Phyllis from All Things Beautiful asked if I could explain our Apollonian pumpkin patch a little bit better, so I will try - keeping in mind that my basic idea today was to get the boys drawing and coloring, not to teach them a math lesson.
To begin with, an Apollonian gasket is a fractal that starts out with three tangent circles being drawn inside another circle, so each circle is also tangent to the outside circle. Then the spaces left outside the three circles are filled in with sets of three tangent circles, and the spaces left by them are filled in with three tangent circles, with all the circles growing smaller, but continuing into infinity - or something like that. Below is a picture of an Apollonian gasket from Wikipedia.
It can be turned into a pumpkin patch, by adding lines in each circle, to suggest the ridges of pumpkins, and then coloring all the circles orange, until you get to the really tiny circles, that can be colored green, to suggest vines and leaves.
Our printer was refusing to load paper this morning, so we had to draw our pumpkin patches freehand, meaning they weren't at all mathematically correct, but still a lot of fun to draw and color.
Of course, if you haven't seen it yet, this would be a good time to check out ViHart's "Doodling in Math Class" video, too.
It's great to be a homeschooler.