The weird and wonderful chemistry of audioactive decay -- Alex Kontorovich, March 23, 2004

<p>

In 1987, Prof. John Conway wrote on the progression of the so-called 
Look-And-Say sequence: 1, 11, 21, 1211, 111221, 312211,... (Can you guess 
the next entry?) He found that the sequence decomposed into certain 
recurring strings. Categorizing these 92 strings and labeling them by 
the atoms of the periodic table (from Hydrogen to Uranium), Conway 
was able to prove that the asymptotic length of the sequence grows 
exponentially, where the growth factor (now known as Conway's Constant) 
is found by computing the largest eigenvalue of a 92x92 transition 
matrix. Even more remarkable is the Cosmological Theorem, which states 
that regardless of the starting string, every Look-And-Say sequence 
will eventually decay into a compound of these 92 atoms, in a bounded 
number of steps. Conway writes that, although two independent proofs 
of the Cosmological Theorem were verified, they were lost in writing! It 
wasn't until a decade later that Doron Zeilberger's paper (coauthored with 
his computer, Shalosh B. Ekhad) gave a tangible proof of the theorem. We will 
discuss this weird and wonderful chemistry, and some philosophical 
consequences. The only prerequisite is basic linear algebra.