Results 1 to 3 of 3

Math Help - Dimension of fractal

  1. #1
    Newbie
    Joined
    Dec 2009
    Posts
    12

    Dimension of fractal

    for a cantor middle-half set K, I think it is one-dimensional because it is on a real line. But the book say that it is 1/2-dimensional.
    Consider KxK, I think it is 2-dimensional because it is a square(containing many small squares) on a plane. But it is a 1-dimensional set. Why?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    Aug 2008
    From
    Paris, France
    Posts
    1,174
    Quote Originally Posted by ldpsong View Post
    I think it is one-dimensional because it is on a real line.
    As a matter of fact, the "dimension" is a precise (and delicate) notion that has a mathematical definition (in fact, several). Maybe you know what the dimension of a vector space is (which is an integer). For Cantor subsets, one obviously needs a different definition, that would account for their different self-similarity.

    But the book say that it is 1/2-dimensional.
    Maybe your book says how it defines dimension. Anyway, you can find lots of ressource on the internet: look for "fractal dimension". In the case of your subset, you can cut it in 2 parts that look like the initial subset up to a scale \frac{1}{4}. Similarly, if we cut a line segment (d=1) in 4 we get parts that look like the initial subset to a scale \frac{1}{4}, if we cut a square (d=2) in 16 we get smaller squares like the first one to scale 1/4, and if we cut the cube (d=3) in 4^3 we get smaller cubes like the first one to scale \frac{1}{4}, and in dimension d, we can cut a cube into 4^{d} smaller cubes that look like half the initial cube up to a scale \frac{1}{4}. So for the middle-half Cantor subset, you would have 2=4^d, hence d=1/2. This is not a proof, since I gave no definition of dimension, but it gives a reason why it should be 1/2.
    Last edited by Laurent; April 5th 2010 at 11:04 AM.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie
    Joined
    Dec 2009
    Posts
    12
    thank you very much
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Fractal
    Posted in the Differential Geometry Forum
    Replies: 5
    Last Post: February 19th 2011, 03:56 AM
  2. fractal dimension of "filled" koch snowflake
    Posted in the Advanced Math Topics Forum
    Replies: 2
    Last Post: June 25th 2010, 09:36 AM
  3. fractal dimension of a set
    Posted in the Calculus Forum
    Replies: 0
    Last Post: May 19th 2010, 05:52 AM
  4. Dimension of cantor middle-half set(fractal)?
    Posted in the Differential Geometry Forum
    Replies: 2
    Last Post: April 5th 2010, 07:03 PM
  5. Fractal Construction
    Posted in the Geometry Forum
    Replies: 2
    Last Post: March 23rd 2010, 02:38 PM

Search Tags


/mathhelpforum @mathhelpforum