A few days ago I made a post on fractals and said I would continue it. Well, this is that continuation. This post is about the Mandelbrot variants. The variants I include here are the exponents of the Mandelbrot. Next time I post on fractals it will be about the absolute value variants. I would like to cover them all here, but that would take too long. This post assumes that the reader already knows about the Mandelbrot Set, for a recap/intro see my last post.

[Disclaimer: Mandelbrot Exponentials is probably not an official term.]

Explanation

This is an explanation of the Mandelbrot Exponentials. If you already know about the Exponentials or are just here for the pretty pictures (or both) then skip to Pictures, although there will be some confusing things that I will refer to that are in this section; so if possible please read this section.

The Mandelbrot Formula is Z^2 + C. Now of course someone after seeing this thought, ‘Why 2?’ and so changed it to 3 to see what happened. This happened.

For comparison, here’s the normal Mandelbrot Set (with a colour scheme change):

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And then they thought, ‘what about 1?’ And they were bitterly disappointed. It’s OK, but…

And then they thought, ‘0’, and then they were sad. Anything to the power of 0 is 1, so it just got stuck at 1 and could never get anywhere, let alone infinity. Next they thought ‘2.5’ and this happened.

If you look closely you can see it’s falling apart at the seams. No thank you, sorry. So that person (whomever they were) was stuck with integers. Negative integers don’t work (they are the same as 0). 1000 is very interesting, but I should leave some of the pictures for the section called ‘Pictures’. So without further ado…

Pictures

Please read the previous post. It’ll be helpful to refer back to that for this section.

Level 1

First we start with the Z^3 + C variant. Zooming in towards the top reveals this interesting antler-like structure (spot the minibrot!) Let’s zoom deeper; into the antlers! (Official name: filaments).

This is the section in the top right, and the answer to the minibrot puzzle from the last picture. Let us go deeper!

Level 2

I have very quickly run out of Level 1 things – the Mandelbrot Exponentials are very interesting.

The largest minibrot in the Exp.3 Mand (shortening for the Exponent 3 Mandelbrot (another way of saying the Z^3 + C Mandelbrot)). It is tied for first place with 3 other minibrots, which are exact mirror images of the first one, flipped over the X axis, Y axis or both axes.

The largest valley of the largest minibrot. If you saw the previous post then you might be able to guess where we’re headed next…

Into the triangular formation at the bottom! And…

And, you’ll NEVER guess what’s coming next…

I’ll zoom in for ages on a completely different nearby triangle that is completely invisible in the previous image. (Seriously, it’s less than a pixel.) Congratulations to the non-real person who got that right. We are rewarded for this seemingly mad decision by this:

This is the inside of that triangle; spot the minibrot. (It’s actually less than a pixel, but its existence can be inferred.)

I think I’m going to need another section for the inside of that circle (actually it’s a nonagon with the vertices missing but close enough).

Level 3

Here we are; the minibrot! Let’s go even deeper.

This is the main valley on the left. One final zoom before we leave the Exp.3 Mand.

And now, to the Exp.4 Mand. (We won’t stay for long, we only want to see the main thing.)

Do you see the pattern yet? Hint: 1, 2, 3… It might be helpful to zoom in on the area to the left.

If this doesn’t help, then the next one should; it is the Exp.5 Mand.

1, 2, 3, 4… On to the Exp.10 Mand.

Note: The size of the Mandelbrot Exponentials are not fluctuating; I’m just varying the size to include the whole thing. Now, it’s time for the grand finale:

A circle! And now we zoom in, on the right (no I am not deranged (I think)):

It’s DEFINITELY a circle and not Exp.1000 Mand.

See? This is perfectly normal circle behaviour. Not a fractal.

Alright, fine. It’s the Exp.1000 Mand. (Tricked ya!) Wait, what do you mean, it was obvious? This was a great trick that I played on you. You must be lying. Anyway, no more of that. We need to zoom in further. Hang on. What do you mean, I’m talking to myself? What? No, you are a perfectly real person who actually is saying all these things, and I am responding despite you being a person and me being an author who can only say things once and cannot reply to you. In short, you are a real person. What? This joke isn’t funny? OK, that might be right. Anyway. We need to zoom in. Stop distracting me, existent person.

I’m going to stay silent now and just zoom in.

I really don’t know what on earth is going on with any of this. It makes no sense to me! There are simply too many minibrots. None of the actual pattern can shine through the chaos. But hopefully this was a good finale. Wait, you wanted me to end on something more beautiful? OK, have it your way.

By the way, this happens to be a Level 3 image that I hopefully will also include in the next post. What? You STILL don’t exist? Really? This isn’t funny? Well, you brought it up.

Anyway, I hope you enjoyed this post and I will make the next one VERY soon. (Seriously it’s going to be next week or something along those lines. I really like fractals.)

See you again then.

What? That sentence has no referent? Okay, I give up.