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
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.
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
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
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.
The "sine" versions have more "hair", and
are more aggressively self-duplicating.
Unfortunately, once you've
used to the "sin Julia" shapes, their "conventional" Julia counterparts
start to look a bit boring.
Reinventing the Circle
The most obvious example of
"sine" Julia fractals differed from their more conventional
the "circle" case. For a conventional Julia Image taken at A=0,
got a simple, single, non-fractal circle. This clearly isn't
case for the "sine" counterpart. For anyone who thought that circles
formula-driven fractals weren't comfortable together, here's
So here it is again, this time
larger and in colour. Behold the Glory of the
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?
Here's the "scroll" image in colour, and a closeup
of one of the scroll spirals:
... 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.
Eric Baird 2009