What is Fractional Dimension and What is a Fractal? (TANTON Mathematics)

Take a picture to a photocopier and set it to scale by a factor of three. Then all lengths in the picture triple in size (factor of three to the one-th power) and all areas increase by a factor of nine (three to the two-th power). But there are objects that scale by fractional powers!

9 thoughts on “What is Fractional Dimension and What is a Fractal? (TANTON Mathematics)

  1. If you only do the erase step in the koch curve (you know, each time you only erace the middle third), you get this:

    size = 2*size*(1/3)^d

    And therefore d = 0.631
    (cool)

  2. “to the one-th power” just means to the power of one. So “factor of three to the one-th power” means “factor of 3^1″ which is a factor of 3 (any number to the power of 1 is itself).

  3. Actually, a factor of a half won’t change matters. Area scales – no matter what specific formula it follows for a particular shape – by k^2 and that is all that is being used.

    E.g. If the sides of a square are doubled, area changes by factor 4.
    If the sides of a triangle are doubled, the area changes by a factor of 4.
    If the radius of a circle is doubled, the area chanegs by a factor of 4.
    If the sides of an irregular dodecagon are doubled, area changes by a factor of 4.

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