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.

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

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?