This chapter is a brief 'anatomy' lesson on the Mandelbrot fractal, covering the conventions I'll be using when talking about it. The Fractal defies an easy description, but I'll try to break it down for you.
Below, picture T1 outlines some of the primary features of the Mandelbrot set:
The heart shaped area shaded in red, known as the 'primary cardioid' (for the cardioid shape it resembles), comprises most of the area of the Mandelbrot set. The two circular regions (circled in blue) lying tangent to the primary cardioid are known as bulbs or atoms. From now on I'll be referring to them as atoms. There are an infinite number of these atoms attached to the main cardioid, each atom with its own group of atoms, and so on all the way down. The atoms are all separated by infinitely thin pinchers of froth (which I refer to as pinchoffs - more on these in chapter 6). The entire Mandelbrot shape I'll refer to as the parent molecule.
The largest atom, left of main cardioid, I call the primary atom. The second largest atom, above the main cardioid, I call the secondary atom. There are two of these secondary atoms mirrored about the real axis.
Finally (still referring to picture T1), the areas circled in green represent what I call the froth. Each atom off the parent Mandelbrot is connected to an area of froth. The Froth lies along the boundary of the Mandelbrot set, and is littered with and infinite number of small molecules resembling the parent molecule. It's a chaotic area where the Mandelbrot set (taking the form of many molecules or lakes) mixes in with the boundary (froth).
Let's take a closer look at one of the atoms, outlined in image T2 below:
Below, picture T3, shows the atom outlined in T2, magnified:
The area circled in blue shows the pinched area between the primary atom (shaded in red) and main cardioid. Blue arrows indicate direction of pinch. The area circled in green shows the five arms of froth associated with the atom shaded in red. Note all the smaller sub-atoms and their associated areas of froth, with all atoms being separated from one another by froth pinchers. Also note the small Mandelbrot molecules (copies or lakes) evident in the outer arms of the froth. These small copies also comprise part of the Mandelbrot set, and are connected to it.
Two more atoms below, images T4 and T5:
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Remember, the froth isn't technically part of the Mandelbrot set, which is displayed in black. Froth takes the form of an infinity of encirclements about the ragged edge of the Mandelbrot set. Froth can be seen as the encircling bands of color (especially evident in picture T3). The froth is usually comprised of several lines or rivers (one or more, see green arrows) which slowly fall forever about points within the froth (circled green areas). Each line of froth is ultimately made up of an infinite lattice or filigree of molecules. Since these molecules are connected on an infinite scale, they will always appear disconnected to us mortals. One of the purposes of the froth appears to be to let us know that there is a connected network of molecules (which belong to the Mandelbrot set proper), even though we can't grasp it in its entirety.
Circled in red is what I call the break molecule. This molecule is the largest molecule in the froth associated with a given atom. It also tends to display the most asymmetry (more on this next chapter). The froth contains an infinite network of molecules, but their respective sizes decrease very quickly. You might note that only a few other black specks are visible in the froth, which represent other molecules.
The encirclements that make up the froth can be viewed as a corrupted radial symmetry. No encirclement of froth will ever cross another. No encirclement is ever broken, though it will be folded and twisted and stretched beyond the limits of human imagination. In theory it's possible to follow a given encirclement around the Mandelbrot set, no matter how far one is zoomed in. Also notice how the froth from each atom never crosses that of another atom, though they can share a very convoluted border.
The general rule is for the the froth to become more twisted and 'furry' as one zooms in, like the path a tortuous river might take. Picture T4 is an example of such an atom and its froth. It uses a color map with a long period to bring out the detail, and this is why the encirclements of color are not as visible in this picture. Also notice how the number of arms making up the froth have increased.
Some of the above points should become clearer if you follow along some of the zoom sequences in the coming chapters. Next chapter, more on the small Mandelbrot molecules (copies or lakes).