Exercise: Finding order in chaos
Part 3. Viewing thematic similarities between Julia and non-Julia fractals
The same numbers and themes also show up in Julia sets whose seeds are in and around these bulbs. To show this, do the following:
1) Zoom into a bulb whose characteristic number you know.
2) On the File menu, click Duplicate as Julia.
Notice how the characteristic number also shows up in the Julia set.
3) Moving through the major bulbs to the right increases the number of arms (and the cycle length) by one. Moving to the left through the major bulbs increases the number of arms by two. And the patterns and rules are subtly different. Why? I don't know.
We've seen how the bulbs in the Mandelbrot set can count, and can do addition. Now let's see them do multiplication.
4) If you go to the three-arm bulb (the fairly large one at the top of the Mandelbrot set), you can see some smaller bulbs coming off of it.
5) If you go into the topmost one of these, you get a cycle of six -- two orbits, each three long.
6) As you move along to other major bulbs, the number of orbits keeps increasing, thus allowing you to multiply any number by three.
If you want to multiply by a number other than three, you'll have to move into a bulb with a different base cycle length.
These regularities can be useful. If you find a picture you like, but you wish it had a few more arms, with a bit of practice and with careful use of the View Location command, you can find other areas that are variations on your picture.