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#1 2006-10-06 13:21:11

John E. Franklin
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Registered: 2005-08-29
Posts: 3,577

infinite barbell, circle, 37777...777773

So I guess a circle with an infinite radius can't exist,
according to Ricky's notion that {1,1,1...2}
equals {1,1,1...}, since you never get to the 2.
I am not so adamant in my beliefs, since I haven't
probably put so much time reading or thinking about
the subject.
What about a barbell with weights on boths ends and
an infinite length bar in the middle??
Can these concepts exist in our mathematical minds??
Ricky seems to think they cannot exist??
Maybe I don't understand him correctly.
Please correct me, and I am sorry for the accusations.
I am just trying to learn from you all.


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#2 2006-10-07 05:56:22

Ricky
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Registered: 2005-12-04
Posts: 3,791

Re: infinite barbell, circle, 37777...777773

So I guess a circle with an infinite radius can't exist,

Sure it can.  A circle with an infinite radius is a line.


"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

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#3 2006-10-07 08:32:05

John E. Franklin
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Registered: 2005-08-29
Posts: 3,577

Re: infinite barbell, circle, 37777...777773

So your saying it's a line, but not a circle, or are you saying
it's a line and a circle at the same time, or what are you saying?


igloo myrtilles fourmis

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#4 2006-10-09 07:50:41

cray
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Registered: 2006-10-05
Posts: 29

Re: infinite barbell, circle, 37777...777773

So your saying it's a line, but not a circle, or are you saying
it's a line and a circle at the same time, or what are you saying?

I believe Ricky must mean that at any point we looked at the circle it would look like a line

I believe (and its a belief not a proof)
that an infinite circle is impossible because by definition a circle is a finite thing
If it were infinite we would not be able to define it

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#5 2006-10-09 08:20:40

Dross
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Registered: 2006-08-24
Posts: 325

Re: infinite barbell, circle, 37777...777773

John E. Franklin wrote:

So I guess a circle with an infinite radius can't exist

Definition of a circle of radius r with centre c in the plane:

The set of all points with distance r from c.

So, can you have an infinite circle? (or, can you have another non-equivalent definition of a circle?) I don't think you can have an infinite circle.

Say there is an infinite circle (with r = "infinity") in the plane. Then take the set of all points that make up the circle and ask where the centre is... apparently, it's "infinitely far away" from all of them. But there is no point on the plane that is infinitely far away from any other pont on the plane, simply because any two points have a finite distance between them.

So you can define the centre on the plane, in which case none of the points of the circle are on the plane (not because they are "infinitely far away" and such... because they are, as previously stated, not on the plane, so there are no points to make a circle with - also, the uniqueness of the centre is then void, and that throws another potential issue).

So I put it to you that you cannot have an infinitely large circle.

...and please, before you go throwing around phrases like "infinitely large" and such, take a second to think about what you mean, and see if you can come up with a clear, precise definition. Otherwise all you're saying really is just meaningless babble. And if you're not clear on anything above, I'll be happy to further discuss it.


Bad speling makes me [sic]

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#6 2006-10-09 12:37:03

cray
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Registered: 2006-10-05
Posts: 29

Re: infinite barbell, circle, 37777...777773

Sure it can.  A circle with an infinite radius is a line.

I just realised you are talking about non-Euclidean Geometry arent you ?
I dont know much about that but I believe people who do know would assert that
the infinite circle could be imagined and that you would be able to imagine walking along
its perimiter, and that a circle of such magnitude, or approaching infinity would appear as a line
In the opposite sense to which two lines that are paralell appear to meet in the distance
this circle would never appear to have rounded edges

Last edited by cray (2006-10-09 12:39:06)

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#7 2006-10-10 00:59:18

Dross
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Registered: 2006-08-24
Posts: 325

Re: infinite barbell, circle, 37777...777773

cray wrote:

Sure it can.  A circle with an infinite radius is a line
I just realised you are talking about non-Euclidean Geometry arent you ?.

