Turning a ball inside out without dissecting

[san_1420]

Is it true that Newton proved that it is possible to turn a ball inside out without dissecting it using calculus.

 

[HallsofIvy]

No, that's not correct. This is way after Newton! What you are talking about a problem in topology. You have to assume that it is possible to pass one piece of the ball through another piece of the ball. If it sounds like it would be easy to do it in that case, think again! As an analogy, take a circle made out of string. Now take one side of the circle and pass it outside the other side. You can do that easily by "cheating"- instead of going through the string just go over it. But notice what happens- you get two little "loops" at each end of the overlap. You can't continue to completely reverse the circle with those loops becoming sharp points- a non-differentiable point that is not allowed.

I remember seeing a movie of this. Imagine that the outside of the ball is red and the inside is blue. Through a series of tricky moves, you get a situation in which exactly half the surface is red and half blue. Then you reverse it but apply the reverse moves to the blue- so you wind up with the blue on the outside!

I wish to Hades I could remember the name of the mathematician who came up with the process. If I remember correctly he was blind!

 

[amcavoy]

When my uncle was in grad. school I remember him telling me about something that seems similar called the "Banach-Tarski Theorem," but I have no idea if it's the same thing you're talking about.

 

[san_1420]

Thank you all for contributing.

I guess I will have to go back to good old textbook on Topology.

Example with the string was real eye opener.

But that leads to another question in a 2- D world ,would
we able to cheat with the string as before

 

[Galileo]

I found this site:
http://www.xs4all.nl/~alife/sphere1.htm

 

[HallsofIvy]

The "Banach-Tarski" Theorem (often called the "Banach-Tarski paradox"!) says that it is possible to divide a unit sphere into subset such that by rigid motions the subsets can be reassembled into two unit spheres! Of course the subsets are very complex- not intervals or anything nice like that.

 

[san_1420]

I found this site:
http://www.xs4all.nl/~alife/sphere1.htm

That explains it .Thanks

 

[HallsofIvy]

Thanks for the link, Galileo.

This is who I was thinking of:

It was not until the 1970s that
the (blind !) mathematician Bernard Morin came up with a visualization,
based on work by Arnold Shapiro.