How can images dimention be larger than the domains?

[vadik]

I'm lost! My book of topology says that its possible to construct a (continuous!) function f:[0,1] -> R^n such that the image is
a ball {x: |x|<=1}
I can't imagine how is it any possible to do such things. The book doesn't give any example or prove of it. It's just a comment. Any ideas? I can't solve this even for n=2.

 

[master_coda]

Those are space-filling curves.

Constructing them usually involves creating a curve with a fractal-style algorithm and showing that as the number of iterations goes to infinity you get a continuous curve that fills the entire space.

 

[HallsofIvy]

It is, of course, a function that is not differentiable at any point so it is really nasty!

 

[kuengb]

It is, of course, a function that is not differentiable at any point so it is really nasty!

I liked it when I first saw that stuff! Perhaps that makes me a nasty guy

Vadik, do a google search on "Peano curve"; those examples request at least a page of calculation, too big to post it in here. But thank god once you've done it for n=2 other examples can easily be made up.

What these functions show is that you can parameterize higher-dimensional manifolds with one parameter - but never in a homeomorphic way.

 

[mathwonk]

this was also thought impossible by mathematicians of the 19th century until peano i guess showed otherwise.

to start just subdivide a square into 4 parts and conect the centers of the 4 quarters.

then subdivide each quarter further into quarters, i.e. 16ths of the original square and connect the centers of all 16 small squares.


continue like this and you will have as a limit a curve that passes through a dense set of points of your square. but since the image of the curve is closed, it will pass through every point of the square!


a good (make that great) book discussing this, and many other wonderful things, is hilbert and cohn vossen, geometry and the imagination, written for the general public!