Mathematical details on fractal trees

Comments on, and Mathematical Details for
an article by Mandelbrot and Frame
Self-Contacting Fractal Trees
by Don West, Department of Mathematics, SUNY Plattsburgh

Abstract
We offer two very different forms of Mandelbrot and Frame's equation (2) for tip contact, and point out some gaps in logic and errors in statements. We give proofs, or at least outlines, for our assertions.

The Spring 1999 issue of The Mathematical Intelligencer offers a ten page article on fractal trees by Benoit Mandelbrot and Michael Frame. The article, which elaborates on Chapter 16 of Mandelbrot's famous Fractal Geometry of Nature, concentrates on self-contacting trees (technically, not trees because they contain non-trivial cycles) and calculates the fractal dimension of the tip set, the canopy, and the shortest path set in these trees. The authors make dozens of claims, both qualitative and quantitative, without proof. They pass off the proofs as "elementary but tedious" and in fairness, including the proofs and derivations would balloon the article from ten pages to an entire issue of the popular journal.
When I asked by e-mail where one could see the details of the investigation, Frame responded "...there isn't, nor is there likely to be, a version of the paper with all details spelled out. It would be a bit long and I think fairly boring." He suggested that a web page might be the appropriate medium for such material. I hope that this page will begin to serve the purpose.
As charming as it is, The Mathematical Intelligencer is not a large circulation journal, so to help readers who do not have the original article available, I shall repeat some of its statements, sometimes with my own comments.
HTML is not designed to reproduce mathematical notation. Indeed, the Math Forum page suggests simply using .gif images to present equations. We used this solution here, with apologies to users of text-only browsers. We also removed several text symbols, such as the angle and triangle symbols, not known to earlier browsers. If you cannot read these pages, but would like to do so, let me know and we will work something out.

MORE STUFF --
Prof. Frame has posted some animations on the on a Union College Mathematics Department web site showing trees sweeping through various changes, take a look!

As alternatives to Mandelbrot and Frame's equation (2):

we offer the cosine equation:

and the closed form:

These pages are in serial order and each one has links for the [NEXT] and [PREV] pages. The contents listed below are sorted by type, not in serial order.

CONTENTS:

Algebraic Derivations

equation (2) implies my cosine equation
cosine equation implies the closed form (4 ways!)

Geometric Derivations

my cosine equation
the closed form

Critical Comments

How tall is a tree, really?
Tip contact iff ________ ?

Other items

Advantages of the Cosine Equation
The Moran Generator Equation
Simplified form of Equation (3)
Properties of the (r, theta) graph

SECTIONS (IN ORDER):

INDEX (this part)
INTRODUCTION and DEFINITIONS
The COS EQ, CLOSED FORM, and comments
GEOMETRIC DERIVATIONS of the
COSINE EQUATION the CLOSED FORM

COSINE EQ. implies CLOSED FORM
Simplified form of EQUATION (3)
Facts and Questions about FIGURE 9
Details of the LEFT END of the (r, theta) graph.

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