Wouldn't a circle be like limit where the ratio of diameter to circumference approaches pi the radius should be the same in any direction.

 

[Josh S Thompson]

Or the other way around

 

[Chestermiller]

Since pi is irrational does that mean that a perfect circle could never be produced?
Wouldn't a circle be like limit where the ratio of diameter to circumference approaches pi the radius should be the same in any direction.

You are aware that materials are made of atoms and molecules, correct. In such a framework, is it possible to have a perfect anything?

Chet

 

[Josh S Thompson]

You are aware that materials are made of atoms and molecules, correct. In such a framework, is it possible to have a perfect anything?

Chet

Um no,
But a circle must be defined by limits while lines do not

 

[mathman]

A perfect circle or straight line or any other geometric shape is a mathematical abstraction. Physically these can't be constructed, although we can make good approximations.

 

[Josh S Thompson]

A perfect circle or straight line or any other geometric shape is a mathematical abstraction. Physically these can't be constructed, although we can make good approximations.

Agreed in a world with atoms and molecules, but the area of a circle is a limit where as area for polygons are not

 

[phinds]

Agreed in a world with atoms and molecules, but the area of a circle is a limit where as area for polygons are not

What does that have to do with whether or not you can have a perfect circle? A circle is a line. Do you think perfect lines are possible in a world of quantized "stuff" ?

 

[DrewD]

Um no,
But a circle must be defined by limits while lines do not

A circle of radius r is all of the points that lie a distance r from some partucular point. There is no need for limits.

 

[DaveC426913]

+1 for DrewD. The collection of all points in a plane that are the same distance from a central point is a perfect circle. QED.

 

[micromass]

You cannot have perfect circles in reality. Neither can you have perfect lines or perfect triangles. This is not only because the world consists of molecules, but also because the universe is curved. So we will never be able to create a perfect Euclidean circle since our world is not Euclidean.

 

[phinds]

You cannot have perfect circles in reality. Neither can you have perfect lines or perfect triangles. This is not only because the world consists of molecules, but also because the universe is curved. So we will never be able to create a perfect Euclidean circle since our world is not Euclidean.

But wait. I agree that "straight" lines in our universe are not Euclidean straight lines but I thought it was possible to chose a position such that a circle IS a Euclidean circle.

Since I'm basing this on my understanding of the Reimann geometry surface used in pop science to "picture" a black hole's effect on space-time, I certainly could be wrong, but if you look at that surface (trumpet shaped) you can see that nowhere on it could you construct a Euclidean straight line, but there IS a way, by circumscribing the "horn", to draw a line that would be a Euclidean circle.

 

[micromass]

But wait. I agree that "straight" lines in our universe are not Euclidean straight lines but I thought it was possible to chose a position such that a circle IS a Euclidean circle.

Since I'm basing this on my understanding of the Reimann geometry surface used in pop science to "picture" a black hole's effect on space-time, I certainly could be wrong, but if you look at that surface (trumpet shaped) you can see that nowhere on it could you construct a Euclidean straight line, but there IS a way, by circumscribing the "horn", to draw a line that would be a Euclidean circle.

Do you mean that the embedding of the set in the ambient space would be a circle in the ambient space? If not, you'll need to include a picture.

 

[phinds]

Do you mean that the embedding of the set in the ambient space would be a circle in the ambient space? If not, you'll need to include a picture.

I have no idea what that means. Here's a pic with a circle shown in blue

 

[micromass]

The area of that circle is not really $\pi r^2$. Neither is the length $2\pi r$. So I don't know if you can call that a Euclidean circle. It certainly is a Euclidean circle in the ambient space. But people living on the manifold will not find this circle very Euclidean. Besides, there is no ambient space in GR.

 

[phinds]

The area of that circle is not really $\pi r^2$. Neither is the length $2\pi r$. So I don't know if you can call that a Euclidean circle. It certainly is a Euclidean circle in the ambient space. But people living on the manifold will not find this circle very Euclidean. Besides, there is no ambient space in GR.

OK. I don't follow that technically, but I believe you. Thanks.

