# mapping the one-cube continuously to the two-cube

Discussion in 'Undergraduate Math' started by Nevertheless, Aug 4, 2010.

1. ### NeverthelessGuest

I took a course in analysis back in 1971 in which the professor showed us a way to map continuously the unit interval onto the unit square.

The proof started with dividing the unit interval into four segments and the square into four sub-squares. Then a U-shaped curve was drawn onto the square, moving through all sub-squares and this was the image of the interval.

The process continued by dividing the interval into sub-sub-intervals and the square into sub-sub-squares and obtaining with each iteration a function which was part of an infinite series of functions uniformly converging to a function that was the required map.

I can reconstruct all of this from memory and explicitly write the function series as well as write the proof; however, I was wondering if I have to do this. Is there a site on the Internet that shows this proof and who first wrote it down? Does a book have this proof?

Charles

Nevertheless, Aug 4, 2010

2. ### Bastian ErdnuessGuest

It's called Hilbert curve. A good start point to a very interesting
topic.

Other keywords: Peano curve, space-filling curve, Sirpinski curve,
Lindenmayer system.

Bastian

Bastian Erdnuess, Aug 4, 2010

3. ### eratosthenesGuest

That is one of my favorite and most useful tools. However, I also
hate it because my analysis class forced a rigidity into my physicists
mind that did not like to be there.

Patrick

eratosthenes, Aug 4, 2010