Exploring Infinite Circles: Can You Have a Finite Distance Without Gaps?

[cragar]

Lets say I take the natural numbers from 1 to infinity and I wrap the positive real line into a circle.
So I have a circle with an infinite circumference and the natural numbers are spaced 1 unit away from each other at the edge of the circle. Now I draw a line from each natural number
to the center of the circle. Now is there a point from the center of the circle at a finite distance such that there are no gaps in a second test circle drawn at this finite distance.
And then can I construct a circle with infinite circumference and have an uncountable number of points and each point is separated from the other by a positive finite distance.
And then draw a line from each point to the center of my circle. will I have another circle inside this that has no gaps in it.

 

[Diffy]

If the circle has an infinite circumference how can it have a center?

How can a circle have an infinite circumference? What does that mean?

Say such a circle exists, and say you have your points 1, 2, 3 on the circle, now say that your point 1 lies on the origin, and the center of your circle lies on the positive x axis. Then what is the angle with respect to the x-axis of the vector passing through 1 and 2? Is it not 0? If it is 0, then you do not have a circle, if it is not 0 then your circumference is not "infinite".

I think you are playing with things that can not exist. I think your circle is not well-defined. Thus you can draw conclusions that make no sense.

Edit: spelling and I made an error.

 

[HallsofIvy]

The difficulty is that you can't do any of the things you say! In particular, you cannot, geometrically, "wrap the positive real line into a circle." And, after that, nothing you say is correct.

 

[Studiot]

A simple proof that you can't do the wrap is: (I claim copyright on this dance)

Consider any circle.
Start anywhere on the circumference and mark zero
Continue marking unit distance for points 1, 2,3,...

But length of the circumference is non integral so your last (joining) marked interval will be of different size from the rest ie non unit.

Thus the original premise is false.

 

[cragar]

you guys are good. Why can't the circumference be integral
why can't my circle exist in the metaphysical realm of the ideals? Just kidding

 

[Studiot]

The real numbers are [STRIKE]countably[/STRIKE] uncountably infinite. That is there are more real numbers than there are natural numbers.

http://en.wikipedia.org/wiki/Countable_set

Be careful about the use of the word 'ideal' it has a special meaning in algebra and number theory.

 

[Mark44]

Edit inline, in blue.

The real numbers are uncountably infinite. That is there are more real numbers than there are natural numbers.

http://en.wikipedia.org/wiki/Countable_set

The natural numbers, integers, and rational numbers are countably infinite; the reals are uncountably infinite. That's probably what you meant to say...

 

[Studiot]

Thank you for the correction, Mark.

I'm glad somebody knew what I meant, because I don't always myself.

 

[cragar]

ok so maybe my circle doesn't exist in the euclidean plane. Why can't I define a new mathematical object and maybe we need to use non-standard analysis

 

[pierce15]

Like someone else said, if you put the center of this "circle" on the x axis, the center occurs at (infinity,0). The radius is not a finite distance

By the way, what is your question?

 

[cragar]

Why can't we construct the circle by taking the limit.
we start with the circle centered at (0,0) with radius 1 and then take the limit as
r goes to infinity. now out at infinity we start by drawing a line from the outer edge back to the center, and then we go like 1 unit over and do the same thing and draw a line back to the center and we do this countably infinite times. Now we make another circle at a finite distance from the center, now where these lines cross the circle of finite radius
will there be gaps in this circle.

 

[Vargo]

Hello Cragar.

Your construction is still not well defined because you can't draw the lines after you blow your circle up out to infinity. However, I will take the liberty of reformulating your question and you can tell me if it matches the idea in your head.

Start with a circle of radius N centered at the origin, where N is a positive integer. The circumference has length 2piN, so we divide it up into N equal arcs of length 2pi each. (I am modifying the spacing to 2pi so that we divide the circle into an even number of pieces). Then we draw a ray from the origin to each vertex creating a system of N rays spaced by equal angles from the origin.

Now we fix another circle of a definite radius R (also centered at the origin), and we mark the points of intersection of this circle with the rays. That set of intersection points depends on the radius N of the big circle. Now we consider that set of points as N grows larger.

Question: As N goes to infinity, are there any gaps left in the circle of radius R?

 

[cragar]

ya that's about it