# Can a perfect circle exist in the real world?

## Main Question or Discussion Point

The (perfect) circle, defined in the Cartesian coordinates as the set of (x,y) pairs that fit the equation x^2 + y^2 = r^2 can "exist" as a mathematical abstraction, I have no problem with that. But can we have a perfect circle in the physical world? Particularly, can an object move in a perfectly circular path? The way I see it, for the object to move in such a way it would have to change direction an infinite amount of times per cycle. Even in a fraction of a cycle it would have to change direction an infinite amount of times; isn't this a logical impossibility? Can we have perfect circular MOTION?

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phinds
Gold Member
2019 Award
It would seem to depend on two things

First, is space-time granular or continuous. This is an open question

Second, are there physical objects that could travel in such an exact path all the way down to the quantum level. I don't know of any.

I think you need to just live with the fact that mathematical shapes are perfect but do not describe the real world down to that level.

rcgldr
Homework Helper
Somewhat of a combination of real world and mathematical model, what about the equal potential sphere around a point charge such as an electron, except that would assume that an electron was a perfect sphere?

Drakkith
Staff Emeritus
No, you cannot have perfect circular motion or perfectly circular objects in the real world. A good example is an orbit. Perfect circular orbits do not exist in the real world since all objects are subject to the gravitational influences of a near-infinite number of objects at varying distances and will not follow the exact circular path through space required for a perfectly circular orbit.

Drakkith
Staff Emeritus
Somewhat of a combination of real world and mathematical model, what about the equal potential sphere around a point charge such as an electron, except that would assume that an electron was a perfect sphere?
Seeing as how the potential around a real electron is influenced by the electric field of other particles, I don't think potential is perfectly spherical.

Staff Emeritus
2019 Award
Considering that the real universe is quantum mechanical, even if you did have a perfect circle, you could never tell that it was in fact perfect. (You can't even tell that it lies in a plane)

Does the ratio of a circle's circumference to its diameter have to equal pi? Or does having its boundary points equidistant from its center qualify?

TumblingDice
Gold Member
Does the ratio of a circle's circumference to its diameter have to equal pi?
Yes.
Or does having its boundary points equidistant from its center qualify?
Yes. I'd say it's the second that defines the circle, which results in the first being true.

My \$.02 towards the OP is that physics is an objective discipline while "perfect" is a subjective term. As V50 pointed out, debatable down to QM and the uncertainty principal.

How about, "Can a "perfect" straight line exist in the physical world?" Easier to contemplate than a circle, yet still maybe something to think of while meditating, not in the physics lab. Jimmy said:
Does the ratio of a circle's circumference to its diameter have to equal pi? Or does having its boundary points equidistant from its center qualify?
I'd say it's the second that defines the circle, which results in the first being true.
If you draw an object on the surface of a sphere which meets the second definition, then the first isn't true*. Could you call that a circle?

* Something other than a great circle...

Edit: I'm considering only the surface of a sphere in contrast to a small circle which meets both definitions. Sorry for any confusion.

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TumblingDice
Gold Member
If you draw an object on the surface of a sphere which meets the second definition, then the first isn't true*. Could you call that a circle?
I'm not sure where you're headed with your recent questions and the introduction of drawing circles on spheres.

Would opening a new thread be better, or are you preparing another perfect circle opinion? I'm gonna' shy away from discussing and pondering the possibility of anything perfect in the physical world. :tongue:

Sorry, I don't mean to be coy. I was considering non-euclidean geometry and whether a circle can even exist in such a geometry. That's why I was trying to pin down the definition of a circle. However, I think there's enough information in this thread to conclude that perfect circles don't exist.

I'm rarely in the mood for internet discussions but sometimes I get the urge. I'll go crawl back in my hole now. :tongue:

TumblingDice
Gold Member
Jimmy: No crawling into holes allowed! It was good to "meet" you in this thread. ("How do you do?" :rofl: ) Looking forward to catching you in future threads and posts.

EDIT: BTW, Since a sphere is perfectly symmetrical, a circle is still a circle when defined on the sphere's surface as perimeter equidistant from its center.

Jimmy: No crawling into holes allowed! It was good to "meet" you in this thread. ("How do you do?" :rofl: ) Looking forward to catching you in future threads and posts.
Ha ha, thanks. I do ok, BTW. Embarrassingly, that expressions seems to be quaint to many ears. I always thought it was a common expression. I guess I'm more posh than I thought. Capital, old bean!

My choice of discussions to join is slightly suspect I suppose.

BTW, Since a sphere is perfectly symmetrical, a circle is still a circle when defined on the sphere's surface as perimeter equidistant from its center.
Thank you.

It would seem to depend on two things

First, is space-time granular or continuous. This is an open question
Yes, I recognize that the nature of space-time is open to debate, however, it seems to me that space and time are abstract coordinates used to describe (label) points in the universe, nothing more, nothing less. Therefore I find it to be a logical fallacy for space-time to be anything other than continuous. Therefore I ask you to assume continuity with regards to the OP.

Second, are there physical objects that could travel in such an exact path all the way down to the quantum level. I don't know of any.
The way I see it, any physical limitations to the question at hand can be divided into two groups:
The first group is the type of limitations that have to do with "what we've got". For instance, what objects do we have? How do they move? If the object is a person, is he drunk? Is he wobbling? If the object is a car, do we have enough fuel to travel the distance? Or are the planets aligned in such a way to even allow for circular motion?
Where in the universe exactly are we, adjacent to the sun or so far away from the furthest boundaries of matter that no perturbation or force has ever set foot?
To be clear, I'm not talking about any of these types of limitations. For instance, we could take all the matter in the universe and consider it as a single object, or just imagine a universe where there's only one object.
The second type of limitation, which is what I'm talking about is whether it is even logical to MOVE in a perfectly circular path. Would it not require changing direction an infinite amount of times for every finite step you take, thus rendering it physically impossible?

My initial verdict on this question was that since you could change directions 3 times, 4 times, 10000 times, etc over a finite distance, but not infinitely, that there could not exist a perfect circular motion. In a forest with finite premises, you can have any finite number of trees, but surely you couldn't have an infinite number of trees in a finite, bounded area of a forest, that would be physically impossible. ( To construct a circle, you would have to have a polygon with infinite number of sides, 3,4 or even a trillion sides won't do. In other words, no amount of finite sides would be a true perfect circle)

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But then I recalled zeno's paradox and how we could have infinite points or divisions over a finite length, provided that the sum of those divisions is finite. In mathematical language, the series could consist of an infinite number of terms but still be summable, i.e. yield a finite sum.

Now let's change our focus from distance to direction (angle). Imagine you are at the origion of the coordinates, facing positive y (north). Now, you could turn 90 degrees counter clockwise to face west, and in doing so, you would've changed your orientation, direction and the angle your orientation makes with the positive x axis an infinite amount of times over the finite 90 degree span. But there is no contradiction, since again the series of changes in angles is summable; the finite span of 90 degrees can be divided into an infinite amount of subdivisions and the span will still be finite, just like a straight line can be divided into an infinite number of segments (and of course still be finite).

So can we somehow combine zeno's paradox regarding displacement and my resolution above to changing directions to explain circular motion, which is a combination of displacement and change in direction?
Is perfect circular motion possible?

phinds
Gold Member
2019 Award
Yes, I recognize that the nature of space-time is open to debate, however, it seems to me that space and time are abstract coordinates used to describe (label) points in the universe, nothing more, nothing less. Therefore I find it to be a logical fallacy for space-time to be anything other than continuous. Therefore I ask you to assume continuity with regards to the OP.
You cannot hand-wave away that it is not known. Regardless of what YOU think I logical or illogical, the universe might not agree w/ you.

The way I see it, any physical limitations to the question at hand can be divided into two groups:
The first group is the type of limitations that have to do with "what we've got". For instance, what objects do we have? How do they move? If the object is a person, is he drunk? Is he wobbling? If the object is a car, do we have enough fuel to travel the distance? Or are the planets aligned in such a way to even allow for circular motion?
Where in the universe exactly are we, adjacent to the sun or so far away from the furthest boundaries of matter that no perturbation or force has ever set foot?
To be clear, I'm not talking about any of these types of limitations. For instance, we could take all the matter in the universe and consider it as a single object, or just imagine a universe where there's only one object.
The second type of limitation, which is what I'm talking about is whether it is even logical to MOVE in a perfectly circular path. Would it not require changing direction an infinite amount of times for every finite step you take, thus rendering it physically possible?
You are missing the point here. All physical objects are either quantum objects or made up of quantum objects at the lowest level and those little suckers are subject to the HUP, so you can't force one to go it an exact circle.

