INTRODUCTION
 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 the trunk". 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 r =.6

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.

 Some Notation 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".

[ NEXT ]      [ PREV. ]      [ INDEX ]      [ FEEDBACK ]