Cardioid Boundary Orbits

Points on the edge of the main cardioid of the Mandelbrot set straddle two worlds. The interior of the cardioid contains points with fixed-point orbits: the orbits eventually settle down to a final value. Points immediately outside of the main cardioid are either divergent (their orbits go to infinity) or periodic (their orbits settle down into a loop). Right on the boundary, if the point is the point of tangency of a disk, then the orbit will combine both fixed point and periodic behaviors--it will converge to a final value, but do so very slowly and through a loop of values. Below are shown the orbits for the points tangent to the 1/4, 2/5, and 5/8 disks.

This coloring is based on points on the edge of the main cardioid that are not tangent points for disks. Since they are not associated with periodic disks, these points have orbits that are chaotic, not periodic. They wander around a well-defined region, but never exactly retrace their steps.

Finding these points is easy, if you know where to look. All points along the boundary of the main cardioid can be characterized by an angle, from 0° at the cusp (c = 0.25), positive counter-clockwise to 120° at the top (tangent to the 1/3 disk, c ~ -0.125,0.65), 180° at the neck (tangent to the 1/2 disk, c = -0.75), 240° at the bottom (tangent to the 2/3 disk, c ~ -01.25, -0.65), and 360° back at the cusp. Alternatively, the angle can be expressed in radians, where one full circle, or 360°, is 2π radians (about 6.28). Thus, the above angles correspond to 0, 2π/3, π, 4π/3, and 2π radians. Another angle unit is the turn, where one full circle is one turn. This is handy because the angle of a tangent point, measured in turns, is exactly the same as the fraction for that disk. So, to find a point that is not a tangent point, we need the angle to not be a fraction of a turn, or not a fraction time 2π radians, or not a fraction times 360°. Since π is an irrational number, as is any fraction of it, one easy way to find such angles is for them to have a whole number of radians, like 1, 2, or 4. The orbits for points at these angles look very different from the orbits of tangent points:

So what? Well, if these points have chaotic orbits, then that means that neighboring orbits diverge very quickly. Or, the orbits of two points that start out very close together will soon move very far apart. We can illustrate this by making a new orbit out of the difference between the orbits of two points, and see where that goes. That’s what this coloring does. Here is the orbit between 1.999 radians and 2.001 radians.

Another way to see the difference between the two numbers is to form their ratio. Here’s what happens with these two angles:

If you can take the difference or ratio of two numbers, you may as well add and multiply them, too, and do all sorts of other fun things.