The Cook Book

Recipe for the week of June 17 - 23

Majority Dynamics on a Torus

This week's soup is definitely a keeper! Using the level set representation that we have featured for the past couple of months, it shows the self-organization of range 7 Box Majority Vote from a symmetric completely random initial configuration until 1's cover the lattice. Since the boundaries wrap, our CA evolves on a torus. As explained last week, in order to achieve such a Good soup, as opposed to a Bad one (all 0's = black here), or an Ugly one (part stripes, part black), opinion 1 must emerge from the 'big bang' as the background state. Last week's soup indicated one way 1's can surround 0's. For those with large desktops, here's another Good one (800 by 600, 342K) using a nice palette with a different feel.

You might think Good soups would be hard to find once the lattice is large. However one can argue mathematically that by choosing a reasonable large range R of interaction, and then using an L by L lattice with L at most a suitable multiple of the square of R, we get a Good soup at least 1 time in 16. If L obeys a larger scaling, then one can encounter a stable hole because balls of the minority opinion with uniformly sufficiently small curvature are invariant (one needs to encounter strictly more disagreement to change).

It is amusing to note that by identifying edges of the array in different ways, Majority Vote can be defined on four manifolds: the torus, the Klein bottle, the Projective Plane, and the Sphere. On the Projective Plane it turns out that Ugly soups are impossible, so by obeying the above scaling constraint we get a Good one half the time. We leave it to our readers to think about Majority Vote on the surface of a Sphere (like ours) and whether Ugly soups can result there...

In fact, I'm so enamored of this week's graphic that it seems appropriate to announce the immediate availability of a new way-cool Postcard Server!!!. The observant among you will note that little envelope buttons are cropping up by the upper right corners of many of our favorite soups here in the Kitchen. To send a picture with accompanying message to a friend, just click on a corresponding envelope icon and fill out the self-explanatory forms. In most cases the buttons appear on the individual recipe pages of our archive (such as this one). Once upon a time I produced a set of real Particle Postcards on cardboard. Major thanks to Dave Kung for making virtual Kitchen Postcards happen. To the rest of you, use the server often and encourage others as well. That way, before long, PSK will be getting more hits than Yahoo.

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