## Main Question or Discussion Point

God knows if I'm posting this in the right place on physics forum but here goes...

If a circle can be thought of as a shape with an infinite number of sides does this then therefore mean that each side would have to be infinitely small?

Within a large circle you can draw a smaller circle inside it. If both the small and large circle has the same number of sides – an infinite number – then does this mean that the large circle has side lengths that are longer than the small circle?

Also, if a value is assigned to the sides of circle then would this then mean that the circle would be infinitely large?

I hope that someone can make sense of my questions and point out where in my thinking that I’m going wrong.

I’ve done some research and found one person claimed that:

When one does calculus, it is assumed by the notion of taking limits for tangents of any curve that a circle can be thought of as a polygon with an infinite number of sides, each of infinitesimal length.

Another person thought that:

The number of "real" sides of a particular circle would be its circumference in units of the Planck length. This would be a big number, but not infinite.

They thought this because The Planck length is the scale at which classical ideas about gravity and space-time cease to be valid, and quantum effects dominate. Therefore this is the smallest measurement of length with any meaning.

Yet another claim was that:

To have a 'side' you need to have 3 collinear points. But a circle doesn’t have 3 collinear points. Thus it has 0 sides.

The 4th claim was:

A circle has 2 sides because the standard form the circular equation is (x-x1)^2+(y-y1)^2=r^2

I don’t know enough mathematics to be able to assess any of these four claims.

So which is it? Do circles have?
a. Infinite number of sides
b. A very large but finite number of sides
c. Zero sides
d. 2 sides

I’ve also read that it all depends on what your definition of a side is.

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chiro
You are encountering the difficulty of working with infinity.

When you deal with infinities you can't use the same logic that you use with finite numbers.

With the circle example both the large and the small can be thought of having infinite sides that are infinitesimally small (ie get very very very close to zero).

There are a lot of "intuitive" problems with infinity, but mathematically we can write systems that work on the concept of infinity (one example is Hilberts Hotel Problem), and mathematically they work out.

Just remember that infinity isn't a "number" per se, so the logical things that apply to numbers don't apply to infinity.

So then it's wrong to say that a large circle has larger sides compared to a smaller circle?
In other words, you can't say both the large and small circle have the same number of sides because infinity is not a number per se?

I've heard that you can have some infinities that are larger than others. This doesn't seem intuitive to me at all, but would this have any significance when it comes to circles of different sizes?

lavinia
Gold Member
So then it's wrong to say that a large circle has larger sides compared to a smaller circle?
In other words, you can't say both the large and small circle have the same number of sides because infinity is not a number per se?

I've heard that you can have some infinities that are larger than others. This doesn't seem intuitive to me at all, but would this have any significance when it comes to circles of different sizes?
- I would think that if the circle did have sides, they would be infinitesimals. Maybe you should look at the theory of infinitesimals - I know nothing about it. In standard geometry, a side is a straight line segment. The circle does not contain any straight line segments.

- it is true that some infinities are larger than others but the infinity of the circle is the same no matter what its size. This infinity is the same size as the infinity of the interval of all decimal numbers between zero and 1 or between 3 and 31 or between any two numbers. The length of the interval has nothing to do with the size of the infinity nor does the size of the circle. This infinity is the same size as the infinity of the entire real line or of all of three dimensional space. It is larger than the infinity of all integers.

It all depends on what you mean with "side".
If you allow infinitesimal sides, then the circle has an infinite number of sides which are all very small.
If you don't allow infinitesimals, then the circle has zero sides.

The other two answers are false:

The number of "real" sides of a particular circle would be its circumference in units of the Planck length. This would be a big number, but not infinite.
Planck length is a notion is physics, it has nothing to do with the mathematical idea of a circle. Of course, if you draw a circle in real life, then we have to deal with Planck length. But mathematicians are not talking about circles in real life: they are talking about idealized circles. Thus the "sides" of the circles can become arbitrary small, even smaller than the Planck length. In real life, this is of course not possible.

A circle has 2 sides because the standard form the circular equation is (x-x1)^2+(y-y1)^2=r^2
This really makes no sense to me at all...

HallsofIvy
Homework Helper
A circle has 2 sides because the standard form the circular equation is (x-x1)^2+(y-y1)^2=r^2
Probably that person was thinking that
$$y= y_1+\sqrt{r^2- (x- x_1)^2}$$
was one "side" and that
$$y= y_1-\sqrt{r^2- (x- x_1)^2}$$
was the other "side".

I'm not sure that makes sense but it is the best interpretation I can come up with.

disregardthat
When doing calculus it is best not to get too mixed up into infinitesimals. It's an analogy at best and may easily be misleading. Standard analysis has nothing to do with infinitesimals. It's frankly absurd to say that the circle has an infinitude of sides, and even more so to say that this has anything to do with the planck length unit.

It should be mentioned that there is something called non-standard analysis which deals with so-called infinitesimals, but this is far from anything in a standard calculus treatment. Note that the non-standard reals, the set which contains in addition to the reals infinitesimals and infinite numbers is an extension of the real numbers, so there is nothing on the real number line corresponding to the infinitesimal or infinite numbers.

The notion of the circle having sides probably stems from that you can approximate the circle curve in a specific way by inscribing polygons, but this has nothing do with the circle having sides. The following:

When one does calculus, it is assumed by the notion of taking limits for tangents of any curve that a circle can be thought of as a polygon with an infinite number of sides, each of infinitesimal length.
is simply wrong. It is not assumed that a circle can be thought of a polygon with an infinite number of sides. It is however a valid form of computing the circumference of the circle by taking the limit of the circumference of incsribed polygons with an increasing number of sides. The amount of sides for each polygon is always finite, there is no such thing as a polygon with an infinite number of sides.

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A circle has two sides: an inside and an outside! ;)