Friday, January 30, 2009

The Line and the Circle

Okay, so regarding the line of infinite length and the circle of infinite circumference; what do you think? Are they related?

Think about what a circle is, a single line bounding a figure equidistantly from a single point, called the centre. The line itself is, however, unbounded. Do you see a way in which the circle might represent a kind of Little Infinity?

Think about the single straight line. Could this be a way of representing Big Infinity stretching out in both directions for eternity?

Are these two shapes so different though? What would a circle with an infinite circumference look like? Here is a picture of a finite circle:
Note that this picture has no particular size associated with it. The circumference is the distance around the outside of the circle. If this distance were infinite it would be impossible to know exactly where to begin the curve of the circle, precisely because there is an infinite distance the circumference must encompass. Once you begin to curve the circle, the shape would demand that the circle be completed (because all circles are the exact same shape with different radii and circumferences; eg. they are all similar) and therefore finite. The best way to demonstrate this might be to look at a very very very very small part of a circle, which might looks something like this if we zoomed way in on a single point:

This looks an awful lot like a line to me. So now, if a circle is representative of Little Infinity and a line is representative of Big Infinity they seem like they might be very much alike indeed.

Do you buy into this? What do you think? Is this convincing you of something more about infinity? Does it make you think maybe there is something interesting at work here that we need to look at more closely?


Baxdragon said...

Hold it now: HIT IT!

No, what I mean to say is slow this fruit truck down. I see what you're getting at but frankly you're talking E=mc2 here. When you look at only a relative protion of the circle you will see only that which is presentable and with an circle big enough you may not be able to see the curvature leading you to see it as an infinite line.

Let's get beyond relativity and talk about the circle as a whole. As a circle it is, by definition, finite. It has to be.

This may not be the case if we're talking about a spiral that only appears to be circle when viewed down the length of it but we're talking about a "circle". Now while it may not be possible to accurately determine where a cirlce ends or begins that doesn't make it infinite, simply without definative origin. After all with one revolution over the circle you're not traversing new parts of the circle but continuing to travel along the same space.

B said...

If the circle is infinite in circumference how would you know where to begin the curve. See all circles curve away from the point of intersection with a tangent line at the same angle. So once you begin the curve it necessitates a circle of finite circumference because all circles are "similar" (the same proportional shape, like equilateral triangles). Thus the only way for a circle to be truly infinite in circumference would be if it "appeared" to be uncurved.

The really interesting thing about this is that in Lobachevskian Geometry, straight lines appear to act as though they are curved, lending an interesting idea that both lines and circles are very much alike.

A line is breadthless length that seems to be unbounded on the ends.

A circle is a figure defined by a boundary line which has no boundaries itself. There is no starting point or ending point on a circle, other than incidentally or arbitrarily.

If they can appear similarly under certain circumstances, perhaps we should expect them to be more similar than we thought when we consider them more closely.