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

[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.

 

[HallsofIvy]

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

You seem to be confused about "mathematical" constructions and "real world" objects. There is NO "error" or accuracy in a mathematical square or circle. They are, by definition, exact.

 

[juan gce]

We messure in straight lines .. pi .. comes naturally to make messures of a circle .. that simple and plain .. The circle's perimeter in euclidean geometry is 2pi r

 

[Saracen Rue]

Just spit balling here, but do we know if the Higgs Boson is spherical or not? Because if it's spherical, it would have to be a perfect sphere as nothing else comprises it. That would mean it's circumference would be a perfect circle.

Please any mistakes I may have made on this matter, I don't really know much about it, the thought just occurred to me and I decided to share it.

 

[HallsofIvy]

The very concept of "shape" simply does not apply to quantum objects.

 

[aikismos]

Actually, π by definition REQUIRES the perfect circle. From dictionary.com: "Perfect: conforming absolutely to the description or definition of an ideal type."

I think the OP should first take note that the circle (whether physical or conceptual) exists independent of π. That's because π is a number (concept, idea, computation, etc.) derived from two other numbers which conceptually have exact values. Back to the poster who said that the definition of a circle as a locus of point which as an "ideal type" are equidistant! That the ratio of the circumference and diameter are such that when rendered as a non-terminating, non-repeating decimal is wholly irrelevant to the existence of the "ideal type" to begin with. From a computational perspective (disregarding the drivel from neo-Platonic mumbo-jumboists) existence of the perfect circle is actually REQUIRED for the calculation of the ratio of it's parts, and not vice versa. You simply can't have π WITHOUT a perfect circle (abstractly, conceptually, computationally, informationally, etc.). Any disagreement should invoke a gentle reminder to reread the definition of perfect listed above.

Can a circle be rendered perfectly in ambient space? That's much more interesting because it depends on your definition of space. Minkowski space CERTAINLY allows for a perfect space (gravitational field around a singularity comes to mind) because Minkowski space is a mathematical construct which even under relativistic considerations allows for the use of the conceptual concept of the circle in calculations. I think the poster who brought in the effects of the wave function of particles has the strongest claim to deny the existence of the perfect circle at the quantum level (though I'm not familiar enough with the mathematics to state with certainty no such mathematical construct exists). Since QP has not replaced (but complements) RP, you can't claim it's NOT possible in space, only not possible at the quantum scale.

As long as the circle is defined as an idea (a hard notion to escape even if 'things' can 'be' round...), then a perfect circle exists both in the mathematical discourse as well as physical discourse.