mapping the one-cube continuously to the two-cube

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

  1. Nevertheless

    Nevertheless Guest

    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
    #1
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  2. 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
    #2
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  3. Nevertheless

    eratosthenes Guest

    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
    #3
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