20 Jul 18
Originally posted by @humyThe zooming in and out implies that rescaling is possible, so by multiplying both x and y by some factor we can rescale so that any given finite circle is shrunk or grown to unit radius. We simply can't do that with an infinite circle as any finite rescaling leaves it unchanged. So I think what you've said is correct, but I'm having problems with translating between your pictorial approach and my formal language approach.
I think I am finally getting what your are saying.
If I now understanding you correctly, you imply that you can define without any contradictions an infinitely large circle (meaning radius = +infinity) and its circumference would be an infinity long straight line and it would be meaningless to say you could "zoom infinitely out" to be able to see it from infi ...[text shortened]... circle as a whole on the finite scale.
Before I more on, have I got that all exactly correct?
20 Jul 18
2 edits
Originally posted by @deepthoughtOK.
The zooming in and out implies that rescaling is possible, so by multiplying both x and y by some factor we can rescale so that any given finite circle is shrunk or grown to unit radius. We simply can't do that with an infinite circle as any finite rescaling leaves it unchanged. So I think what you've said is correct, but I'm having problems with translating between your pictorial approach and my formal language approach.
I now have two new questions;
With your definition of an infinitely large circle (meaning radius = +infinity), does that allow for more than one defined center point to be in each infinitely large circle?
The reason why I ask is because there are an infinite number of points that can be specified to be a finite arbitrary distance and direction away from any one defined center point and each of those points also fits your definition of it being the center of the infinitely large circle thus an infinitely large circle has an infinite number of different center points!
I suppose the question I really want to ask here is; Does the valid definition of any circle, even if it is for an infinitely large one, necessarily imply it must have one and only one unique center point to be a circle?
And, with your definition of an infinitely large circle (meaning radius = +infinity), does that allow for it being the same shape as some other shape that would be a different shape if finite in size but the same shape if infinite in size?
The reason why I ask is because there are an infinite number of shapes that would be different by definition if finite in size but the same shape as an infinitely large circle. For example, there are an infinite number of different irregular ovoid shapes you can define if all finite but they will all fit your same definition as your infinitely large circle if all of of those ovoid shapes were infinite in size.
20 Jul 18
7 edits
continued...
Just had another thought;
Also, wouldn't each and every point 'outside' an infinitely large circle ('outside' here would be the other side of the infinitely long straight line that is its circumference) also fit your definition of being one of its center points? If so, wouldn't that mean a single infinitely large circle would also be more than one infinitely large circle because there be one on each side of that infinitely long straight line?
20 Jul 18
Originally posted by @humyMy misedit;
The reason why I ask is because there are an infinite number of shapes that would be different by definition if finite in size but the same shape as an infinitely large circle.
extend the end of that sentence with the words "... if infinite in size."
20 Jul 18
Originally posted by @humyThat's a good point. Every point on the open real plane fulfills the property of being a centre of the circle.
OK.
I now have two new questions;
With your definition of an infinitely large circle (meaning radius = +infinity), does that allow for more than one defined center point to be in each infinitely large circle?
The reason why I ask is because there are an infinite number of points that can be specified to be a finite arbitrary distance and direction away f ...[text shortened]... efinition as your infinitely large circle if all of of those ovoid shapes were infinite in size.
This was my point that an infinite square has the defining property of a circle. A semicircle would consist of a straight line joining two points at infinity and the set of points at infinity on one side of it. So I don't think it's a requirement that all of the shape's points are points at infinity.
20 Jul 18
Originally posted by @humyThere are no points outside the infinite circle. The points at infinity are the limit points of the extended plane which are not interior points.
continued...
Just had another thought;
Also, wouldn't each and every point 'outside' an infinitely large circle ('outside' here would be the other side of the infinitely long straight line that is its circumference) also fit your definition of being one of its center points? If so, wouldn't that mean a single infinitely large circle would also be more t ...[text shortened]... infinitely large circle because there be one on each side of that infinitely long straight line?
20 Jul 18
Originally posted by @deepthoughtThis is obscured by technical language. There are no points to the right of +infinity on the extended real line. So on the corresponding plane which has points at infinity on its perimeter, there are no points outside the perimeter.
There are no points outside the infinite circle. The points at infinity are the limit points of the extended plane which are not interior points.
20 Jul 18
1 edit
Originally posted by @deepthoughtOK. Thanks for all that.
This is obscured by technical language. There are no points to the right of +infinity on the extended real line. So on the corresponding plane which has points at infinity on its perimeter, there are no points outside the perimeter.
From our conversations I think I can now conclude that;
A said infinitely large circle or a said infinitely large sphere or any other said infinitely large 'shape' of a kind that has a said center point and has all of its point on its 'circumference' or 'surface' defined as being an infinite distance from this center point;
(1) has no outside because there are no points that can be more than an infinite distance away from that center point.
(2) each has an infinite number of center points with all center points having identical properties.
(3) all such types of infinitely large 'shapes' in the same number of dimensions so defined as having all its points infinitely far from a center point also all have the same defining properties as each other. Thus an infinitely large square is also an infinitely large circle etc. And an infinitely large cube is also an infinitely large sphere etc.
But doesn't (3) above make it meaningless to call such an infinitely large shape a "circle" or a "square" precisely because there is nothing to distinguish between them thus begging the question why call it a "circle" rather than a "square" or an "egg shape" or any one of the infinite number of other kinds of shapes you can equally legitimately but totally arbitrarily call it?
In fact, would that make it meaningless to call such a said infinitely large shape a "shape" at all!?
Couldn't you just as legitimately call it an infinitely large shapeless construct?
21 Jul 18
Originally posted by @humyI wouldn't say that an "infinitely large circle" is the same as an "infinitely large square."
OK. Thanks for all that.
From our conversations I think I can now conclude that;
A said infinitely large circle or a said infinitely large sphere or any other said infinitely large 'shape' of a kind that has a said center point and has all of its point on its 'circumference' or 'surface' defined as being an infinite distance from this center point;
(1) ha ...[text shortened]... pe" at all!?
Couldn't you just as legitimately call it an infinitely large shapeless construct?
For example, consider a square with sides 2R and a circle with radius R. The ratio of the circumference of both shapes is 4 over pi. Now take the limit R -> \infty. The ratio of the circumference remains the same, but the ratio of circumference of the circle with respect to the circle is 1. So in this limiting procedure, the two aren't the same.