• August 14, 2022, 09:22:19 PM

Author Topic:  A spiral is only fractal if you zoom into the center - Is there a name for this?  (Read 938 times)

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Offline Fractal Institute

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While we keep struggling with finding a simple to understand explanation and definition for what makes out a fractal we came across this question.

A spiral is only fractal if you zoom into the center.

If you zoom anywhere else, you either reach emptiness or a line with infinitely small curvature.
So where you zoom is absolutely crucial to the question of "is this mathematical object a fractal"
Is this a special class of fractals?

In the Mandelbrot-Set this is called a Misiurewicz_point
Other fractals have similar behaviour. The Cantor-Set for example must be zoomed into the ever dividing "bottom line".

What are your thoughts on that?






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« Last Edit: August 16, 2017, 09:39:36 AM by Fractal Institute »

Offline Sockratease

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Really?

My opinion?

I guess I can talk about this since it's Mathematical Objects and not "real-world" objects in question.

This is a point I make often when being skeptical about fractals.  I don't agree that the spiral is a fractal if it is just a spiral and not part of a larger structure like those Misiurewicz points found in the Mandelbrot Set which you mentioned.

A "plain" spiral is not self similar - it is self identical!  To me, it is structurally identical to a straight line, or the border of a circle, in terms of what happens when you "zoom in" to it.

And therefore not a fractal.

I have been run out of discussions for saying that The Cantor Set is too much like a spiral to be Fractal.  While not self identical, there is simply not enough variation to qualify as fractal in my personal definition.  But it is a lot closer to fractal than a spiral!

I'd call these things "Semi-Fractal" or "Quasi-Fractals" since they lack the diversity I feel is needed to rise above self identity and into the realm of self similarity.

Similarity requires diversity if it is to be distinguished from Identity.
Study stupidity, kids. It'#039#039s not getting smarter out there - Frank Zappa

Offline Fractal Institute

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Interesting standpoint.

Stating that the Cantor-Set doesn't qualify to be a fractal shows that you have a very different perception of the topic.


The Cantor set is the prototype of a fractal.
It is a purely mathematical object, strictly self similar to infinity with a Hausdorff dimension of 0.6309 and is made of small copies of it self.

It basically doesn't get more fractal than that.


The following reasoning is far too little to overthrow the common textbook knowledge.
..there is simply not enough variation to qualify as fractal in my personal definition.
..they lack the diversity I feel is needed to rise above self identity and into the realm of self similarity.

Offline Sockratease

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I know.

And I always end up relenting and granting the Cantor Set the right to be called a fractal - but only over my strenuous objections!

It just doesn't seem like it should qualify to me, so I argue.

And every once in a while, my stubbornness pays off.

Not with Fractals, but I've gotten a few weird chemical formulas on the market because they just felt like they'd work for something and despite them "not being for that purpose" I eventually got my boss to foot the bill and have my formula tested  (no small achievement since the tests in question are needed to get some products legally able to be sold and cost a couple thousand dollars just for the test!).

So yeah, if you insist - I'll allow the Cantor Set to be called a fractal.

But I wont believe it is one!  For a scientist, I place an unscientifically high value on my "gut feelings" and personal views   O0

Offline hobold

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Oh by the way, the name for something staying the same after (a particular depth of) zooming is "scale invariance". Literally "it doesn't vary if you scale".

Individual isolated points of larger objects being scale invariant is not so special. Things get more interesting if scale invariant points are dense in the object (meaning there are infinitely many such points, and any small disk covering a continuous (non-zero weight) part of the object will contain such points). Fractals are among the few objects that contain a dense subset of scale invariant points.

A simple straight line is also scale invariant in any of its points, so that alone does not suffice for being fractal.

Offline Chris_M_Thomasson

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While we keep struggling with finding a simple to understand explanation and definition for what makes out a fractal we came across this question.

A spiral is only fractal if you zoom into the center.

If you zoom anywhere else, you either reach emptiness or a line with infinitely small curvature.
So where you zoom is absolutely crucial to the question of "is this mathematical object a fractal"
Is this a special class of fractals?

In the Mandelbrot-Set this is called a Misiurewicz_point
Other fractals have similar behaviour. The Cantor-Set for example must be zoomed into the ever dividing "bottom line".

What are your thoughts on that?

What makes you think about that line of thought?
It's a Fractal Life, 247... ;^)

Offline youhn

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A line is a fractal, made out of line segments, made out of smaller line segments, etc. Just all in the same direction.

A point can be a fractal, giving the same shape zooming into the point itself.

Nothing is a fractal, since you see the same from any angle or zoom level. The ultimate fractal.

Everything is also a fractal, because any subgroup of the all-universal EVERYTHING is basically (in practical sense) also Everything, which contains more everythings, which contain more things.

Things can be fractal, since things can be collections of other similar things.

Ideas are things, so all above applies to to all physical things and abstract ideas.

Basically everything on all levels, from all angles is fractal.

Except of course the atom, or unit, or base, whatever you name it.

Which then conflicts with the fact that EVERYTHING is a fractal, so we must conclude that there is no such thing as a single thing. Which can be generalized to simply saying "There's nothing".

Enough said.
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