19 trillion maps of the United States Neil Freeman, 2010-11

mapping (edition of 19,468,490,047,488)

All of the ways to make a map of the United States with four colors: Every time this page is reloaded, it shows a different version of a map with four colors.

A well-known theorem proves that any map can be colored with no more than four colors so that no color touches itself. (The map has to have a few reasonable features, like all the "countries" must be continuous). This isn't of surpassing use to cartographers, who generally use more colors for aesthetic reasons. However, it is an amazing property for a map to have.

The next step is to ask how many ways there are to accomplish this simple task. For the contiguous United States, there are over nineteen trillion different maps that use only four colors. Even the tightest constraints can yield near-infinite variety.

This map above is unique, and will never appear again, but you can get a physical copy! Order a custom hand-colored postcard of this map ($7).