Banach�Tarski paradox

[dannyday5821]

okay, i know NOTHING about geomotry, but this really interests me...

I cant understand how physically you could get two balls (yeah yeah i said balls - i mean a sphere for those immature people!) out of one...i mean...how?

id love to know, especially since these guys apparently sussed it! im jelous to be honest, i want to know!

also, i read somewhere about how they figure they can get infinite balls out of one....what!?

could someone explain it to me in the simplist possible terms, if that is possible...as its such a complex topic i appreciate! i think its so unfair interesting things like this go unmissed by people like me who dont understand them because of their complexity. why cant someone publish a book explaining the most interesting complex things but in the simplist way?

anyway, id appeciate it if someone could help!

[jake-the-peg]

I think this related to the fractal star of David.

In this you take an equilateral triangle and divide each side into 3 and draw an equilateral triangle on the middle third of each side you get a star of David.

If each side of the first triangle was of length x the circumference of the triangle was 3x

Now it's 3(x-1/3x+2/3x) = 4x

Now do the same again and again like this:
http://library.thinkquest.org/26242/full/types /images/22.gif

Do this an infinite number of times and you have a figure with an infinite circumference the will still fit within a circle and hence has a finite area.

In the same way the pieces that the sphere must be subdivided into parts that are infinitely complex and hence only possible in a mathematician's mind

[kwicky]

http://en.wikipedia.org/wiki/Banach-Tarski_par adox