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.
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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