My initial verdict on this question was that since you could change directions 3 times, 4 times, 10000 times, etc over a finite distance, but not infinitely, that there could not exist a perfect circular motion. In a forest with finite premises, you can have any finite number of trees, but surely you couldn't have an infinite number of trees in a finite, bounded area of a forest, that would be physically impossible. ( To construct a circle, you would have to have a polygon with infinite number of sides, 3,4 or even a trillion sides won't do. In other words, no amount of finite sides would be a true perfect circle)
Now you're back into pure math, it seems to me, and your question was about the real world.

phinds
Gold Member
2019 Award
Is perfect circular motion possible?
Not of an actual physical object, for the reasons in my previous post.

No, you cannot have perfect circular motion or perfectly circular objects in the real world. A good example is an orbit. Perfect circular orbits do not exist in the real world since all objects are subject to the gravitational influences of a near-infinite number of objects at varying distances and will not follow the exact circular path through space required for a perfectly circular orbit.
What if we only have one object?

Considering that the real universe is quantum mechanical, even if you did have a perfect circle, you could never tell that it was in fact perfect. (You can't even tell that it lies in a plane)
I don't see how this is to the point. I'm not asking whether we as part of the universe (insiders of the universe) could determine with certainty if the path is a perfect circle, I'm asking if there could ever exist one to be determined (or mistaken for not being circular by us)
In other words, is it POSSIBLE that the path in question may be truly circular? (even if we can't measure it with utmost accuracy)

TumblingDice
Gold Member
What if we only have one object?

In a fictional universe with only one object, assuming there was a way to observe this one object, circular motion would not be possible. According to Newton's first law, every object in a state of uniform motion tends to remain in that state of motion unless an external force is applied to it.

You cannot hand-wave away that it is not known. Regardless of what YOU think I logical or illogical, the universe might not agree w/ you.
Yes, this is why I asked YOU to assume continuous space-time. In other words, are there any other reasons apart from the probable discreteness of space-time to think that perfect circular motion may not be possible?

You are missing the point here. All physical objects are either quantum objects or made up of quantum objects at the lowest level and those little suckers are subject to the HUP, so you can't force one to go it an exact circle.
What do you mean if EYE could force one, why would EYE have to force it? Are you referring to the centripetal force? What if nature creates the force, without me having to make any interventions, measurements or observations? If there's no observer, there's no HUP. Think omnipotent. It's there to be observed. Or it isn't. Either way, we're not gonna do anything about it. Just speculate whether it could possibly be there or not.

Now you're back into pure math, it seems to me, and your question was about the real world.
Yes, in a sense you are right. If it helps, consider it as a purely mathematical inquiry and answer from that viewpoint. Is it logically consistent or logically flawed to assume perfect circular motion for an object or dot?

In a fictional universe with only one object, assuming there was a way to observe this one object, circular motion would not be possible. According to Newton's first law, every object in a state of uniform motion tends to remain in that state of motion unless an external force is applied to it.
Ok, consider it a mathematical question.

My issue is with the infinite number of changes in direction over a finite distance needed to traverse a perfect circle. Can we somehow extend (the resolution to) zeno's paradox to changes in direction combined with displacement (as in circular motion)?

TumblingDice
Gold Member
What do you mean if EYE could force one, why would EYE have to force it? Are you referring to the centripetal force? What if nature creates the force, without me having to make any interventions, measurements or observations? If there's no observer, there's no HUP. Think omnipotent. It's there to be observed. Or it isn't. Either way, we're not gonna do anything about it. Just speculate whether it could possibly be there or not.
Omnipotence isn't an appropriate postulate in physics, and speculation is inappropriate here at PF. Please refer to the forum rules you agreed to when you joined. Also, PF and its members can be very helpful when you have a problem understanding something, and often provide a reference with your question to get the best guidance. However just posting questions and thinking of new questions without doing any research or studying on your own will not get very far here. Just trying to be helpful and want you to enjoy PF!

Dale
Mentor
Closed pending moderation