Friday, July 27, 2007

Voronoi Cells, Personal Space and Boundary Functions

"Equations are the devil's sentences."
-Stephen Colbert

What do fractals have to do with personal privacy? Maybe nothing. Maybe a lot.

In a recent Grits post summarizing a workshop at the national Restorative Justice conference in Kerrville, I briefly discussed fractals and Twila Hugley Earle's theories about how relatively recent higher math enabled by computers (a set of approaches which she broadly labeled "chaos theory") implied new ways of thinking about crime, punishment, and identifying just, situation-specific outcomes in the criminal justice system.

So I was interested to discover through this post over at Ernie's 3-D Pancakes a new (to me) blog called Flight404, and in particular this post on Voronoi cells and Magnetism. What's a Voronoi cell? "Voronoi cells represent an enclosure where all the points within that enclosure are closer to the seed than any other seeds." I know, I know, it makes perfect sense, right? Here's Flight404's example:

Writes the author, "In the above image, I have created 3000 seeds. The boundaries are the Voronoi cells. All the pixels within a specific cell are closer to the parent seed than any other seeds in the image. Okay, so I just said the exact same thing twice. I'm sorry."

For me, an interactive explanation worked better, so check out this applet created by Paul Chew in 1997 - just click anywhere in the frame to create the first "seed" point, then each additional click will divide up the terrain in accordance with the Voronoi-based mathematical calculations. Go try it now.

Why do I think this might have applications describing human behavior? Because I think people move in packs but also behave as individuals, that we are at once individual decisionmakers responsible for our own actions and simultaneously gravitate toward family, peers, the people closest to us with power, charisma, traditional authority, or other "parent seeds," in this analogy. They themselves revolve around larger centers of gravity the way moons revolve around planets that revolve around suns that revolve around galaxies, etc..

We are each both motivated and constrained by others, and also independent free agents. It's not an either or, thing, both are true, simultaneously. Indeed, such nonlinear truth is why I think social scientists have trouble crafting predictive mathematical models, much less workable solutions, for many extremely human concerns like crime, migration, or in this train of thought, privacy.

I don't have a clue about the math behind this concept, but I can appreciate its utility and dynamism, and think the idea may well have broader applications. To get a clearer picture of how such math might describe human interactions, see this video and explanation of an installation art called "Boundary Functions" from 1998 by Scott Snibbe based on the Voronoi cell concept.

I'm totally spitballing, here, but in Snibbe's video in particular you can see how this mathematical approach might be used to describe nonlinear human interactions in ways that might, theoretically, enable a dynamic mathematical definition of realms of "personal privacy," a definition previously left to the domain of jurists, not math geeks.

Am I reaching? Probably. And as a commenter warned me in the post about Twila Earle's talk, one should beware of mathematics by analogy. But those were the thoughts that filled my head as I learned more about Voronoi cells, FWIW, and I thought them interesting enough to share.

Let me know in the comments what other implications or possible applications come to mind from the Voronoi patterns? See also Scott Snibbe's Voronoi Portrait Series, and the Flight404 author's gallery installation of Voronoi-generated pieces. And in parting check out the Voronoi Beat video that first attracted me in Ernie's post: It's really cool:

Voronoi from flight404 and Vimeo.


Anonymous said...


Gritsforbreakfast said...

The devil's sentences, I tell you.

Anonymous said...

Is it just me...or does Voronoi's cell make your eyes go nuts and render your brain unable to read words near the image? Oh! The solution to crime: Your own personal space!

Gritsforbreakfast said...

Well, the "solution to crime" might be preventing violations of personal space - e.g., crimes against person or property, etc. - enforcing boundaries and providing restoration for victims upon their violation, might be how I'd put it, to continue in this vein. best,

Anonymous said...

I can't wait for your application of Einstein's special theory of relativity. Is it possible that because time and space have no fixed point or location that all crimes are relative to the ability to detect them? If this is the case, then we might be able to predict the most likely event based upon Voronoi's cell!

Gritsforbreakfast said...

Nurit wrote: "all crimes are relative to the ability to detect them"

That's really a truism on the ground, isn't it? (It made me think offhand of Barry Bonds allegedly using undetectable designer steroids and the fallout.) The assumptions behind law and punishment are so deeply ingrained, I think we don't necessarily realize when they become anachronistic. The definition of crime is "it's against the law." And you can pass a law, theoretically, against anything. But can you enforce it? That's a function of can you detect it? And that gets us back to your question.

As for potentially predicting crime that way, though, you've either a) completely blown my mind or b) proposed a circular model, since if crime is relative to its detection, then nearly by definition it is going to be found where you look.

Crimes with victims are easily detected, obviously, by the victim. But crimes with no victims, I think, is where your observation comes more starkly into play. Quien sabe? I'm very much out of my depth, here. Interesting stuff to think about, though, huh?

Anonymous said...

Indeed, Scott.

Very interesting.

Anonymous said...

Anon. at 4:08. "Huh?"

I recommend you read it until you get it. I think that getting my head around it might have busted loose some of that plaque that messes with people's brains.



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