Rotational Symmetry

Robert P. Munafo, 2003 Dec 16.

There is much (of what appears to be) rotational symmetry in the filaments. Rotational symmetry manifests itself in three different forms:

Surrounding Branch Points with order corresponding to the denominator of the internal angle of a mu-atom through which one passes to get from the branch point to R2a. For examples, see R2F(1/3)B (3-fold rotational symmetry), R2F(5/8)B (8-fold) and R2F(1/2(1/3B1)B) (2-fold and 3-fold).

Surrounding Islands with an order that is a power of 2, doubling as you get closer to the island (see Bifurcation).

In the embedded Julia sets and their paramecia. Since Embedded Julia Sets surround islands, this symmetry also has a power of 2 order and bifurcation. See 2-fold embedded Julia set and n-fold embedded Julia set.

In all cases, the rotational symmetry is only approximate. For example, look at R2F(1/3)B and R2F(5/8)B. Aside from the obvious fact that the arms are not the same length, and one of them has the entire rest of the Mandelbrot Set hanging off of it, there are other somewhat more subtle differences:

- The arm leading to the Mandelbrot Set is perfectly straight. (Well, almost: again, if you examine closely enough you find that it is slightly curved). The arm leading away is fairly straight too. The others are crooked.

- There is a sort of mirror symmetry in the crookedness, with the two "straight" arms as a centerline. The arms on the "right" take a sharp left turn just before the islands and a sharp right turn afterwards; the arms on the "left" bend the other way.

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