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© Copyright 1993 Mario Hilgemeier


Consider these strings: 1, 11, 21, 1211, 111221, 312211, ... what is the next item? If you think its 13112221, you're right. Why? Speak aloud, "one" 1, "this is one one" 11, "now we have two ones" 21, "thats one two and one one" 1211 - you get the idea. Therefore Conway has called the iteration rule "audioactive". I've called it "Gleichniszahlenreihe" (GZR, see glossary), because of its analogy with meaning (what does 11 mean? 1 ), description (what is 11? 21), and growth because the string length (i.e. the number of characters) grows regularly (exponentially) with a factor of about 1. 303577269 (in the limit), according to Conway [13] (Figure 7.2).

Figure (7.)2


Conway was the first to discover that these strings, growing exponentially in length, do split "naturally" at certain points. After such a split, each of the fractions develops without influencing the other any more; then it splits again. Such a fraction is called an "element". An example of this split is shown in figure 3. The last string of the primordial elements, 13112221, splits into 131 and 12221; this is indicated by a space. In the following audiactive decay the pair 11 (where the split occurs) is changed to 21. But the resulting two parts 11132 and 13211 can be considered as independent in the next stages. For instance, the string called 71Lu in figure 3 develops from 72Hf; we do not need to know that the whole string is longer.

In analogy to the decay of radioactive elements, Conway called the splitting process "decay". If you start with the number 1 (or any other character) and let it grow for some hundred iterations, you get only 92 (mostly instable) elements of the audioactive decay. Because there are also 92 known (stable) chemical elements in nature, and some of them change by radioactive decay, Conway named them uranium (U), protactinium (Pa) down to helium (He) and hydrogen (H) . Each of these elements (except hydrogen) transmutes into another or splits into two or more elements. In this "nuclear-chemical" view, the Gleichniszahlenreihe starts with a "primordial element" 1, which rapidly develops into the elements Hafnium (Hf,11132) and Tin (Sn, 13211). A "primordial element" (see figure 3) is an element that never recurs in audioactive decay. Figure 4 shows the complete development scheme for all 92 elements.

Figure (7.)3

Figure (7.)4

In table 1 (a table of elements) you find a column denoted "element abundance". The abundance of an element is defined as the number of atoms of each element per million atoms. The appropriate measuring unit is "ppm", parts per million. How can we compute the abundance of an element - why do they have particular abundances anyhow? We will consider this question below.

Figure (7.)5

The sawtooth-like form of the element abundances when drawn as a function of element number (figure 5) is caused by the decay process. There are certain development chains in which one element decays into exactly one other element. One instance of such a loop is the chain starting at 61Pm (middle left in figure 4). Because generally the development is from higher to lower element numbers, and because the total string length grows with each decay step, the higher element numbers in a decay chain are "better fed". Because of the many decay chains the sawtooths in figure 5 occur. You can see this also in table 1: abundances of elements with only one successor are exactly 1.303577269 (the growth factor) times greater than their successor (e.g. 60Nd and 59Pr). When the abundances of the elements are ordered by rank (starting with the highest abundance) nothing special can be seen (figure 6).

Figure (7.)6

A context-sensitive string-rewriting system can be defined in this way: Every character of the previous string is transformed according to its context (its neighboring ciphers or characters). This is exemplified in the GZR by the fact that a 1 doesn't always develop in the same way, depending on the ciphers that are before and after it in the string. In contrast, in a context-free grammar each character develops in a certain way, independent of its neighbors (See exemplary L-system below). By defining his elements, Conway achieves something remarkable: a context-sensitive grammar can be seen as a context-free grammar by going to the higher level of the "elements". Both grammar types are present in the same iterated system. At the start, when primordial elements develop, everything is context-sensitive. But as soon as the first elements appear, the context-free view becomes possible.

Stop and Think

Is the ordering of the elements in Table 1 really forced; that is, is it the only one possible? When you look at figure 4, you will find elements that have two or more outgoing arrows (e.g. 2He, 4Be, 31Ga, 32Ge). These elements develop into two or more elements. At these junctions one could change Conway's ordering. I would be interested to hear from readers who find an alternative ordering. This is the "labyrinthine" aspect of the GZR.

The generated strings 1, 11, 21, 1211 etc. can be interpreted as a number system with the base 3 (but without the zero). When position is counted from right to left, the first position has the multiplier 1(= 3^0), the second 3(= 3^1), the third 9(= 3^2), the forth 27(= 3^3) and so on. The number 10 in this system is written as 31(= 3*3 + 1*1), the decimal number 23 can be expressed as 212 (= 2*9 + 1*3 + 2*1). A numerical interpretation of the Gleichniszahlenreihe (GZR) can be found in table 2. Can you find any interesting number interpretations of the GZR-strings? Are there significant number-theoretic aspects?

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