Saturday, May 12, 2018

Dendrite and cross fractal families

Here are two fractal families that are probably not well known as families despite having well known members.

The first I'll call the dendrite family because it contains the attractive Pentadendrite. The six sided version is the Hexaflake and the four sided version is the Vicsek fractal and the two sided version is just a line:

The Hausdorff dimension is log(n+1)/log(3) where n are the number of sides, for the n=even case.
The n=odd case is a little higher. The three sided fractal I haven't seen before, so I'll call it the tridendrite. here's a close up:

The fractal is built up from a three pointed star by swinging a copy of itself 180 degrees around each outer point:
For even pointed dendrites the new outer point is simply the new farthest, for odd pointed dendrites like this, there are two equidistant new outer points, we choose either the clockwise, or the anticlockwise and stick with it, repeating the process ad infinitum.

Consequently there are two resulting fractals, one being a mirror image of the other. So while all the dendrites have n-fold rotational symmetry, the even pointed dendrites also have mirror symmetry.

A variation is to replace the shape each iteration with n copies, rotated 180 degrees and attached by their outer points to (0,0):
and repeated, giving a denser "cross" family:

The n=3 case is sometimes called a Fudgeflake, or can be built from three terdragons. The four pointed case is just a 90 degree Koch curve or Greek cross fractal. The larger n cases include substantial self-overlap.

By the way, here is a close up of the pentacross, the density variation has some nice patterns: