On Fri, Jun 19, 2015 at 6:15 PM, David MacQuigg <macqu...@ece.arizona.edu> wrote:
> Hi Kirby, > > Nice work. I've used Mandelbrots to introduce Python to engineering > students at U of A. > engr.arizona.edu/~edatools/ece175/Lecture/ > The images are breathtaking in their beauty. > engr.arizona.edu/~edatools/ece175/Lecture/Mandel00.html > engr.arizona.edu/~edatools/ece175/Lecture/Mandel_zoom_14.jpg > > If I do this lecture again, I'll try a variation like yours with PIL to > generate the images. I never could get pygame working right. In > principle, gaming routines should provide fast rendering of complex images, > flashing to the screen as fast as I can generate them in C, but the > overhead of my function calls seems to be ruining the potential for > creating a Mandelbrot movie. > > David MacQuigg, Business Manager > Benson Animal Hospital > purl.net/macquigg > > Nice and colorful, which mine was not. As one runs z = z * z + c over and over, more and more of the points "peel away" i.e. spiral into divergence and thereby leave the set. It's like sandblasting: the more you cycle, the more flecks blast away -- but then how long did they linger? That's where the coloring comes in, and I didn't use any (this time). Before it scrolls out of scope I should link to a thread on math-teach where I do some followup and contextualizing: http://mathforum.org/kb/thread.jspa?threadID=2716437 (re Mandelbrot as a core curriculum topic) Where I went with that thread was in favor of "Gnu Math" at the high school level, by which I mean breaking out of the TI mold in which many still find themselves. Do like they do in South Africa with the TuxLabs: let math collide with bash. Mix it up more with free software. Coding is still mostly on the other side of a firewall in math class, STEM notwithstanding, in many a public or private high school. However, that sorry state of affairs has been changing, slowly but surely. We started out in Oregon by re-purposing an old Discrete Math, making that more computational while still eligible for maths credit (three needed for a HS diploma in Oregon, similar in other states). I've more recently been consolidating some of these advances, in my own postings, using a different heuristic: Delta versus Lambda Calculus (fine to use the Greek letters). Delta Calc: our familiar pre-calc / calc (differential / integral calc), the staple of high school Lambda Calc: meant as a doorway to a more semi-numerical skill set (e.g. SQL and regular expressions), i.e. the Gnu Math, something different and relatively new in the high school classroom. The stuff I'm lumping under lambda calc is what we might expect to find under CS (CompSci) but again, a goal is to stay on track for the three math credits and a diploma. CS has been treated as recreational / extra-curricular in that sense. By making this stuff a "calculus" and assigning a Greek letter, we make it harder to push off stage. However, I do get more specific, in the above math-teach thread, in how I think the gateway to a lambda calculus could be framed. I'm not always so hand-wavy. We start off along this trajectory, around Algebra, by introducing a lesson about: Re-purposing numeric operators, multiplication and exponentiation (e.g. * and ** in Python), to mean: "composition" of functions i.e. (f * g)(x) == f(g(x)) and (f ** 3)(x) == f(f(f(x))). I've posted about this topic on math-teach a lot already. And what does any of this have to do with the Mandelbrot Set? Given a patch of complex plane (a square of "cloth" or "skin" C), we compute M(M(M(M...M(C)...))) i.e. (M**n)(C) where M is our repeated "sand blasting" operation. That's just one edge of connection. Kirby
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