LOGO-L> What is a Fractal
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LOGO-L> What is a Fractal

;Hello Yehuda Katz!

Quote:

>I wrote to you directly SEVERAL times in the past, but I don't know if
>those mails ever reached you, as I never got a direct reply from you.

;Dear Yehuda !
;I have big trouble with my internet service provider,many of your and
;other's past messages were lost ;but  usually I can see all your vived
;activities on comp.lang.logo.
;question "is a circle a fractal ?. is a straight line a fractal ?"
;May be the following will leed to an answer to this question:

;In The Byte Magazine of august 1987 William A. McWroter Jr. and
;Jane Morrill Tazelaar published a paper "Creating Fractals" which
;contained many interresting facts about space filling curves, since
;these curves may be fractals or non fractals; the following is quoted
;from that article:

;The space filling curves are those curves which arrise from an infinitly
;repeated construction process. strangely few regular fractals appeared
;outside Mandelbrot's book "fractal Geometry of Nature".
;Felix Hausdorff had set a dimension to fractals which is something in
;between the dimension of a line and that of an area i.e. a number more
;than one and less than two. By hausdorff's definition the dimension of
;the Koch's snow-flake fractal is 1.2618 while that of the Sierpinski
;carpet is 1.8928. While conventional dimension can only determine if
;an object does or does not fill a space, the Hausdorff dimension can
;measure what fraction of the space an object covers (something like
;measuring the density of a cloud). Fractals live in a nether world
;between conventionally dimesioned spaces.
;So koch's snowflake and Sierpinski's carpet live in a world between their
;one-dimensional parts and their two-dimensional home. For example the
;space which the Sierpinski carpet can cover on a square land will not
;exeed 1.8928 of its area whatever was the number of repetiton levels.
;The Hilbert curve which is the most famous of the space-filling curves
;is not a fractal since its Hausdorff dimension is 2, more than any other
;space filling curve.
;On the other hand a dragon is defined as an organism of cells arranged
;according to a genetic code. It begins life as a single cell and then,
;by daily cell devision, grows into a creature with a shape and character
;governed by the "DNA" of its genetic code. This defintion includes both
;regular fractals and space filling curves. It is based on Dekking's
;notion of recurrent sets having its origins in Lindimayer's study of
;cell development and cellular automata. The most famous automaton
;is Conway's game of Life.
;Here are the names of the most famous space-filling curves:

;Hilbert curve          Koch's snow-flake      Heighway's dragon boundary
;Sierpinski's carpet    Gosper's curve         Heighway's dragon
;Heighway's interior    Heighways curd         Mandelbrot's arrow-head
;Brick curve            Pentigree curve        Lace curve
;Brick interior         Moor's necklace        Chrismas tree
;Mandelbrot's quintent  Dekking's church
;
;so according to the above definition a straight line has a husdorff
;dimension of one the same thing applies to the circle and the ellipse.
;Also the Hilbert curve is not a fractal since its dimension is 2

;Best Regards
;Mhelhefny.
;------------------------------
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Thu, 25 May 2000 03:00:00 GMT

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