Mandelbulb set

The Mandelbulb is a set generated similarly as the Mandelbrot set in two dimensions. These pictures were generated with Mandelbulb3D version 1.9 running on Mac OS 10 (the original Windows program was packaged by the Mandelbubl3D developers using Wine and runs nicely on the Mac (for some reason, it does not display on my macbook (and support logs seem to point to graphics card issues), but the following pictures were rendered on the Imac using that program)). Click on a picture to see a 7680 x 5760 resolution version. There is a nice citation (used as a logo) on that MB3D website:

"Mighty is geometry; joined with art, resistless."

As pointed out in this blog of a nameless math teacher, the quote is often attributed to Euripides (480-406 BC) but it has been deformed. Euripides wrote in his work ``Hecuba": "Numbers are a fearful thing, and joined to craft a desperate foe". [ We can read that Euripides was a tragic poet, whose work tragically has been mostly lost (to 80 percent) and (also tragic) was mocked by comic poets. Aristotle called him the "the most tragic of poets". It would be only consistent that also his work has been tragically deformed and distorted by future generations. ] If the etymology assessment of the blog is correct, then the quote deformed ``numbers" to ``geometry", ``craft" to ``art" and ``fearful" to ``mighty". In this quotation, the writer attributes the citation to Morris Kline "Mathematics for the Non-mathematician (1967)". Indeed, that is where it appears in 1967 [PNG]. The unknown Math teacher furthermore tells in that blog: This just proves how our culture has shifted. Today, it's all about the one-liner, the punchline. But most worrisome is the lean toward the quick and easily digestible without contextualization or scholarship. I agree that one should give attributions where due. I myself do not know the work of Euripides and definitely can not answer whether the quote has indeed appeared in some of his work. Morris Kline (1908-1992) was a legendary mathematician and writer. Whether he overreached with that quote is not clear. [His assessments about math education from the 70ies have been vastly confirmed (like An example: Why Johnny can't add: the failure of new math or (even better) Why the Professor Can't teach.] Reverberating maybe bit with Morris Kline, I don't quite agree with the Hetpadecagon blog critique about the one-liner. I think that we need the punchline. We need the easily digestible in education. These can be an entry points. The Mandelbulb is of that kind. There is not much contextualization, nor much scholarship yet about this set, simply because there is nothing known about it. In this collection of theorems [PDF] there is only one theorem about the Mandelbulb set: and the theorem tells "there is no theorem about the Mandelbulb set!". The Mandelbulb is not only beautiful, it has huge potential for new mathematics which currently does not appear to exist yet. A starter would be the analogue of the Douady-Hubbard theorem about the connectedness of the Mandelbrot set. And what is more exciting about having some objects we do know nothing about? A bit about the history of the Mandelbulb is given in this talk [PDF].