 

[Cruz Martinez]

I have no idea what that means. Here's a pic with a circle shown in blue

View attachment 85548

That thing looks like an ordinary circle in the ambient euclidean space it is embedded. What you're saying is very strange, i.e. it makes no sense to me.

 

[phinds]

That thing looks like an ordinary circle in the ambient euclidean space it is embedded. What you're saying is very strange, i.e. it makes no sense to me.

Quite possibly that's because, mathematically, I don't know what I'm talking about (see my previous post).

 

[Cruz Martinez]

Quite possibly that's because, mathematically, I don't know what I'm talking about (see my previous post).

You're thinking about an ambient space. Are you familiar with the fact that spacetime does not naturally live in any ambient space as far as we can tell? It's justa basic GR thing, you don't need a lot of math for that.

 

[micromass]

OK. I don't follow that technically, but I believe you. Thanks.

Don't believe me just like that. Here's a question: on the figure you linked, can you show me what the center and the radius of the circle is?

 

[WWGD]

Maybe we could use the area form (of the universe, given the assumptions) to add to Micromass' argument about the area and diameter of the circle.

 

[phinds]

You're thinking about an ambient space. Are you familiar with the fact that spacetime does not naturally live in any ambient space as far as we can tell? It's justa basic GR thing, you don't need a lot of math for that.

Don't know what an "ambient space" is. I'll check it out.

 

[phinds]

Maybe we could use the area form (of the universe, given the assumptions) to add to Micromass' argument about the area and diameter of the circle.

Good point. I see exactly what you mean. Focusing on the circumference isn't all that meaningful, which I didn't think about until you pointed it out. Thanks.

 

[WWGD]

Don't know what an "ambient space" is. I'll check it out.

It is the space where the object "lives". Take a circle , the circle can be a circle in the plane, or a circle in higher dimension, or a circle that is contained in a space, the ambient space.

 

[phinds]

It is the space where the object "lives". Take a circle , the circle can be a circle in the plane, or a circle in higher dimension, or a circle that is contained in a space, the ambient space.

So, if I understand it correctly, a line of longitude on the Earth, considering only the Earth's surface, is NOT a circle because there is no center of that "circle" in that ambient space. But if you consider the sphere as part of a 3D ambient space, then the same line IS a circle, yes?

 

[WWGD]

So, if I understand it correctly, a line of longitude on the Earth, considering only the Earth's surface, is NOT a circle because there is no center of that "circle" in that ambient space. But if you consider the sphere as part of a 3D ambient space, then the same line IS a circle, yes?

Correct, modulo some technical details. Please give me some time and I will try to come up with a clearer explanation. You see, a circle can be seen from the point of view of topology, geometry, etc. In topology, distances do not matter, and any object you get by stretching, bending and doing "continuous transformations" of your standard circle is still a circle. In geometry, distance does matter. So let me see...

 

[Josh S Thompson]

A circle of radius r is all of the points that lie a distance r from some partucular point. There is no need for limits.

You need limits to define the area of a circel

 

[Josh S Thompson]

+1 for DrewD. The collection of all points in a plane that are the same distance from a central point is a perfect circle. QED.

How do you define "point in a plane"? you must do it with limits.

 

[lavinia]

How do you define "point in a plane"? you must do it with limits.

Points can be defined as intersections e.g. of two straight lines.

 

[DrewD]

JST, you do need limits (or similar ideas) to compute the are of a circle. In geometry, the existence of points is usually taken as an axiom. If you choose instead to define the plane as $\mathbb{R}^2$, then you are correct, since the construction of the reals requires limits.

But none of that really matters if you are considering the physical construction of a perfect circle. If you want to actually physically construct any object, it will always have flaws (or at least there will be uncertainty about the perfection of the object). You recognized that. So it has nothing to do with whether or not limits are needed. It doesn't matter whether $\pi$ is rational or irrational. It is true that limits are needed to compute the area of a circle. It is true that a perfect circle cannot be physically construct. The two facts are not related.

 

[DaveC426913]

How do you define "point in a plane"? you must do it with limits.

Yes. What's your point?

You asked if a perfect circle can exist. Are you asking if a perfect circle can be physically rendered?