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.

;Your Discussions about fractals is very interresting. Specially your

;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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