Technically impossible. 😎

Getting a true fractal tattoo would take forever

I'm going to leave the jokes about doing a fractal tattoo the same way a fractal is plotted (iteration by iteration) aside.

What you're going to want to do is find a tattoo artist that does a lot of very fine detail work, ideally using shading and membrane needles. Shading is probably the best way to depict a fractal image on skin, because, by definition, doing it point by point just isn't going to cut it. Ultimately it's an approximation on a rather low res medium (skin).

You're going to want to pick a fractal that will print well onto tattoo transfer paper. I don't know if freehanding it is going to be a good idea, or even feasible. My guess is that Sierpinski fractals would be your best bet.

If you're set on a Mandelbrot, maybe look for an artist that does a lot of oceanic depictions. It seems like if you're familiar with tattooing tentacles a lot, tattooing those parts of a Mandelbrot might not be too different.

What I would do is have the tattoo "signed" with the equation and parameters used to generate the fractal in question. But that's just me, do what you like.

There's a few ways to find the right artist.

1 - Word of mouth. do you have any friends with tattoos? Who'd they go to? I found my favorite artist this way.

2 - Social Media. Most artists maintain Instagram pages showcasing their work. Do a web search for shops in your area and then look at the artists previous work. This will help you start to find someone who fits the style your looking for.

3 - Go to some local shops and talk to the people there. More often than not you can find an artist that specializes in the type of work your looking for pretty easily this way. A good shop will either have someone who can do what you're looking for or give you some recommendations for others in the area that do.

These aren't hard, fast rules as different artists / shops have different cultures, but generally speaking they should be cool with helping you out. If you get a bad vibe or the people your talking to seem shitty or judgemental, move on. Most folks in the body mod community are pretty easy going. Don't put up with bullshit. That's not the norm. Don't be afraid to walk away and go somewhere else.

Good luck!