19.
The most obvious part of the Mandelbrot set is the large cardioid and the infinite sequence of tangent disks, with the largest at the left. Each of these tangent disks has an infinite sequence of tangent disks attached to it, and so on ...
The image at the location shown below is of the second largest disk that is attached to the largest disk that is attached to the main cardioid. It shows that the sequence of attached disks continues. This creates infinitely many chains of tangent disks that leave at a point of tangency to the main cardioid. But the image also shows that the these chains of disks that start at a tangency to the the main cardioid do not make up the whole Mandelbrot set. They make up a large part that we can call the main "mandelbug." there are many copies of the mandelbug in the Mandelbrot set that are attached to the main one by filaments.
(The color images obtainable from this page range in size from 5,500 bytes to 98,000 bytes.)
c= -1.1953125+0.328125i, radius=0.1171875, skip=5, bands/color=1.
19.
The largest secondary mandelbug is centered on the real axis with cusp at -1.75. This mandelbug is placed in its image below to approximately match the placment of the the full Mandelbrot set in its image.
c= -1.7546875+0i, radius=0.0358632651, skip=7, bands/color=1.
20.
It is seen that the secondary mandelbug does not match the main mandelbug exactly. The main cardioid is distorted, the attached disks are not proportional, and the filaments are straighter and proportionally longer. Thus the copies of the main mandelbug are not copied in exact proportion.
There are other variations that are found as one considers different parts of the Mandelbrot set. The filaments emanating from the top "bud" of the main cardioid branch so that three filaments leave each branch point.
c= -0.10112096+0.95627072i, radius=0.30555, skip=2, bands/color=1.
21.
c= 0.33+0.59i, radius=0.12, skip=4, bands/color=1.
At the next largest "bud" to the right, four filaments leave each
branch point.
22.
c= 0.415+0.215i, radius=0.04, skip=7, bands/color=2.
Two "buds" farther down, 6 filaments leave each branch point.
23.
The next color image combines the top "bud" and filaments of the secondary mandelbug (the one with cusp at -1.75+0i) with the top "bud" and filaments of the main cardioid pictured in image 19 above. The "bud" on the secondary mandelbug is on the left half of the color image. The similarities and the differences are striking. The locator in-line image and the image data are for the left half of the color image.
c= -1.7576944919+0.0176803446i, radius=0.0056, skip=13, bands/color=3.
24.
Except for the symmetry across the real axis, it turns out that each point on the boundary of the Mandelbrot set has a unique "neighborhood." While different parts of the Mandelbrot set may resemble each other up to a scale factor, in fact they are quite different. It may take much magnification to see the difference however.
Forward to What is the
Mandelbrot set - Page VI (9 embedded images).
Back to What is the
Mandelbrot set - Page IV (12 embedded images).
Back to The Mandelbrot Set and Julia Sets.
