From the Simple to the Complex

We will now see what happens to various Complex points 'C' when we iterate them through the equation Z_n+1 = Z_n² + C. This exploration will explain the concept of Periodicity

Periodic Orbits

Click in the left panel to choose a value of C, represented by a white cross. The right panel shows the points that represent the evolving value of Z for the chosen value of C. If you check the 'Points' checkbox, it will draw a dot at every point that Z passes through. Uncheck the 'Animate' checkbox if you want to instantly see the behavior of a single point instead of animating the journey to that point Use the arrow keys to nudge the cursor by one pixel at a time.
Click mouse to choose a value of "C" <Alt>Click mouse to Zoom In (either panel) <Ctrl>-Click mouse to Zoom Out.

Click on the image above which lies in the Complex Plane. Play around with the applet. Wherever you click in the left panel, you are choosing that Complex value for C and iterating it through the equation. The right panel shows the Orbit, or trajectory of the value of Z as the equation is iterated 100 (?) times.

Click in the black area inside the main body of the Mandelbrot Set, and you can see the orbit converge to a finite value. Click just outside the edge of the Mandelbrot Set somewhere, in the colored area, and you will see the orbits diverge, and spiral away to infinity.

Try clicking inside and around some of the bulbs attached to the main body. Look for certain shapes in the orbit pathways, triangles, squares, pentagons, star shapes, etc.

Periodicities

Click inside the Period-3 bulb (See the map in Chapter 3 if you've forgotten where that is) and describe the shape made by the orbit of Z. You should see a triangle. We call this the Period-3 bulb because the value of Z behaves periodically, returning to the same point every 3 iterations.

If you visit the next large bulb over, to the right, you will find the Period 4 bulb, in which the orbits follow a square trajectory. Continuing clockwise down the large valley on the right, you'll encounter pentagonal behavior in the Period 5 bulb, you'll find hexagonal orbits in the Period 6 bulb, and you can continue all the way down the valley, discovering ever higher-order polygons, corresponding to the periodicities of the bulbs.

Question:
What are the coordinates of the Period 6 bulb?Real: [ ] Imaginary: [ ]

Return to the Period-3 bulb, and explore in the clockwise direction this time. You'll come to the Period-5 bulb, but this time, instead of a simple pentagonal orbit, you'll see that the vlue of Z follows a star-shaped pathway. The same is true for the Period-7, Period-9 etc bulbs, all the way down the Seahorse Valley. Interestingly, the major bulbs in this region have only odd-numbered periodicities, which makes some sense, since the simple star-shaped orbit requires an odd number of points.

Twisting

<Alt>-click to zoom in near the Period-3 bulb. Click in the main body, just below the Period 3 bulb, and you will see a triangular tunnel in the orbit diagram, converging on the stable point inside the bulb. As you move your mouse to the left or right of the center, you can see the triangular tunnel twist to the left or right. This is a demonstration of the Spiraling Rule seen at the level of the orbits.

Note the complex periodicities in between the major bulbs, and see how they follow the Periodicity Rule for addition.

Now that you've learned your way around the surface level, you're ready to go deeper! <Alt>-click to zoom in and look at some of the close-up details along the edge of the Mandelbrot Set. See what happens to the orbits in the Seahorse Valley. You can <Alt>-click to zoom in to the orbit diagram on the right as well, to see its structure in more detail too.

Question:
Test the Periodicity Rule using orbits.

Zoom in and and find the orbit for the largest bulb between the Period-3 bulb and the Period-5 bulb. What is its periodicity? (How many points are on the star? [ ]