Notes for the Lifebox, the Seashell, and the Soul - Rudy Rucker

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Notes for The Lifebox, the Seashell, and the Soul, by Rudy Rucker how easy it would be to build a virtual computer out of a StarWars pattern, it is reasonable to call it class four. It’s also my impression that if you click one of the dead pixels into the firing state, the effects may propagate outwards to an arbitrarily large degree. It could be that the cascade sizes obey a histogram power law, in which case StarWars would serve as a class four rule which supports inverse power law statistics. The StarWars image was made with Mirek Wójtowicz’s downloadable Windows executable Mirek’s Cellebration, which can be obtained from his http://www.mirekw.com/ . Note that Mirek has a Java version of his program online as well, although the standalone executable seems to run a bit faster. Mirek discovered the StarWars rule. It has 4 states which we think of as being the ready state 0, the firing state 1, and the 2 resting states 2 and 3. The update rule is as follows. If a cell in the ready state 0 has exactly two firing neighbors it goes to the firing state 1. If a cell in the firing state 1 has three, four or five firing neighbors then it stays in state 2. Otherwise it goes to the resting state 2. A cell in the resting state 2 goes to the resting state 3. A cell in the resting state 3 goes to the ready state 0. Examples of News Stories Nearly everyone knows about the big ones. The invention of automobiles. The Second World War. The fall of the Berlin Wall. The dot com bubble. Harry Potter. 9/11. Life is Hard at Every Level Returning once again to my concern about being a mid-list non-best-selling author ⎯ here I am stuck out on the fat-but-not-all-that-fat tail of the gravy train. Criticality means that at least things are boiling at every level. So it’s never going to be dull. In fact it’s not easier to send your stories to very low-circulation magazines or take jobs you’re over-qualified for. There’s just as much hassle at every scale where you’re competent. Power Law Notes Natural phenomena often obey a certain scaling property called a power-law distribution. To set the stage, we first break some phenomenon into components of various sizes, and then for each size level, we evaluate how much the components of this size contribute to the phenomenon as a whole. In a power-law distribution, there is some characteristic exponent D such that the relative contribution of pieces of a given size R is proportional to the reciprocal of R raised to the power D. Intensity of contribution from the pieces of a given size R ~ 1 / R D More formally, there will an exponent D and some constant factor C such that if P(R) is the intensity or strength of the contribution from the pieces of size R, then P(R) = C / R D Let me illustrate with a simple example, the sizes of rocks on a mountainside, which obey a power-law distribution with D = 1. To begin with, we’ll measure the sizes of the p. 86
Notes for The Lifebox, the Seashell, and the Soul, by Rudy Rucker rocks in terms of their volume. And we’ll measure the “intensity” of the rocks of a given size by counting how many rocks of that type there are. For simplicity’s sake, I’ll describe a way of getting rocks of different size levels by repeatedly making copies of the existing rocks and break the copies in two. • Level 0. Start with a big rock of size 1. • Level L+1. Make copies of the Stage L rocks and break each of the copies into two equal pieces. Admittedly, the notion of copying rocks is artificial. But as you can see in the figure below, we could also think in terms of starting with an appropriate number of big rocks and breaking each of down to a different level. [Figure (not drawn)] We can reach L levels of division by starting with L+1 big rocks and shattering each of them through successively more steps. If we go down to level L, we then have: • 1 piece of volume 1, • 2 pieces of volume 1/2, • 4 pieces of volume 1/4 ... and • 2 L pieces of volume 1 / 2 L . Now in each row, the volume of the pieces is the quantity R = 1/ 2 L on the right, and the number of pieces of that volume is the quantity P(R) = 2 L on the left. If I combine the two equations I get: P(R) = 1 / R So in this case, as promised, we have a power law with D = 1. Now let’s ring in a change. Suppose we think of the rocks as cubes, and that we measure their size in terms of their edge length instead of their volume? And let’s also suppose that when we break a cube in two, we can, without changing the total volume, knead the pieces like clay so that they too are little cubes. In this case, we’d have: • 1 piece of edge 1, • 2 pieces with an edge equal to the cube root of 1/2, • 4 pieces with an edge equal to the cube root of 1/4 ... and • 2 L pieces with an edge equal to the cube root of 1 / 2 L . If we write R for the edge length, then at level L, R 3 equals 1 / 2 L , and the reciprocal of this is the same as the intensity P(R) = 2 L . So we get this kind of power law. P(R) = 1 / R 3 If you don’t like the idea of kneading the pieces into cubes, you could instead be cutting each cube into eight pieces. 1 piece of edge 1, 8 pieces of edge 1/2, 64 pieces of edge 1/4 ... and 8 L pieces of edge 1 / 2 L . Now here, since 8 L is the same as (2 L ) 3 , we again have P(R) = 1 / R 3 p. 87
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Notes for the Lifebox, the Seashell, and the Soul - Rudy Rucker

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