The Cook Book

Recipe for the week of September 9 - 15

Belousov's Brew, continued

A few years ago, during a visit to the University of Toronto, I was able to try out last week's BZ reaction recipe myself. Ray Kapral and the good folks there set me up with a lab coat, protective goggles, Petri dish, pipette, and all the necessary ingredients. Aside from sucking too hard at one point, thereby ingesting a bit of Ferroin, the cooking went flawlessly. Sure enough, some sweet little spirals developed. To see counterparts of my computer experiments arise in the 'real world' of a chemistry lab was truly inspirational.

That said, there are so many advantages to computer simulation of nonlinear physical dynamics that even applied types are increasingly drawn to virtual reality. Toronto's Michael Menzinger sums it up thus:

For instance, the CoLoS Project at Oxford University uses both workstation- and Web- based interactive visualization to study a wide assortment of problems in physical and theoretical chemistry. Here is a snapshot of target patterns generated by their BZ simulation tool.

This week's soup, produced by our WinCA software, shows a Greenberg-Hastings (GH) rule after 100 updates started from a completely random configuration of 8 colors. Though we offered several GH models during the first few months the Kitchen was open (see Recipes 2, 4, 14, 17), we have not yet featured parameters that illustrate prototypical nucleation of spiral pairs (ram's horns) in excitable media. The neighborhood here is range 3 Box, with a threshold of 5 excited neighbors needed to activate a resting cell. The palette depicts excitation as a fire ranging from fully excited = orange, through successively darker shades of red for the refractory period, to black for resting (excitable). By the time shown this process invariably locks into a locally periodic state in which every cell cycles with period 8.

GH dynamics are simple enough that they are rather surprisingly amenable to rigorous mathematical analysis. On the downside, they lack some of the subtler physical features of the actual BZ reaction and related excitable media. For this reason many applied researchers implement more complicated computer algorithms, with additional parameters, either to better fit empirical data or to capture additional detail. Perhaps the best known variant is the Hodgepodge Machine of Gerhardt and Schuster, popularized in a Scientific American article by A. K. Dewdney in the late 1980's. There is a page of CA graphics from a Parallel Computation Group in Madrid that features a striking 2d Hodgepodge type image. Computers have also been used, by Arthur Winfree and others, to study the analogs of ram's horns in three dimensions: scroll rings and more intricate topological structures. Jörg Heitkötter offers a rather nice 3d Hodgepodge graphic.

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