Self-Similarity

The first day of school this year, we drew two fractals: Cantor Dust and a Koch Snowflake.  To refresh your memory, Cantor Dust was formed when we drew a segment, then removed the middle third of the segment.  Then, for the next iteration, we removed the middle third of the remaining two segments.  We repeated the process several times, noting a paradox: if we could continue this process infinitely, we would have an infinite number of "segments" with a total length of zero.

We also drew a Koch Snowflake.  To refresh your memory, from a regular triangle, we removed the middle third of each side of the triangle and constructed a smaller regular triangle using the removed segment as the base.  We repeated this process several times.  We noted that the perimeter would be infinite, but the area would approach a specific value, thus a paradox similar to that found with Cantor Dust.

http://www.stsci.edu/~lbradley/seminar/fractals.html

What's amazing about fractals is that they have a property called "self-similarity."  Given an infinite number of iterations, if you zoom in on a piece of the fractal, it looks identical to the original. The fractal is composed of infinitely smaller shapes each similar to the larger shape.

The word "fractal" was coined in 1975, so the study and experience of fractals is relatively new.  Computers facilitated understanding of these shapes. 

And also common culture has given us accessibility to these ideas:

Your task for this blog could be one of  a variety of activities.  (1) you could go to "First Friday Fractals" at the Museum at Natural History (This Friday is a first Friday of the month, so you could go that early!) and write very briefly about not just the spectacular art, but something about the mathematics behind the art (that's discussed in the show).  (2) You could read about the history of the development of fractals and discuss that.  (3) You could find and discuss another fractal (Sierpinski triangle or Menger Sponge for examples)  Regardless of which of the three you chose, you need to include the topic of similarity in your blog  -- after all that is the topic we've been studying.

(As always, if you come up with a different idea for this blog, ASK!  I'm always inclined to say yes, but before you chose to deviate from this topic, please ask to verify that your idea is an ok one to use for this blog post. )