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Mercredi, 04 Janvier 2012 18:13

## The Fractal Dimension of ZIP Codes

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ZIP codes have been around for less than fifty years. In addition to allowing future historians to ascertain that the television show Beverly Hills 90210 must have been created no earlier than 1963, ZIP codes have become part of our culture, organizing our locations and determining the flow of mail.

But ZIP codes are not created randomly. There is an order and a structure to this mail system. Let’s use this nearly-fifty-year anniversary as an opportunity to examine the quantitative aspects of ZIP codes.

One quick way to look at ZIP codes is by seeing how each part of a ZIP code defines a part of our country. Ben Fry, of Fathom, created a simple visualization called zipdecode to do just this. As you type each successive digit of a ZIP code and see what regions of the United States it describes. For example, if you’re typing in 64110 (Kansas City), you can see what parts of the United States begin with ‘6’.

But let’s think about ZIP codes a bit more abstractly. ZIP codes must serve every resident of the United States in an efficient manner, much like how the circulatory system serves every cell within an animal. Circulatory systems, however, all have a certain shape: they are self-similar. Parts of these branching circulatory systems look like the whole.

And it turns out that branching structures, whether circulatory systems or not, are fractals. Essentially, that means that that they have fractal (or fractional) dimensions, since they fill space or a surface, but are built up using lines. The circulatory system fills a three-dimensional space using tubes (which are essentially two-dimensional), and the Peano Curve fills a plane while only being a long and snaking one-dimensional line. These objects, that don’t quite obey regular shapes, all have fractal dimensions.

Play with space-filling curves here

In a similar spirit, I decided to explore the dimension of the ZIP Code system and see if it has a similar type of fractal dimension. I did this using the wonderful images created by Robert Kosara called ZIPScribbles, which connect the coordinates of sequential ZIP codes (02445 is connected to 02446, 02446 is connected to 02447, and so forth). As you can see below, there is a geographically hierarchical nature to it. ZIP codes divide the population first into states, and then divide into little scribble regions even further, in a self-similar fashion.

So, I set out to measure the fractal nature of the ZIP code system. I used one of the simplest methods, called the box-counting method, which estimates the self-similarity of a shape by looking to see how many boxes in a series of ever-smaller grids are required to cover a shape. Doing this, I was able to calculate the fractal dimension of the ZIP Code system, using the ZIPScribble: 1.78.

(Kosara has also constructed ZIPScribbles for other countries; calculating the fractal dimensions for these other countries’ mail systems is left as an exercise for the reader.)

ZIP codes serve every resident without having a code for every point on the plane and, at least according to this method, describe a shape that is somewhere between a line and a two-dimensional plane, though closer to a two-dimensional surface. While we can take the comparison between the circulatory system and ZIP codes even further—mail carriers are red blood cells, letters are oxygen molecules—I think the amount of junk mail we receive might make this a difficult analogy to swallow.

Analogies aside, fractals are all around us. We often think of them in the natural world, whether within our bodies, in the shapes of trees, or the shape of coastlines. But often, engineered systems, once they have reached a certain level of complexity, can take on the properties of something a bit more organic. The ZIP code system is one of these.