Contents

Reproduced by Henry Bottomley with permission

### Audioactive

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).

### Decay

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.

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.

### Table 1 Abundances, lengths and strings of Conway's elements

```Element    abundance in ppm  length     string
92 U       102.56285249      1          3
91 Pa      9883.5986391      2          13
90 Th      7581.9047124      4          1113
89 Ac      6926.9352045      4          3113
88 Ra      5313.7894999      6          132113
87 Fr      4076.3134078      10         1113122113
86 Rn      3127.0209328      12         311311222113
85 At      2398.7998311      7          1322113
84 Po      1840.1669683      10         1113222113
83 Bi      1411.6286100      10         3113322113
82 Pb      1082.8883286      9          123222113
81 Tl      830.70513293      12         111213322113
80 Hg      637.25039755      14         31121123222113
79 Au      488.84742983      18         132112211213322113
78 Pt      375.00456739      24         111312212221121123222113
77 Ir      287.67344775      28         3113112211322112211213322113
76 Os      220.68001229      34         1321132122211322212221121123222113
75 Re      169.28801808      42         111312211312113221133211322112211213322113
74 W       315.56655252      27         312211322212221121123222113
73 Ta      242.07736666      32         13112221133211322112211213322113
72 Hf      2669.0970363      5          11132
71 Lu      2047.5173200      6          311312
70 Yb      1570.6911808      10         1321131112
69 Tm      1204.9083841      14         11131221133112
68 Er      1098.5955997      9          311311222
67 Ho      47987.529438      7          1321132
66 Dy      36812.186418      12         111312211312
65 Tb      28239.358949      16         3113112221131112
64 Gd      21662.972821      11         13221133112
63 Eu      20085.668709      7          1113222
62 Sm      15408.115182      6          311332
61 Pm      29820.456167      3          132
60 Nd      22875.863883      6          111312
59 Pr      17548.529287      8          31131112
58 Ce      13461.825166      10         1321133112
57 La      10326.833312      5          11131
56 Ba      7921.9188284      6          311311
55 Cs      6077.0611889      8          13211321
54 Xe      4661.8342719      14         11131221131211
53 I       3576.1856107      18         311311222113111221
52 Te      2743.3629717      13         1322113312211
51 Sb      2104.4881933      7          3112221
50 Sn      1614.3946687      5          13211
49 In      1238.4341972      8          11131221
48 Cd      950.02745645      10         3113112211
47 Ag      728.78492056      12         132113212221
46 Pd      559.06537945      18         111312211312113211
45 Rh      428.87015042      24         311311222113111221131221
44 Ru      328.99480576      21         132211331222113112211
43 Tc      386.07704943      15         311322113212221
42 Mo      296.16736852      20         13211322211312113211
41 Nb      227.19586752      29         11131221133221131112211312221
40 Zr      174.28645997      23         12322211331222113112211
39 Y       133.69860315      7          1112133
38 Sr      102.56285249      7          3112112
37 Rb      78.678000089      10         1321122112
36 Kr      60.355455682      14         11131221222112
35 Br      46.299868152      16         3113112211322112
34 Se      35.517547944      20         13211321222113222112
33 As      27.246216076      26         11131221131211322113322112
32 Ge      1887.4372276      23         31131122211311122113222
31 Ga      1447.8905642      17         13221133122211332
30 Zn      23571.391336      3          312
29 Cu      18082.082203      6          131112
28 Ni      13871.124200      8          11133112
27 Co      45645.877256      5          32112
26 Fe      35015.858546      8          13122112
25 Mn      26861.360180      12         111311222112
24 Cr      20605.882611      5          31132
23 V       15807.181592      8          13211312
22 Ti      12126.002783      14         11131221131112
21 Sc      9302.0974443      16         3113112221133112
20 Ca      56072.543129      2          12
19 K       43014.360913      4          1112
18 Ar      32997.170122      4          3112
17 Cl      25312.784217      6          132112
16 S       19417.939250      10         1113122112
15 P       14895.886658      12         311311222112
14 Si      32032.812960      7          1322112
13 Al      24573.006695      10         1113222112
12 Mg      18850.441227      10         3113322112
11 Na      14481.448773      9          123222112
10 Ne      11109.006821      12         111213322112
9 F        8521.9396539      14         31121123222112
8 O        6537.3490750      18         132112211213322112
7 N        5014.9302464      24         111312212221121123222112
6 C        3847.0525419      28         3113112211322112211213322112
5 B        2951.1503716      34         1321132122211322212221121123222112
4 Be       2263.8860324      42         111312211312113221133211322112211213322112
3 Li       4220.0665982      27         312211322212221121123222112
2 He       3237.2968587      32         13112221133211322112211213322112
1 H        91790.383216      2          22
```

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).

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. [HB Note: there are seven such complete orderings - see Henry Bottomley's demonstration]

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?

### Table 2 Numerical interpretation of the Gleichniszahlenreihe (GZR)

```string     number                factorized
1          1                     1
11         4                     2^2
21         7                     7
1211       49                    7^2
111221     376                   2^3 * 47
312211     886                   2 * 443
13112221   4777                  17 * 281
...

```

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Reproduced by Henry Bottomley with permission