NCTM 2021

What happens when three plants orbit around each other in space?

Can you draw a path that crosses every bridge once, but not more than once, without entering the water? You can start and end on any area of land.

Can you connect each of these utility companies to each of the houses, without any of the lines intersecting?

In this diagram, we highlight all points x0 on the complex plane, for which the recursive sequence xn=xn−12+c is bounded (it doesn't diverge).

For Julia sets, we chose a fixed value for c, and then changed the position of x0 to colour the plane. Now let’s fix the value of x0=0, and instead change the value of c:

xn=xn−12+${complex(c)}

x0=${complex(x0)}

x1=${complex(x1)}

x2=${complex(x2)}

x3=${complex(x3)}

…Bounded!Diverges!

Like all fractals, we can “zoom into” the Mandelbrot set forever, finding new patterns at every scale. Here you can zoom into a part of the Mandelbrot set called the Seahorse valley:

Scale: ${pow(scale)}

The chaos game is a way to generate fractals using a simple rule. You start with a point, repeatedly pick a random vertex of a polygon, and then mark the midpoint of the line from your original point to that vertex. Then you continue from that new point. What shapes can you make?