These are screenshots from the iOS app Fractals from Wolfram. It aims less at aesthetics and more at insight. That is, if you know what you’re doing, because it’s always a surprise what could come out of your input. I understand that this is no different for professional mathematicians.
Directly above you see a Julia set, to the top a screenshot with info.
It seems the app might stop working after the upcoming iOS 10.3 update with the new file system that dumps 32 bit apps. Hopefully all apps will be updated in time. You can find more information on the Wolfram website, that has more than Mathematica. The app is 1.99, euros as well as dollars.
And as a bonus a real life fractal. A picture by the Juno spacecraft of Jupiter’s clouds. It’s amazing.