Originally posted by @deepthoughtNot sure if I am asking exactly the right questions here but:
Not quite, all shapes all of whose points are on the points at infinity of the plane formed by the product of the extended real line with itself are circles because they fulfil the defining property of a circle. However, it doesn't necessarily follow that a circle is also a square. A square is defined by some property and a set of points either has it ...[text shortened]... and D be sets of points. That P(C) & P(D) is true does not entail that Q(D) follows from Q(C).
In what way could the defining property of an infinite circle be different from the defining properties of being an infinite square or vice versa?
What property could one have that the other doesn't?
I currently cannot think of any example of such a difference.
ANYONE; can you give an example of such a difference?
Originally posted by @deepthoughtYes, that was sort of one of the things I was trying to say to him but just couldn't figure out how to just say it because only had it on the purely intuitive level.
Taking limits won't help here, since we never leave the normal real number line during the limiting procedure.
I often find I have that kind of problem.
Originally posted by @humyI recommend you take a course in high school mathematics, which should cover the basics of limits.
ANYONE;
Is it meaningful to call infinity/infinity a "ratio"? Or is that nonsense?
Note infinity/infinity can equal any number!
So perhaps infinity/infinity itself is nonsense? because, for example, can have;
infinity/infinity = 1
and
infinity/infinity = 2
which seems to imply the nonsense of 1 = 2.
Originally posted by @kazetnagorraI suggest reading the thread. He actually does mean infinity over infinity and not the limit as we are using the extended real number line.
I recommend you take a course in high school mathematics, which should cover the basics of limits.
Originally posted by @deepthoughtPluralis maiestatis?
I suggest reading the thread. He actually does mean infinity over infinity and not the limit as we are using the [b]extended real number line.[/b]
Originally posted by @humyWe need a definition of a square that is correct for finite squares and is not broken in the infinite case. Wikipedia gives a bunch of equivalent definitions all of which use the word quadrilateral, which is going to cause problems. In the infinite case the four straight edges give us problems, so how about:
Not sure if I am asking exactly the right questions here but:
In what way could the defining property of an infinite circle be different from the defining properties of being an infinite square or vice versa?
What property could one have that the other doesn't?
I currently cannot think of any example of such a difference.
ANYONE; can you give an example of such a difference?
A plane figure is a square if and only if it's perimeter is straight almost everywhere and is left unchanged by rotations if and only if the rotations are integer multiples of 90°.
A circle will never fulfill this as it is symmetric under rotations other than integer multiples of 90°.
I need to think about it to decide if any infinite figure can fulfill this definition. We need 4 privileged points (the vertices) which break full rotational symmetry in a natural way. It's not obvious to me one can just assert them into existence.
Originally posted by @kazetnagorrahumy is assuming that system as well, so it is the common we.
Pluralis maiestatis?
Originally posted by @deepthoughtTo the best of my knowledge, the only one in this thread who has any understanding in handling "extended real lines" is you. Obviously, one should have a grasp of basic calculus before approaching such topics. In the standard way of dealing with limits and infinity, the infinitely large square and circle are different (at least) in the manner indicated.
humy is assuming that system as well, so it is the common we.
Originally posted by @kazetnagorraMy experience with it is limited to knowing it exists and this thread. My reason for introducing it is that the OP specified an "infinite" circle. There is no circle whose radius is a member of the set of reals which is not finite. We have to be careful taking limits, consider a saw-tooth function where y = x - int(x) where int(x) <= x is the integer part of x. In the limit x -> 1 y = 1, but at x = 1, y = 0. I think there is a similar problem taking limits if the extended real line (and it's extension to the plane) is in play, so that we need to define what shapes are in a way that is correct for every finite instance and doesn't just break for the infinite case. Avoiding concepts like measure and curvature as far as is possible is a good idea here.
To the best of my knowledge, the only one in this thread who has any understanding in handling "extended real lines" is you. Obviously, one should have a grasp of basic calculus before approaching such topics. In the standard way of dealing with limits and infinity, the infinitely large square and circle are different (at least) in the manner indicated.
The alternative approach is just to have a single point at infinity a la Riemann (c.f. one point compactification) and then all infinite shapes consist of a single point and are guaranteed to be identical with each other.
Originally posted by @humyOne has to be careful with this, but using x to mean any positive finite quantity and inf to mean infinity then:
ANYONE;
Is it meaningful to call infinity/infinity a "ratio"? Or is that nonsense?
Note infinity/infinity can equal any number!
So perhaps infinity/infinity itself is nonsense? because, for example, can have;
infinity/infinity = 1
and
infinity/infinity = 2
which seems to imply the nonsense of 1 = 2.
x + inf = x*inf = inf
inf/inf = 1
1/inf = 0
1/0 = inf
inf*0 = 1
Those equals signs should be read as "defined as". Don't get carried away with the above, the circumference of an infinite circle is infinity. Divide that by the radius and you'll get 1. You'll also get 1 if you divide by the diameter so don't start trying to do arithmetic with these quantities, they are limit points that close the set of points on the plane.
See the Wikipedia page on the extended real line.
Originally posted by @deepthoughtI like that.
We need a definition of a square that is correct for finite squares and is not broken in the infinite case. Wikipedia gives a bunch of equivalent definitions all of which use the word quadrilateral, which is going to cause problems. In the infinite case the four straight edges give us problems, so how about:
A plane figure is a square if and only if ...[text shortened]... ill never fulfill this as it is symmetric under rotations other than integer multiples of 90°.
So perhaps the strategy here should be;
(1) Make a definition of a square that is correct for finite squares and is not broken in the infinite case.
(2) Make a definition of a circle that is correct for finite circles and is not broken in the infinite case.
(3) Now compare the two definitions for the infinite case and see if they are in anyway different.
I fear the conclusion from (3) might depend entirely on the arbitrary definitions made in (1) and (2) making my question of whether an infinitely large circle is an infinitely large square objectively unanswerable and only subjectively answerable with both the answers 'yes' and 'no' legitimate depending on which arbitrary definitions you choose.
I've just had another thought;
Lets say we accept as valid the definition of an infinitely large circle as being the set of all points in an infinitely large 2D plain that is at infinity distance from an arbitrary chosen center point.
There would obviously be an infinite number of such points that would make up the 'circumference' of such an infinite circle because each one would be in a different direction from that center point and there are an infinite number of such directions thus an infinite number of such corresponding points.
If we were to arbitrarily pick two such different directions from that center point that are arbitrarily 'extremely close' to being the same direction (so that, for example, one is only a trillionth trillionth trillionth of one degree different from the other), no matter how 'close' they are to the same direction, the two corresponding points on the infinite circumference would be infinitely far away from each other.
But, on the other hand, if we now slightly change one of those two very slightly different directions so that now it is the same direction as the other direction, the two corresponding points on the infinite circumference would be zero distance from each other i.e. they would be the same point because those two directions will now equate i.e. be the same direction and there is only one unique point infinitely far away defined for the same given direction.
So, either way, there cannot be two points on this infinite circumference that are a finite distance away from each other!
Another way of thinking about that is if we first suppose it makes sense to say there was two points on this infinite circumference that are a finite distance away from each other, that would mean they must be in exactly the same direction from the center point. But there can only be one, not two or more, points infinitely far away from the center point because else there would be nothing to distinguish one such point from the other because they are in exactly the same direction from the center point.
But doesn't that means you cannot meaningfully call the circumference of an infinitely large circle a straight line or a curved line or a zigzag (thus with vertices) etc? Because what meaning would any of those things have if all points on them are either zero distance from each other thus the same point or if they are all an infinite distance from each other, whatever that's supposed to mean!?