Orbits in the Mandelbrot Set

What is an orbit?

The orbit feature of the Fractal Microscope allows you to visualize the calculations involved in determining which points are included in the Mandelbrot Set. When you select "Orbit" from the pop-up menu and click on the image, the Fractal Microscope will return an image that traces the path of the actual calculation involved in this process. The name "orbit" is nrelated to the periodic nature of the paths traced by the calculations. This picture shows the orbit of a point within a lobe at the top of the Mandelbrot Set. The orbits of the points within this main lobe converge to orbits of period 3.

How does the actual calculation work?

A visual representation of the Mandelbrot Set is similar to the graph of a function. The horizontal axis represents Real numbers, and the vertical axis represents Imaginary numbers. The origin (zero on both axes) is very close to the lower point of the triangle in the above picture.

In the Mandelbrot Set z**2+c, each point is squared and added to itself N times (indicated by "depth"). If this new number that we get after N iterations leaves the circle of radius 2 centered about the origin, we say that the point we started with is outside the Mandlebrot Set and we color it in. Otherwise, we leave the point black. The different-colored bands surrounding the Mandelbrot Set signify that the points within each band left the circle of radius 2 after the same number of calculations.

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