LOGOL> What is a Fractal?
Author 
Message 
Yehuda Ka #1 / 20

LOGOL> What is a Fractal?
Hello Turtlers, I wonder what do we mean by Fractal: * Is every infinitly selfrepeating image a Fractal? * Are Fractals only images which have a fractional dimension? * Is a circle a Fractal? * Is a spiral a Fractal? * is a straight line a Fractal? I'll appreciate any help provided to make things clear for me. Regards... [[Yehuda]] 

Mon, 22 May 2000 03:00:00 GMT 


Tom Woo #2 / 20

LOGOL> What is a Fractal?
Quote: >I wonder what do we mean by Fractal:
Most of what I know about mathematics has been selftaught, so my understanding of fractals is probably skewed somewhat. When I think of a fractals I think of things which, no matter how closely you look at them, no matter how greatly they are magnified, there is always variation. Hence, clouds are fractal shapes, as are coastlines, leaves, and the view of the mountain I see as I look out my window. To me, lines and curves would not be fractals because they lack variation. This does not fit well with "selfrepeating" edges such as Koch snowflakes. Perhaps others can help me round out my working definition. Tom 

Mon, 22 May 2000 03:00:00 GMT 


Olga Tuzo #3 / 20

LOGOL> What is a Fractal?
Thu, 04 Dec 1997 03:52:17 +0200
Quote: > Hello Turtlers, > I wonder what do we mean by Fractal: > * Is every infinitly selfrepeating image a Fractal? > * Are Fractals only images which have a fractional dimension? > * Is a circle a Fractal? > * Is a spiral a Fractal? > * is a straight line a Fractal? > I'll appreciate any help provided to make things clear for me. > Regards... > [[Yehuda]]
It's rather... interesting(?), isn't it. We have had so many programs for drawing fractal curves here and now Yehuda's question seems to catch us by surprise. Here is a beginning of the article from http://shum.cc.huji.ac.il/~radai/fracwhat.htm  What is a Fractal? Copyright (c) 1995, 1996 by Yisrael Radai. All Rights Reserved. A fractal is a certain type of geometric figure. Below we will explain how fractals differ from "ordinary" figures (i.e. from the figures studied in Euclidean geometry and other traditional branches of mathematics). Those who are not so interested in the explanations should at least be aware that fractal images are not designed in advance. The shapes come out naturally as the results of simple mathematical processes. The only things which are chosen by humans are the particular region of interest, the particular colors used to fill in the shapes produced by the process, and possibly some minor variations on the basic algorithm. (Actually, one does occasionally find "fractal" images which are designed in advance to some degree. Typically these are landscapes which merely make use of some fractal techniques but are not what one would ordinarily call fractals. See, for example, Musgrave's work EECS News: Fall 1994: Building Fractal Planets. I call such images "artifracts".) Regards, Olga. 

Mon, 22 May 2000 03:00:00 GMT 


Jef #4 / 20

LOGOL> What is a Fractal?
Benoit Mandelbrot coined 'Fractals' and 'Fractal Geometry'. He describes a fractal as 'rough but selfsimilar', on the one hand NOT Euclidean, but on the other, NOT geometrically chaotic. The endeavour of fractal geometry is to sort noneuclidean shapes and curves into 'orderly' and 'disorderly' chaotic. References are: 'The Fractal Geometry of Nature', Mandelbrot; 'The Beauty of Fractals', Peitgen&Richter; and for a Turtle Geometry spin, 'The Algorithmic Beauty of Plants', Lindenmayer. cheers Jeff Richardson 

Mon, 22 May 2000 03:00:00 GMT 


Yehuda Ka #5 / 20

LOGOL> What is a Fractal?
Tom and Jeff, I appreciate your answers. How do I apply them to find if a binary tree is a fractal or not? What about a spiral, a circle, a straight line? Can a fractal be a degenerated one? Regards, [[Yehuda]] Quote:
> Most of what I know about mathematics has been selftaught, so my > understanding of fractals is probably skewed somewhat. When I think of a > fractals I think of things which, no matter how closely you look at them, no > matter how greatly they are magnified, there is always variation. Hence, > clouds are fractal shapes, as are coastlines, leaves, and the view of the > mountain I see as I look out my window. > To me, lines and curves would not be fractals because they lack variation. > This does not fit well with "selfrepeating" edges such as Koch snowflakes. > Perhaps others can help me round out my working definition.
> Benoit Mandelbrot coined 'Fractals' and 'Fractal Geometry'. He describes > a fractal as 'rough but selfsimilar', on the one hand NOT Euclidean, but > on the other, NOT geometrically chaotic. The endeavour of fractal > geometry is to sort noneuclidean shapes and curves into 'orderly' and > 'disorderly' chaotic.


Mon, 22 May 2000 03:00:00 GMT 


Yehuda Ka #6 / 20

LOGOL> What is a Fractal?
Quote:
> It's rather... interesting(?), isn't it. We have had so many programs > for drawing fractal curves here and now Yehuda's question seems to > catch us by surprise. > Here is a beginning of the article from > http://shum.cc.huji.ac.il/~radai/fracwhat.htm >  > What is a Fractal? > Copyright (c) 1995, 1996 by Yisrael Radai. All Rights Reserved. > A fractal is a certain type of geometric figure. Below we will > explain how fractals differ from "ordinary" figures > (i.e. from the figures studied in Euclidean geometry and other > traditional branches of mathematics). Those > who are not so interested in the explanations should at least be > aware that fractal images are not > designed in advance. The shapes come out naturally as the results of > simple mathematical processes. > The only things which are chosen by humans are the particular region > of interest, the particular colors used > to fill in the shapes produced by the process, and possibly some > minor variations on the basic algorithm. > (Actually, one does occasionally find "fractal" images which are > designed in advance to some degree. Typically these are > landscapes which merely make use of some fractal techniques but are > not what one would ordinarily call fractals. See, for example, > Musgrave's work EECS News: Fall 1994: Building Fractal Planets. I > call such images "artifracts".) > Regards, > Olga.
Hi Olga, Thank you for that quote (written by a researcher at the Hebrew University of Jerusalem Israel). Unfortunately my questions are still open: Is a binary tree a fractal? Is a spiral a fractal? What about a circle? A straight line? What are the minimal demands from an image to deserve the title of Fractal? Or: Maybe "Fractal" is only a convention, with a vogue meaning of an infinitely selfrepeating image. Regards... [[Yehuda]] 

Mon, 22 May 2000 03:00:00 GMT 


Jef #7 / 20

LOGOL> What is a Fractal?
Quote: > How do I apply them to find if a binary tree is a fractal or not? What > about a spiral, a circle, a straight line? Can a fractal be a > degenerated one?
Yehuda Mandelbrot specifically excludes Euclidean figures, which do exhibit some selfsimilarity. This is done using something called the 'Hausdorff Dimension' to separate 'smooth' from 'chaotic'. A fractal has a noninteger Hausdorff dimension. The Hausdorff dimension is derived from set theory, measuring the number of sets needed to cover THE set of the fractal(or otherwise) figure. This is the limit of my understanding. The boundary condition for the onset of 'nonfractal' chaos is as yet undefined. Jeff 

Mon, 22 May 2000 03:00:00 GMT 


Yehuda Ka #8 / 20

LOGOL> What is a Fractal?
Quote:
> > How do I apply them to find if a binary tree is a fractal or not? What > > about a spiral, a circle, a straight line? Can a fractal be a > > degenerated one? > Yehuda > Mandelbrot specifically excludes Euclidean figures, which do exhibit some > selfsimilarity. This is done using something called the 'Hausdorff > Dimension' to separate 'smooth' from 'chaotic'. A fractal has a > noninteger Hausdorff dimension. The Hausdorff dimension is derived from > set theory, measuring the number of sets needed to cover THE set of the > fractal(or otherwise) figure. > This is the limit of my understanding. The boundary condition for the > onset of 'nonfractal' chaos is as yet undefined. > Jeff
Jeff, What puzzles me is that Harvey, in his book (vol I, ed.1, p. 134) calls binary trees "Fractals". Now, a binary tree doesn't fit, to my limited understanding, in the above definition(s). So maybe there are more than one definition for "Fractal", or am I wrong somewhere? Regards, [[Yehuda]] 

