written after a day spent exploring fractals (again)...

[kirby urner]

In my "focal points" for a STEM curriculum (stealing jargon from NCTM
of that day), I included Mandelbrot Set for one of the four.

I deliberately chose what I'd call "complex topics" i.e. when you zoom
in, you get lots of detail, could go on for hours (but don't have to).

Like another of the four was Pascal's Triangle, which fleshes out to
Pascal's Pyramid and more -- all sorts of number patterns there.

Back to fractals, they've continued to explode and lots of work has been done.

Computer science has help make computational thinking an experimental
exercise, complete with feedback loops, lots of trial and error.

What would be the equivalent of the Mandelbrot Set in volume (I'll
deliberately not say "in three dimensions" for reasons of my own).

Interestingly, the aesthetic quality of having lots of rich and varied
detail no matter how far one zoomed, was the key filtering criterion,
so candidate "3D Mandelbrots" were dismissed on aesthetic grounds.

An offshoot or branch in this research is what's now called the
Mandelbulb, a discrete / specific shape one could come to recognize
much as the flat 2D shape has become familiar.

Here it is: http://krzysztofmarczak.deviantart.com/art/z7-b-3D-fractal-2-138950342

The software for generating and filming such beasts is out there, and
I'm trying to work backwards from an environment that's at least
conducive to playing with such tools.

They're downloadable legally free of charge in many cases. Mainly one
just needs an excuse, a set aside social time, for using tools of this
genre.

http://www.mandelbulber.com/download.php

Ideally, that's where schools come in, whereas in practice we often
make it less practical to play in this way, by herding kids and to
buses and taking them away from their on-line university connection.

Adults don't get any time either but that should change, as
"retirement homes" prepare for those geeks who'd like nothing better
than to tinker with Linux for hours.

Anyway, that's an example of a foray which would need to be backed up
and reinforced other other kinds of hands on activities, such as just
playing with complex numbers interactively.

More exhibits:

https://www.youtube.com/watch?v=is-dFvEZDlA&feature=related

http://youtu.be/bO9ugnn8DbE

Focal Points: http://mybizmo.blogspot.com/2006/09/focal-points.html

http://www.skytopia.com/project/fractal/mandelbulb.html

The idea the public schooling might also be a function in the home
environment is not new: it's called homework.

More homework of this kind, with guardians encouraged to set aside a
study space in the house, and/or provided as community spaces, like
coffee shops, is what to expect.

http://mathforum.org/kb/message.jspa?messageID=7830736

Kirby

[Donald Cohen]

Kirby, take a look at the applet http://www.ies.co.jp/math/java/comp/itoi/itoi.html which gets fractals from raising (a+bi)^(a+bi).. or iterates (a+bi)..  ies found i^i^i..  and its graph in my worksheet book and made the applet much more interesting and really spectacular.

Don

Kirby

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[Alexander Bogomolny]

You may want to have a look at another applet that illustrates the behavior of iterations in the Mandelbrot set. The fascinating aspect of the exercise is that the iterations correspond to the near-by bulbs. The interface is simple - just click for a new starting point. (You may have to press the Clear button initiate the applet.)

 

Alex

[kirby]

Apparently these Mandelbulbs don't need complex numbers. They're in
R3, in spherical coordinates. An 8th power is involved, though that
itself is a variable.

I don't yet understand the solution space, but I'm happy with the
visuals. What I consider pre-geared for those with academic
inclinations.

Related blog post: http://controlroom.blogspot.com/2012/06/at-toms.html
(has a picture of this 8th power beast -- beautiful beast may I add).

Kirby

[kirby]

Here's a rendering of the classic Mandlebulb I was able to get within
a few minutes of installing free software.

http://www.flickr.com/photos/17157315@N00/7331167528/in/photostream

Again, in terms of my four focal points, this fits the bill as
homework for a flipped classroom, where we meet to project our own
portfolios in lightning talk / ignite show & tell format.

Does this sound a lot like elementary school then (show & tell)? You
bet it does.

I have no problem with public funding for schools being about making
the home space a safe and well equipped one.

Alaska is a pioneer because being out on the road can be dangerous,
whereas the Internet might still be working fine. Cell towers don't
need the roads to stay clear.

Hi Anna.

Kirby