Is a Tangent Line Defined by a Single Point or Its Surroundings?

[PcumP_Ravenclaw]

Halo, my question is what is the shape of a single point? Is it round? When we find the tangent line it is in relation to the surrounding points around the point of interest. For a circle tangent line is line that cuts through a single point only but in a parabola this cannot be true because the tangent "scrapes" the neighboring points also. So I mean to say that a single points tangent depends on the points surrounding it.

 

[BvU]

Point has no shape. Anything finite that you let shrink to size 0 becomes a point.

When we find the tangent line it is in relation to the surrounding points around the point of interest

I think so, yes...

Tangent line is well defined. Look it up.

For a circle tangent line is line that cuts through a single point only but in a parabola this cannot be true because the tangent "scrapes" the neighboring points also

Huh ?

 

[Samy_A]

Single points don't have a shape.
It is correct that the tangent to a curve at a point depends on the behavior of the curve in the neighborhood of the point (as the slope of the tangent is given by a derivative).
But the tangent to a parabola in a point P doesn't "scrape" any neighboring point on the parabola: the only point in the intersection of the tangent and the parabola is the point P.

 

[fresh_42]

There are many somehow strange facts when it comes to dimensions, zero in this case. E.g. our concept of points lead to a homeomorphism between a single and two balls, all three of the same size. (One needs the axiom of choice as well, but the concept of points and orbits is crucial.) Another example is that one can paint out a 2-dimensional square by a 1-dimensional line. Our concepts of geometrical objects are idealized and don't match with real objects. Even there is no such thing as a smooth object in reality. Nevertheless, circle and parabolas are smooth.

 

[micromass]

There are many somehow strange facts when it comes to dimensions, zero in this case. E.g. our concept of points lead to a homeomorphism between a single and two balls, all three of the same size. (One needs the axiom of choice as well, but the concept of points and orbits is crucial.)

What the hell are you talking about? There is no homeomorphism between a single and two balls. Are you trying to explain the Banach-Tarski paradox? That has nothing to do with what you just said.

 

[fresh_42]

What the hell are you talking about? There is no homeomorphism between a single and two balls. Are you trying to explain the Banach-Tarski paradox? That has nothing to do with what you just said.

Yes, and yes homeomorphism was wrong. Sorry, wrong mapping.
I remember a colloquium on this paradox I've been to some decades ago. And I remember the discussion on it afterwards, in which the professor, who basically sketched the proof as in Wiki, pointed out that in his opinion the concept of a point is more important to the result than the axiom of choice. I never had a reason to doubt his assessment for he was certainly more qualified than me. Don't you think "has nothing to do" is a little abrasive? How would you classify this paradox if not on points, orbits and the discrepancies between a rigorous proof and our geometric intuition?

Edit: Homeomorphism isn't plain wrong but trivial, so it's not a major aspect.

 

[micromass]

Edit: Homeomorphism isn't plain wrong but trivial, so it's not a major aspect.

No, it is plain wrong. There is no homeomorphism between "one ball" and "two balls".

 

[fresh_42]

No, it is plain wrong. There is no homeomorphism between "one ball" and "two balls".

As far as I remember the two shared exactly one point where they touched which makes it homeomorph. If my memory is false, of course then it's not.

 

[micromass]

As far as I remember the two shared exactly one point where they touched which makes it homeomorph. If my memory is false, of course then it's not.

You mean "one ball" and "two balls sharing a point"? Those are not homeomorphic.

 

[fresh_42]

You mean "one ball" and "two balls sharing a point"? Those are not homeomorphic.

Spheres are not. Balls are. 1 ball ↔ 2 touching balls.

 

[micromass]

Spheres are not. Balls are. 1 ball ↔ 2 touching balls.

Prove it.

 

[fresh_42]

Prove it.

In this case I claim the opposite. However, using polar coordinates ... It's certainly easier than it is to make a coffee cup out of a ring.
https://en.wikipedia.org/wiki/Homeomorphism#/media/File:Mug_and_Torus_morph.gif

Edit: Maybe mirroring it point by point would do.

 

[micromass]

Can you please link me to a proof that says "one ball" is homeomorphic to "two touching balls"? Or did you say it's actually false?

 

[fresh_42]

I'm searching. "Ball" isn't a pleasant keyword.

Edit: 1) The mapping in the proof of the Banach-Tarski paradox might not be a homeomorphism, likely something piecewise isometric or so.
2) However, a homeomorphism between one ball and two touching balls can be established by stretching all diameters of the first ball by a factor of two and making the center of the first ball the touching point of the two balls. The diameters then become the projection on the boundaries through the origin, the touching point, of the two balls left and right the origin. The straight must therefore be suitably stretched, compressed, resp. This would lead to a singularity in the origin, which is avoided by mapping it onto itself. This can be done continuously as long as the balls are touching.

 

[WWGD]

Spheres are not. Balls are. 1 ball ↔ 2 touching balls.

Maybe if by touching you mean they share a sort of tubular neighborhood that joins the two, meaning that the intersection set is uncountable. If the two balls are just tangent to each other (meaning the two intersect at a single point) , then a standard argument for why the two are not homeomorphic is that the two tangent balls can be disconnected by removing just one point (the point of tangency), while there is no similar point for the single ball. But I do get your point about searching " a ball and two balls touching" yikes for those search results.

 

[fresh_42]

Maybe if by touching you mean they share a sort of tubular neighborhood that joins the two. But I do get your point about searching " a ball and two balls touching" yikes for those search results.

One point is enough. You double the ball and then compress it like with a belt into two balls sharing a common point in 0. If I were better on spherical coordinates I could write down the mapping. (How far (euclidean metric) is a point on a sphere with radius 1 and center in (1/2,0,0) away from (0,0,0)? If that is d then you get the image points by 1/(2d) times all diameters in the domain ball. The singularity can be continuously filled by mapping 0 to 0.)

 

[micromass]

One point is enough.

It's not. WWGD's proof shows this.

 

[WWGD]

One point is enough. You double the ball and then compress it like with a belt into two balls sharing a common point in 0. If I were better on spherical coordinates I could write down the mapping. (How far (euclidean metric) is a point on a sphere with radius 1 and center in (1/2,0,0) away from (0,0,0)? If that is d then you get the image points by 1/(2d) times all diameters in the domain ball. The singularity can be continuously filled by mapping 0 to 0.)

But how do you address the 1-connectivity issue? k-connectivity is a homeomorphism invariant.

 

[fresh_42]

It's not. WWGD's proof shows this.

I mapped the center of the single ball onto the touching point. I can't see where it shouldn't be continuous or not one to one. Ah, I think I got it. One cannot compress [0,1] on {1/2}. Damn, that was a tough one. Thanks for your patience.

 

[WWGD]

I mapped the center of the single ball onto the touching point. I can't see where it shouldn't be continuous or not one to one. Ah, I think I got it. One cannot compress [0,1] on {1/2}. Damn, that was a tough one. Thanks for your patience.

Yes, that counters injection, aka 1-1-ness.