By Kollar J., Lazarsfeld R., Morrison D. (eds.)
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Extra resources for Algebraic Geometry Santa Cruz 1995, Part 2
3: If A is even, then replace A by A/2 and go to step 2. 4: If A is odd, then replace A by 3A + 1 and go to step 2. Let us try a few choices for A: • 3, 10, 5, 16, 8, 4, 2, 1, STOP • 34, 17, 52, 26, 13, 40, 20, 10, 5,16, 8, 4, 2, 1, STOP • 75, 226, 113, 340, 170, 85, 256, 128, 64, 32, 16, 8, 4, 2, 1, STOP The obvious conjecture is: the algorithm comes to a stop no matter what the initial A is. Well, it seems that the larger the initial A is the more steps we have to run until we arrive at 1. Let us try A = 27 to verify that guess.
The second surprise is that the copy machine paradigm is not just a way to recover 'mathematical monsters' like the Sierpinski gasket or its relatives (soon we will see many of them). Let us ask what the images are which we can obtain this way. What can they look like? The answer is simply incredible. For many more natural pictures there is a copy machine of the above kind which generates the desired picture. How does one design the machine for a given picture? Well that is a difficult problem, as one can guess.
In any event, this extremely simple feedback system demonstrates very dramatically how complicated structures can be the result of very simple feedback. In a way, this is the theme of the book. Our next set of experiments tries to bring more of a systematic light into this world of exciting phenomena. The basic principle is the same as with video feedback: an initial image is processed and then the resulting image is reprocessed by the same machine over and over again. 2 The Multiple Reduction Copy Machine We now tum to a set of experiments which will provide us with a very intuitive access to the language of fractal geometry.
Algebraic Geometry Santa Cruz 1995, Part 2 by Kollar J., Lazarsfeld R., Morrison D. (eds.)