Mathematical Details (and comments) for|
an article by Mandelbrot and Frame:
Self-Contacting Fractal Trees.
by Don West, Department of Mathematics, SUNY Plattsburgh
"A trunk of length 1 divides into two branches of length r, each of which
makes an angle theta > 0° with the linear extention of
the trunk. Each branch then divides by the same rule."
Benoit Mandelbrot and Michael Frame begin their article with this brief discription
of the construction of a binary fractal tree. The animation illustrates the construction in case theta=30° and
r=.553. (Turn off the animation with the "stop" button before it drives you nuts.)
Definitions and Notation
Constructing a self-similar binary tree
A tree will be defined by three parameters:
- a length t (of the trunk)
- an angle theta (of the branches)
- a ratio r (of successive branch lengths)
We start with a segment (the trunk) of length t
(t>0) think of it as vertical.
(Usually, we consider a canonical tree with a trunk of length one.)
The bottom of the trunk is
the root, at the top we affix two branches, each of length tr. Each
of the branches "makes an angle theta with the linear extention of
To the free ends of each of these branches we affix two
branches of length tr2, again, making an angle theta
with the linear extension of the previous branch.
We continue in this way, adding at
the n-th stage, 2n branches of length trn.
A tree, constructed through stage 3 with
angle theta = 40 degrees and ratio
The union of larger and larger trees is not a closed set, the
completed tree is the closure of the union of all the finite trees.
When we take the closure, we add limit points of actual branch points.
We call these limit points branch tips and refer to their
aggregate as the tip set.
In a binary tree, the unique path from the root to any node (end
of a branch) can be given by specifying a sequence of right (R) or
left (L) turns. (The top of the trunk is located by the empty sequence,
don't ask how to designate the root.) So a sequence such as RLR not only
locates a node in the tree, it also describes the unique path from the
root to that node. Addresses of branch tips are infinite sequences
of "L" and "R".
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