Mon, 22 May 2000 03:00:00 GMT 


Brian Harv #9 / 20

LOGOL> What is a Fractal?
Quote:
>What puzzles me is that Harvey, in his book (vol I, ed.1, p. 134) calls >binary trees "Fractals". Now, a binary tree doesn't fit, to my limited >understanding, in the above definition(s). So maybe there are more than >one definition for "Fractal", or am I wrong somewhere?
It's entirely possible that my understanding of the word "fractal" is flawed. I used it to mean "selfsimilar figure." (And of course to be truly selfsimilar the tree would have to be infinitely many levels deep. The Logo program draws an approximation to a fractal.) But I'm not quite sure why this is such an interesting thing to worry about. A line segment is a degenerate case of an ellipse. Is a line segment an ellipse? Well, for most purposes, I'm sure most people would say "no." But occasionally it's useful and/or interesting to see what happens if we apply our ideas about ellipses to a line segment. Similarly, there might be some situations in which it would be interesting and/or useful to think of a line segment as a fractal, but ordinarily one wouldn't.

Mon, 22 May 2000 03:00:00 GMT 


Jef #10 / 20

LOGOL> What is a Fractal?
Quote: > > Mandelbrot specifically excludes Euclidean figures, which do exhibit some > > selfsimilarity. This is done using something called the 'Hausdorff > > Dimension' to separate 'smooth' from 'chaotic'. A fractal has a > > noninteger Hausdorff dimension. The Hausdorff dimension is derived from > What puzzles me is that Harvey, in his book (vol I, ed.1, p. 134) calls > binary trees "Fractals". Now, a binary tree doesn't fit, to my limited > understanding, in the above definition(s). So maybe there are more than > one definition for "Fractal", or am I wrong somewhere?
'Classic' Turtle Geometry trees are very much fractals, they arise from iteration theory, which is where Mandelbrot began. Jeff 

Mon, 22 May 2000 03:00:00 GMT 


Olga Tuzo #11 / 20

LOGOL> What is a Fractal?
Dear Yehuda, Thank you for rising these questions. I've got a lot of pleasure looking through articles in Internet. I think, you and those who are interested will like the very clear article by M.Connors at http://www.math.umass.edu/~mconnors/fractal/fractal.htm Regards, Olga. 

Tue, 23 May 2000 03:00:00 GMT 


Yehuda Ka #12 / 20

LOGOL> What is a Fractal?
Quote:
> Dear Yehuda, > Thank you for rising these questions. I've got a lot of pleasure > looking through articles in Internet. > I think, you and those who are interested will like the very clear > article by M.Connors at > http://www.math.umass.edu/~mconnors/fractal/fractal.htm
Olga Dear, Thank for that. Unfortunately I couldn't enter that URL. I'll try it again tomorrow. Shabat Shalom, [[Yehuda]] 

Tue, 23 May 2000 03:00:00 GMT 


gub.. #13 / 20

LOGOL> What is a Fractal?
Quote: Yehuda Katz writes:
> > http://www.math.umass.edu/~mconnors/fractal/fractal.htm Try: http://www.math.umass.edu/~mconnors/fractal/fractal.html 

Tue, 23 May 2000 03:00:00 GMT 


Dale R. Re #14 / 20

LOGOL> What is a Fractal?
Quote:
> > Dear Yehuda, > > Thank you for rising these questions. I've got a lot of pleasure > > looking through articles in Internet. > > I think, you and those who are interested will like the very clear > > article by M.Connors at > > http://www.math.umass.edu/~mconnors/fractal/fractal.htm > Olga Dear, > Thank for that. Unfortunately I couldn't enter that URL. I'll try it > again tomorrow.
Worked for me and it is a very interesting site. I have already forwarded it to other educational discussion lists. But it did not work the first time so, as I have learned to do, I started deleting the last part of the address until it worked and then surfed forward from there. Dale  


Wed, 24 May 2000 03:00:00 GMT 


Yehuda Ka #15 / 20

LOGOL> What is a Fractal?
Quote:
> Yehuda Katz writes: > > > http://www.math.umass.edu/~mconnors/fractal/fractal.htm > Try: > http://www.math.umass.edu/~mconnors/fractal/fractal.html
Thank you, I managed to enter the site. Seems very attractive, and I'll revisit it tomorrow (the time here is about 4 after midnight...) Bye, [[Yehuda]] 

Wed, 24 May 2000 03:00:00 GMT 


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