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?

 

[WWGD]

Yes. What's your point?

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

Doesn't that depend on what "can" can be (though not on what "is" is)?

 

[DaveC426913]

Doesn't that depend on what "can" can be (though not on what "is" is)?

Yes. Which is why I'm trying to get the OP to more clearly define the parameters and terms of the question.

As it stands, with no restrictions, the answer is: yes a perfect circle can exist. It is simply all points equidistant in a plane from some central point.
Now, whether you can draw them or point to them accurately - that is another question.

 

[WWGD]

Yes. Which is why I'm trying to get the OP to more clearly define the parameters and terms of the question.

As it stands, with no restrictions, the answer is: yes a perfect circle can exist. It is simply all points equidistant in a plane from some central point.
Now, whether you can draw them or point to them accurately - that is another question.

Agreed. This is one of those questions that degenerates quickly without clear definitions.

 

[Josh S Thompson]

Sorry for the misunderstanding I don't know why I left out some words and can't edit it.

The question is: can you observe a perfect circle in the world
and I'm saying you need to define what a point is as an infintly small box if a circle can exsist on a plane.

 

[Josh S Thompson]

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.

these two facts are related
limits are needed to define circles on a plane of real numbers. a perfect circle cannot be physically constructed.

 

[Mark44]

these two facts are related
limits are needed to define circles on a plane of real numbers. a perfect circle cannot be physically constructed.

The first statement is not a fact. A circle is the locus of all points that are equidistant from a central point. This is the definition of a circle. No limits are required.

 

[FactChecker]

The fact that π is irrational has nothing to do with whether a perfect circle is possible. Just because something is hard to write down or represent in rational numbers does not mean it is any more or less possible. That would be like asking: "Because photography is not possible in the dark, does the world cease to exist when you turn the lights out?"

 

[DaveC426913]

The fact that π is irrational has nothing to do with whether a perfect circle is possible. Just because something is hard to write down or represent in rational numbers does not mean it is any more or less possible.

True. The hypotenuse of a 1x1 right triangle is also irrational.

 

[Laurence Cuffe]

True. The hypotenuse of a 1x1 right triangle is also irrational.

A fact which can have lethal consequences, as Hippasus found out! Harking back to the original question, I wonder how many digits of pi are correct for circles close to the surface of the earth?

 

[micromass]

A fact which can have lethal consequences, as Hippasus found out! Harking back to the original question, I wonder how many digits of pi are correct for circles close to the surface of the earth?

The smaller the circle on the surface of the earth, the better the approximation of the perimeter as $2\pi r$. For large circles on the surface of the earth, you have a large deviation from this quantity.

 

[TheDestroyer]

Since pi is irrational does that mean that a perfect circle could never be produced?

You're saying this as if not having an infinite precision number is the only problem of having a perfect circle.

I think many students do this and confuse physics with math. There's a very thick line separating physics from math, and that's where you're wrong.

A circle is a mathematical construct. It could be defined in many ways. For example, a 2D closed object that consists of a line, whose points have the same distance from a single point in the middle. Now with this, we're still talking math, since you're setting rules and spaces. When you try to draw a circle, you're starting there to gloss on physics, and when you gloss on physics, there are basic constraints that you can't overcome due to the limitations of our physics world. The limitation include, for example, that our world is discrete. Even if you manage to align all these atoms/molecules in a perfect way, you still have to consider quantum effects dictated by the uncertainty principle, where atoms and molecules are not localized and their position oscillates all the time.

Even if you manage to overcome all that, which is already impossible, you have to prove you're right by doing a measurement of that circle. Let me tell you that the most accurate measurement in the history of science that shows agreement between theory and experiment is the measurement of the g-factor of the electron using QED. The relative precision is about 10^{-12}. Hence, no perfect measurements are ever possible!

My point it: Your postulation of the problem is incorrect, because you have to understand the difference between math and physics, and you have to consider the limitations of the physics framework, in which you can even prove what you want to do.

 

[DaveC426913]

My point it: Your postulation of the problem is incorrect, because you have to understand the difference between math and physics, and you have to consider the limitations of the physics framework, in which you can even prove what you want to do.

In the OP's defense, I would argue that he does understand the difference between math and physics, and sees the discrepancy, which is why he is posting this question, asking for confirmation of his suspicions.

 

[Chrono G. Xay]

Just because of the wording of the question I feel drawn to post this:

A Perfect Circle is very possible:

 

[write4u]

If we can construct a perfect circle in mathematics, the answer is "Yes, we can".
If we can construct a perfect circle in physics, the answer is "No, we cannot".

Does that sound reasonable?

 

[Josh S Thompson]

so constructing a square and circle in math can be done with equal precision?

 

[DrewD]

so constructing a square and circle in math can be done with equal precision?

In my opinion the word "constructing" is a little confusing because it could refer to "physically" constructing or Euclidean construction with straight edge and compass. A circle and a square can be constructed with equal precision by the second definition*. Mathematically speaking, a square and a circle can be defined and constructed without reference to limits.*if you require that the circle and square share particular traits, eg. area or perimeter, this is not true. If you require nothing or want a circle with a radius equal to the side of a square, then the two can be constructed easily by Euclidean methods.

 

[Josh S Thompson]

What is Euclidean construction with straight edge and compass?

 

[Josh S Thompson]

Basically, I'm saying you need a right angle to construct a square so constructing one would produce similar error to a circle.

But if you have a plane of real numbers then circle would be harder then square

 

[Josh S Thompson]

In my opinion the word "constructing" is a little confusing because it could refer to "physically" constructing or Euclidean construction with straight edge and compass. A circle and a square can be constructed with equal precision by the second definition*. Mathematically speaking, a square and a circle can be defined and constructed without reference to limits.*if you require that the circle and square share particular traits, eg. area or perimeter, this is not true. If you require nothing or want a circle with a radius equal to the side of a square, then the two can be constructed easily by Euclidean methods.

Ok let's just talk 2d shapes in a plane of real numbers. Sorry for the confusion

 

[rootone]

Yes a circle it is a true mathematical construct, involving the universal constant π
No it it is not something which could be 'perfectly' constructed in a physical sense.
Because whatever you try to construct it with with have size and shape properties, even at molecular level.
Having said that we can make very useful mirrors and lenses and other stuff relying on the idea of π and it's consequences being true.