Evolution of Conway's 92 Look and Say audioactive elements

The Look and Say sequence generates a increasingly long string of digits. It soon becomes obvious that some patterns in the string seem to repeat themselves. Conway has shown that the sequence soon breaks down into elements whose evolution do not affect the evolution of other elements in the string. 92 of these elements dominate the evolution, in that they either soon make up the whole string, or if not then as the string lengthens to infinity, they make up "almost all" (a measure theory term) of the string. The number 92 lead to an analogy with the atomic elements up to Uranium in the Periodic Table. There are seven possible complete orderings of the 92 audioactive elements (see Notes below). This table allows you to follow the evolution of individual elements through hyperlinks in the classic ordering starting from Uranium.

Element Evolution String String Evolution
1 H H (stable) 22 22
2 He Hf Pa H Ca Li 13112221133211322112211213322112 11132132212312211322212221121123222112
3 Li He 312211322212221121123222112 13112221133211322112211213322112
4 Be Ge Ca Li 111312211312113221133211322112211213322112 3113112221131112211322212312211322212221121123222112
5 B Be 1321132122211322212221121123222112 111312211312113221133211322112211213322112
6 C B 3113112211322112211213322112 1321132122211322212221121123222112
7 N C 111312212221121123222112 3113112211322112211213322112
8 O N 132112211213322112 111312212221121123222112
9 F O 31121123222112 132112211213322112
10 Ne F 111213322112 31121123222112
11 Na Ne 123222112 111213322112
12 Mg Pm Na 3113322112 132123222112
13 Al Mg 1113222112 3113322112
14 Si Al 1322112 1113222112
15 P Ho Si 311311222112 13211321322112
16 S P 1113122112 311311222112
17 Cl S 132112 1113122112
18 Ar Cl 3112 132112
19 K Ar 1112 3112
20 Ca K 12 1112
21 Sc Ho Pa H Ca Co 3113112221133112 132113213221232112
22 Ti Sc 11131221131112 3113112221133112
23 V Ti 13211312 11131221131112
24 Cr V 31132 13211312
25 Mn Cr Si 111311222112 311321322112
26 Fe Mn 13122112 111311222112
27 Co Fe 32112 13122112
28 Ni Zn Co 11133112 31232112
29 Cu Ni 131112 11133112
30 Zn Cu 312 131112
31 Ga Eu Ca Ac H Ca Zn 13221133122211332 11132221231132212312
32 Ge Ho Ga 31131122211311122113222 132113213221133122211332
33 As Ge Na 11131221131211322113322112 31131122211311122113222123222112
34 Se As 13211321222113222112 11131221131211322113322112
35 Br Se 3113112211322112 13211321222113222112
36 Kr Br 11131221222112 3113112211322112
37 Rb Kr 1321122112 11131221222112
38 Sr Rb 3112112 1321122112
39 Y Sr U 1112133 31121123
40 Zr Y H Ca Tc 12322211331222113112211 11121332212311322113212221
41 Nb Er Zr 1113122113322113111221131221 31131122212322211331222113112211
42 Mo Nb 13211322211312113211 1113122113322113111221131221
43 Tc Mo 311322113212221 13211322211312113211
44 Ru Eu Ca Tc 132211331222113112211 111322212311322113212221
45 Rh Ho Ru 311311222113111221131221 1321132132211331222113112211
46 Pd Rh 111312211312113211 311311222113111221131221
47 Ag Pd 132113212221 111312211312113211
48 Cd Ag 3113112211 132113212221
49 In Cd 11131221 3113112211
50 Sn In 13211 11131221
51 Sb Pm Sn 3112221 13213211
52 Te Eu Ca Sb 1322113312211 1113222123112221
53 I Ho Te 311311222113111221 13211321322113312211
54 Xe I 11131221131211 311311222113111221
55 Cs Xe 13211321 11131221131211
56 Ba Cs 311311 13211321
57 La Ba 11131 311311
58 Ce La H Ca Co 1321133112 11131221232112
59 Pr Ce 31131112 1321133112
60 Nd Pr 111312 31131112
61 Pm Nd 132 111312
62 Sm Pm Ca Zn 311332 13212312
63 Eu Sm 1113222 311332
64 Gd Eu Ca Co 13221133112 11132221232112
65 Tb Ho Gd 3113112221131112 132113213221133112
66 Dy Tb 111312211312 3113112221131112
67 Ho Dy 1321132 111312211312
68 Er Ho Pm 311311222 1321132132
69 Tm Er Ca Co 11131221133112 3113112221232112
70 Yb Tm 1321131112 11131221133112
71 Lu Yb 311312 1321131112
72 Hf Lu 11132 311312
73 Ta Hf Pa H Ca W 13112221133211322112211213322113 11132132212312211322212221121123222113
74 W Ta 312211322212221121123222113 13112221133211322112211213322113
75 Re Ge Ca W 111312211312113221133211322112211213322113 3113112221131112211322212312211322212221121123222113
76 Os Re 1321132122211322212221121123222113 111312211312113221133211322112211213322113
77 Ir Os 3113112211322112211213322113 1321132122211322212221121123222113
78 Pt Ir 111312212221121123222113 3113112211322112211213322113
79 Au Pt 132112211213322113 111312212221121123222113
80 Hg Au 31121123222113 132112211213322113
81 Tl Hg 111213322113 31121123222113
82 Pb Tl 123222113 111213322113
83 Bi Pm Pb 3113322113 132123222113
84 Po Bi 1113222113 3113322113
85 At Po 1322113 1113222113
86 Rn Ho At 311311222113 13211321322113
87 Fr Rn 1113122113 311311222113
88 Ra Fr 132113 1113122113
89 Ac Ra 3113 132113
90 Th Ac 1113 3113
91 Pa Th 13 1113
92 U Pa 3 13

