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Exploring the Mandelbrot Set    

Robert P. Munafo, 2023 Jul 1.

Most people who encounter the Mandelbrot set and begin to explore it discover its various features in a particular order. Each is discussed separately. In order of discovery, these features are:

1. The Continent (and self-similarity).

2. The Filaments and dwell bands.

3. The Island mu-molecules.

4. Filament Symmetry including self-similarity and two types of rotational symmetry (related to mu-atom Periods and bifurcation).

and then, at roughly the same time:

5a. Nested Filament Symmetries
5b. The unique attributes of Cusp filaments

Then one reaches a bit of a lull. To most observers, these properties seem to account for everything in the structure of the Mandelbrot Set. In fact, most people will stop here, and some technically-minded enthusiasts have devised Naming Systems based on these features alone (plus possibly the External Arguments).

Persistent explorers and those with a more mathematical background will also discover:

- The External Arguments.

- The Embedded Julia Sets.

Depending on exploration style and the availability of a viewer that uses extended precision, one of the following is likely to be discovered next:

- Embedded Julia sets tuned by sparse Fatou dusts

      which transition gradually into

- The Second-Order Embedded Julia Sets.

When deeper zooming (typically by magnifications of 1030 or usually much more) the explorer is likely to discover Julia morphing and Leavitt navigation. Exotic structures such as polaftis and Julia "trees" are just one of the huge variety of things that have been found.

See also:

algorithms, for discussion of how to write a Mandelbrot program.

history, for a brief history of the Mandelbrot Set's exploration.

R2 for a guide to the largest features of the Mandelbrot Set.

enumeration of features for a discussion of the various number sequences that are discovered when the Mandelbrot set's features are counted.

See also features, Feigenbaum point, paramecia, R2.C(1/2)

revisions: 20020418 oldest on record; 20230701 add Leavitt navigation, Polaftis, Julia morphing

From the Mandelbrot Set Glossary and Encyclopedia, by Robert Munafo, (c) 1987-2024.

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