Hopping Midpoints

Put down three points on a piece of paper. Choose one of them as your “starting point”. Now, randomly choose one of the three points and hop from your starting point, halfway over to the chosen point. Mark down where you’ve landed. Then repeat: randomly choose one of the three starting points, and move halfway from your newly marked point to this new chosen point. Mark where you land. And on, and on, to infinity.

What pattern will arise? Watch and see!

E to increase points/second.
Q to decrease points/second.
Click and drag the red points to move them around.
Pressing a number key will make a polygon with that number of sides.

Exploring Julia Sets

Here’s a natural follow-up to my last post on the Mandelbrot set – an interactive Julia set explorer!

The Julia set corresponding to a particular point c = x + iy in the complex plane is defined as the set of complex numbers z that stay finite upon arbitrary iterations of the following function: fc(z) = z2 + c. The Mandelbrot set, by comparison, is defined as the set of complex numbers c such that the value obtained by starting with 0 and iterating the function fc arbitrarily many times converges.

What’s remarkable is now beautiful and complex the patterns that arise from this simple equation are. Take a look for yourself: just hover over a point to see its corresponding Julia set!

Resolution is preset at a value good for seeing lots of details and loading at a reasonable speed, but should you want to change it, controls are ‘E’ to increase it and ‘Q’ to decrease it. To reset to default, press ‘SPACE’.