The
Belousov-Zhabotinski
Reaction
a description by Professor Jack Cohen
and the kit you can buy to do it yourself
Malonic acid is oxidised by bromate, producing carbon dioxide, but in acid solution the reaction cannot proceed; there must be a latent period while the CO2 diffuses away. The solution is 'poised'; foci of blue appear randomly, and spread as circles. New foci appear at circle centres, so that a series of bulls-eyes (not spirals) progresses.
You can display the reaction very well by placing the petri dish onto an overhead projector (as long as the projector's fan is not unbalanced and vibrating).
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