# Is a perfect circle possible?

1. Jul 4, 2015

### 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.

2. Jul 4, 2015

### Josh S Thompson

Or the other way around

3. Jul 4, 2015

### Staff: Mentor

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

4. Jul 4, 2015

### Josh S Thompson

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

5. Jul 4, 2015

### 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.

6. Jul 4, 2015

### Josh S Thompson

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

7. Jul 4, 2015

### phinds

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" ?

8. Jul 4, 2015

### DrewD

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.

9. Jul 4, 2015

### 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.

10. Jul 5, 2015

### 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.

11. Jul 5, 2015

### phinds

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.

12. Jul 5, 2015

### micromass

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.

13. Jul 5, 2015

### phinds

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

14. Jul 5, 2015

### 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.

15. Jul 5, 2015

### phinds

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

16. Jul 5, 2015

### Cruz Martinez

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.

17. Jul 5, 2015

### phinds

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

18. Jul 5, 2015

### Cruz Martinez

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 dont need a lot of math for that.

19. Jul 5, 2015

### micromass

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?

20. Jul 5, 2015

### 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.

21. Jul 5, 2015

### phinds

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

22. Jul 5, 2015

### phinds

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.

23. Jul 5, 2015

### WWGD

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.

24. Jul 5, 2015

### phinds

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?

25. Jul 5, 2015

### WWGD

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...

Last edited: Jul 5, 2015