LOGO-L> What is a Fractal?
Author Message LOGO-L> What is a Fractal?

Hello Turtlers,

I wonder what do we mean by Fractal:

* Is every infinitly self-repeating 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]]

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Mon, 22 May 2000 03:00:00 GMT  LOGO-L> What is a Fractal?

Quote:
>I wonder what do we mean by Fractal:

Most of what I know about mathematics has been self-taught, 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 "self-repeating" edges such as Koch snowflakes.
Perhaps others can help me round out my working definition.

Tom

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Mon, 22 May 2000 03:00:00 GMT  LOGO-L> 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 self-repeating 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
--------------------------------------
What is a Fractal?

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
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.
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Mon, 22 May 2000 03:00:00 GMT  LOGO-L> What is a Fractal?

Benoit Mandelbrot coined 'Fractals' and 'Fractal Geometry'. He describes
a fractal as 'rough but self-similar', on the one hand NOT Euclidean, but
on the other, NOT geometrically chaotic. The endeavour of fractal
geometry is to sort non-euclidean 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
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Mon, 22 May 2000 03:00:00 GMT  LOGO-L> What is a Fractal?

Tom and Jeff,

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 self-taught, 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 "self-repeating" 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 self-similar', on the one hand NOT Euclidean, but
> on the other, NOT geometrically chaotic. The endeavour of fractal
> geometry is to sort non-euclidean shapes and curves into 'orderly' and
> 'disorderly' chaotic.

---------------------------------------------------------------

Mon, 22 May 2000 03:00:00 GMT  LOGO-L> 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
> --------------------------------------
> What is a Fractal?

> 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?
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 self-repeating image.

Regards...

[[Yehuda]]

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Mon, 22 May 2000 03:00:00 GMT  LOGO-L> 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
self-similarity. This is done using something called the 'Hausdorff
Dimension' to separate 'smooth' from 'chaotic'. A fractal has a
non-integer 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 'non-fractal' chaos is as yet undefined.
Jeff
---------------------------------------------------------------

Mon, 22 May 2000 03:00:00 GMT  LOGO-L> 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
> self-similarity. This is done using something called the 'Hausdorff
> Dimension' to separate 'smooth' from 'chaotic'. A fractal has a
> non-integer 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 'non-fractal' 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]]

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Mon, 22 May 2000 03:00:00 GMT  LOGO-L> 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 "self-similar figure."  (And of course to
be truly self-similar 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  LOGO-L> What is a Fractal?

Quote:
> > Mandelbrot specifically excludes Euclidean figures, which do exhibit some
> > self-similarity. This is done using something called the 'Hausdorff
> > Dimension' to separate 'smooth' from 'chaotic'. A fractal has a
> > non-integer 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
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Mon, 22 May 2000 03:00:00 GMT  LOGO-L> 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.

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Tue, 23 May 2000 03:00:00 GMT  LOGO-L> 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]]

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Tue, 23 May 2000 03:00:00 GMT  LOGO-L> 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  LOGO-L> 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
--
--

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Wed, 24 May 2000 03:00:00 GMT  LOGO-L> 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]]

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Wed, 24 May 2000 03:00:00 GMT

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