Paint can cardioids

As I looked into a nearly empty paint can, I noticed a familiar heart-shaped graph:


I'm used to seeing the cardioid-shaped region of the Mandelbrot set (what my students have sometimes called "the butt crack"), but also noticed it in my Spirograph program, where a family of graphs can be generated that seem similar to the paint can cardioids (or caustics):



You can get the same cardioid graph by drawing a cycloid around a circle of the same diameter (an epicycloid):

Speaking of Fibonacci

Updated Teacher Soundboard

Here's a classroom soundboard that I have found useful.  It's 2mb, so be patient.

Fibonacci right outside

After all the fun I've had with simulated sunflowers, I thought it was time to grow some real ones.  They're probably 8 feet tall now.



I wondered if the Fibonacci sequence (1, 2, 3, 5, 8, 13, 21, 34, 55...) really showed up in the seeds, or whether it was just a close approximation.


Opening this image in Paint and putting a dot on each of the spirals, I found Fibonacci numbers in full bloom: 55 clockwise spirals and 34 counterclockwise!

Ian Stewart's Nature's Numbers gives a brief but sufficient explanation for this phenomenon in plants, and doesn't go overboard with finding the Golden Ratio everywhere, as I have been annoyed to see some do.