Note first the mildly astonishing result that this ordering, when starting with U (uranium - the string "3"), evolves each element in turn (generally as part of a compound), as far as the stable element H (hydrogen - the string "22") after 91 steps. These 92 natural elements suggest the connection with the atomic elements (only a pedant would suggest Tc - technetium - has never been found naturally on Earth). In fact the string "22" arrives after only 5 steps (in the middle of Fr) and the element H appears after 14 steps from U (by which time the elements in the compound are Dy Eu Ca Co Nd K Cu Cl Cr Si La H Ca Co Pt).

Note second, that this is not a unique ordering with this property.
The alternative examples are similar, but with the pattern
U-Ta, Hf-Er, Ho-Gd, Eu-Sm, Pm-Y, Sr-Ga, Zn-Sc, Ca-He, H
replaced by
U-Ta, Hf-Er, Pm-Y, Sr-Ga, Eu-Sm, Zn-Sc, Ho-Gd, Ca-He, H;
U-Ta, Hf-Er, Pm-Y, Sr-Ga, Zn-Sc, Ho-Gd, Eu-Sm, Ca-He, H;
U-Ta, Ca-He, Hf-Er, Ho-Gd, Eu-Sm, Pm-Y, Sr-Ga, Zn-Sc, H;
Sr-Ga, Eu-Sm, Zn-Sc, Ho-Gd, Ca-He, Hf-Er, Pm-Y, U-Ta, H;
Sr-Ga, Zn-Sc, Ho-Gd, Eu-Sm, Ca-He, Hf-Er, Pm-Y, U-Ta, H; or
Sr-Ga, Zn-Sc, Ca-He, Hf-Er, Ho-Gd, Eu-Sm, Pm-Y, U-Ta, H.
A proof of this is here.

Note third, that the elements cannot be combined arbitrarily. For example U Ca is the string "312" which behaves as Zn rather than as two separate elements. The first step is Cu (which as "131112" looks like Pa K), then Ni (which as "11122112" looks like Th Ar), but at the third step it becomes Zn Co (the string "31232112") rather than Ac Cl (the string "3113132112").

Note fourth, that Mario Hilgemeier has drawn a pictorial graph of the transition table above, as part of his writings in Fractal Horizons, and has also produced a table of asymtopic abundances. The table can clearly be turned into a rank 92 transition matrix, and its largest eigenvalue is then the asymtopic rate of growth of a Look and Say string - this is Conway's Cosmological Theorem.

© Copyright July 1999 Henry Bottomley

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