ART MATRIX PO 880 Ithaca, NY 14851-0880 USA (607) 277-0959, Fax (607) 277-8913 'The Paths of Lovers Cross in the Line of Duty.' IN DEFENSE OF THE MASTER Copyright (C) 1991 Homer Wilson Smith All rights reserved. 'In the preface to The Science of Fractal Images, Mandelbrot suggests that fractal geometers also use computer graphics to develop hypotheses and conjectures. But the difference is that the hypotheses and conjectures are (like the objects which they study) self- referential. One generates the pictures to learn more about the pictures, not to attain deeper understanding. That the pictures have occasionally inspired fine mathematicians to prove good theorems seems serendipitous at best.' Steven G Krantz, The Mathematical Intelligencer, Vol 11, No 4. I have probably burned more CPU cycles than most in the search for pretty pictures, thus I can sympathize with Krantz's deploring such use of computer time, however I have also worked long and closely with Dr. Hubbard during many of the years that he was first interested in fractal mathematics and there is something that needs to be said here. In the first place Dr. Hubbard is the first to tell his graduate students that pretty pictures are a waste of time unless they can PROVE something about them. Proof, it would seem, is the coin of the realm. I know this for a fact because I have had to listen to the endless woes of poor graduates students who can produce the most amazing pictures, but who have a very hard time proving anything. 'Hubbard wants me to PROVE something', they complain. Further my own experience working with Dr. Hubbard has given me a direct and personal insight into the relationship between pretty pictures and deep mathematical cognition or proof. In the first place, no picture can ever prove a conjecture, just as one example can never prove a hypothesis. However one measly picture CAN DISPROVE a conjecture in no time flat. Dr. Hubbard is a fine one for coming up with endless conjectures about iteration theory, and he uses the images that I make for him to scan anxiously for the one that will prove the conjecture wrong in an absence of immediate analytical ability to prove or disprove it formally. It is an enormous waste of time trying to prove something right that is indeed wrong, and if one can bypass that effort by making a few pictures to see if the disproof is easily forthcoming, then making such pretty pictures is well worth the time. Secondly, we were recently working on a problem of the intersection of two quadratics and the behavior of Newton's Method in this space to find the points of intersection. Dr. Hubbard directed me to make a whole slew of movies, each movie had a 1000 frames, that scanned the parameters from low to high. During the viewing of these movies it became apparent that there was a line in the space that Dr. Hubbard could not immediately prove should be there. Thinking about it some he suddenly came up with the conjecture that this line was invariant under Newton's method which meant that any iteration starting on the line would forward iterate to another point still on that line. He got very animated suddenly, dragged me over to a table and said 'This is amazing, I can't believe how stupid I am to not have thought of this before, I wonder if I can PROVE it is true!' He then proceeded to drag me through the entire procedure of working through the proof, possibly for the first time in human history. Now THIS is the stuff of real mathematics, all of Steven Krantz to the contrary. And I was there. Homer