Ah, but there are many types of non-Euclidean geometry, so you should really specify what sort you're using. For example, you would be hard-pressed to imagine an infinite circle if you're using spherical geometry - since all your points are on a sphere, there is most certainly a limit to how large a circle could get!


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#8 2006-10-10 01:51:46

Ricky
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Registered: 2005-12-04
Posts: 3,791

Re: infinite barbell, circle, 37777...777773

Not exactly.  Take a piece of paper, and draw a fair sized circle.  Now double or tripple the size.  Continue doing this, drawing as much of the circle as you can.  You should see the curve of the circle start to become straighter and straigher.  It's with this observation that you can reach a conclusions that if we take the limit of a circle as it's radius approaches infinity, it becomes a line.

Not rigorous mathimatics, just something to think about.

And you most certainly need an infinite circle when dealing with improper integrals in polar coordinates, the most famous example is the Gaussian integral.


"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

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#9 2006-10-10 02:10:18

Dross
Member
Registered: 2006-08-24
Posts: 325

Re: infinite barbell, circle, 37777...777773

Ricky wrote:

It's with this observation that you can reach a conclusions that if we take the limit of a circle as it's radius approaches infinity, it becomes a line.

Indeed - but this does not mean that an infinite circle exists. Just because

exists, that does not mean that f(c) exists - f(x) could well be undefined at c.
So although the limit of larger and larger circles may indeed be a line, this does not mean that there exists an infinitely large circle.


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#10 2006-10-10 03:39:26

cray
Member
Registered: 2006-10-05
Posts: 29

Re: infinite barbell, circle, 37777...777773

For example, you would be hard-pressed to imagine an infinite circle if you're using spherical geometry - since all your points are on a sphere, there is most certainly a limit to how large a circle could get!

I wish I'd listened to my teacher more when I was at school because I would be taking a rough ride to nowhere to try and prove you wrong with my knowledge but it sure sounds like an obsfucation to bring in circular geometric planes !
I am not disputing your word, just making clear I havent a clue what youre talking about ! HA !

Anyway my simple points were that
1) I dont think that a infinite circle can exist in practice simply because by definition every aspect of it would be infinitely spaced apart - just the fact that logic alone would tell you it is impossible, as a circle would have a defined radius that cannot be infinitely long. A circle is a defined article so to speak.

2) If one were to begin the mathematical excersise of defining an infinitely large circle the definition would necessarily prove that the circle was not infinite since the implication that something is circular implies that it is also curved and to define a curve that must be done in a finite space

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#11 2006-10-12 11:36:07

John E. Franklin
Member
Registered: 2005-08-29
Posts: 3,577

Re: infinite barbell, circle, 37777...777773

I really like your logic, Cray.
What about an infinite spiral??
Can you have that??


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#12 2006-10-13 03:38:09

Ricky
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Registered: 2005-12-04
Posts: 3,791

Re: infinite barbell, circle, 37777...777773

Dross wrote:
Ricky wrote:

It's with this observation that you can reach a conclusions that if we take the limit of a circle as it's radius approaches infinity, it becomes a line.

Indeed - but this does not mean that an infinite circle exists. Just because

exists, that does not mean that f(c) exists - f(x) could well be undefined at c.
So although the limit of larger and larger circles may indeed be a line, this does not mean that there exists an infinitely large circle.

Does x^2 exist at infinity?  Or is that a meaningless question?


"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

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#13 2006-10-13 15:02:49

John E. Franklin
Member
Registered: 2005-08-29
Posts: 3,577

Re: infinite barbell, circle, 37777...777773

I don't know if x^2 exists at infinity.  What about x - 5 at infinity??  or ln(x) at infinity?
Are they all the same thing? infinity?
If everything about infinity is infinity, then you can't have an object like a circle or a drawing of a house the size of infinity.
But if infinity is a world unto itself, but separate from the real world, then maybe all these things could exist there.
From the real world, it all looks like infinity, one undefined number.
But from the infinite world, then things are as diverse as they are here in the real world.


igloo myrtilles fourmis

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