From Bubbles to Sand

Redistricting based on soap bubbles wasn’t good enough. Finite element analysis to simulate surface tension of the bubbles was too computationally intensive. Such redistricting also did not take into account county or city boundaries. Although the starting and growing from a seed point seemed correct, a better apportioning method was needed.

I considered a pile of sand. If the sand was coming from a funnel in the sky, it would fall to the surface and form a cone. The base of the cone would grow in a circular manner similar to the soap bubble analogy. Instead of a completely flat surface, consider what would happen if there was a wall wherever there was a county or city boundary. The sand would hit the wall and stop expanding there, causing the base of the sandpile to assume the shape of the boundary. Eventually, the sandpile would overflow the wall and start filling the next compartment. Software using this analogy could grow districts in a compact manner, but also take into account the county and city boundaries. The key question then was how high the wall should be. After much thought, I concluded that what was really desired was districts with a set maximum value of eccentricity. For example, a city could have its wall low enough that the widest part of the base would not exceed double the narrowest part for an eccentricity of two. County walls would be higher with an eccentricity not exceeding three.  It is easy to visualize a sandbox with slightly higher walls for county boundaries, and within those county boundaries are slightly lower walls for city boundaries. Sand falling from the funnel in the sky still has a roughly circular base, but modified by the wall boundaries. The eccentricity determines that roughness.

With software we don’t have to strictly follow the model of our physical world. When a particular cone of sand reaches a wall, the height of the wall can be set for the desired eccentricity. If a different cone of sand reaches the other side of the same wall, a different height will be computed. Thus, the inside of the wall will have a different height from the outside of the wall. For those who demand an analogy, see the explanation for TARDIS in the Dr. Who episode “Robots of Death” from the BBC.

One of the nice things about soap bubbles was that they would push away from each other on contact. To keep this nicety, when two sand piles touch, vector forces can move the sky funnels away from each other. Actually the software would keep the funnels fixed in place while the vector forces accumulate, and then all the funnels would move to their new location at the same time.  The movement would be triggered by events, such as the program having assigned to districts a certain percentage of the State’s population.

But time marches on, bringing progress both good and bad whether we want it or not. The next computer revolution changed everything yet again.

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