Cool Stuff: Fractals

2009

# Alternative Julia-type Sets:

### 3: "Standard Julia Set" vs. "sine Julia set"

Because the sine function is intrinsically related to the the properties of right-angled triangles, and because these proportions are given by Pythagoras' Theorem ("the square of the hypoteneuse equals the sum of the squares of the other two sides"), we're dealing with powers of two, as with the conventional Julia images. And it turns out that if we examine a certain part of the sine parameter range, we find counterparts of the conventional Julia images, evolving in the same basic pattern.

Parallel evolution of the standard Julia and "sine Julia" sets, D=0
standard Julia fractal image "Sine Julia" counterpart
linear
pointy
domes
circle
four-leaf
scroll

On the left we have the conventional progression of Julia set images for D=0 (see also this progression as a video), and on the right we have the corresponding evolution running through part of the "Sine Julia set" parameter range.

While the fundamental shapes are the same in each case, the "Sine" version contains branching and intersecting threads that emerge from each feature on the shape, and aggressively duplicate the pattern. This is the sort of behaviour that we're more used to seeing with the Mandelbrot Set than with conventional Julia set images.

As with the conventional Julia set images, varying the imaginary component ("D") causes any solid regions to break up, and twists parts of the image clockwise or counterclockwise. This gives us counterparts of other Julia set images, and again, the basic shape is repeated strongly in the form of little islands surrounding the main figure.

D<>0
... zoom ...
http://www.relativitybook.com/

The "sine" versions have more "hair", and are more aggressively self-duplicating.

http://www.relativitybook.com/

Unfortunately, once you've gotten used to the "sin Julia" shapes, their "conventional" Julia counterparts start to look a bit boring.

### Reinventing the Circle

The most obvious example of how the "sine" Julia fractals differed from their more conventional counterparts was the "circle" case. For a conventional Julia Image taken at A=0, B=0, we got a simple, single, non-fractal circle. This clearly isn't the case for the "sine" counterpart. For anyone who thought that circles and formula-driven fractals weren't comfortable together, here's the counterexample.

So here it is again, this time larger and in colour. Behold the Glory of the Circley Fractal!

Since a circle doesn; thave any corners to act as obvious focal points for threads and bulbs, this image is a little surprising. Are these circle-like features actually threaded together (which would mean that their outlines weren't truly circular), or are they proper, perfect closed circles, that just happen to be butted up against each other in rows, to form the appearance of threads?

Dunno.

### Colour Versions

Here's the "scroll" image in colour, and a closeup of one of the scroll spirals:

In Colour
... zoom ...

where the usual Julia set produces a simple fractal spiral scroll consisting of a line of smaller similar spiral scrolls, the outline of this version is pierced by criss-crossing "hairy" threads that spawn new versions of the shape around each junction.

www.relativitybook.com