Mr. Voronoi, meet the US state boundaries

In the Scholars’ Lab we are working with remarkably detailed datasets showing changes to US political boundaries over time.  We’ve all been fascinated with visualizations where the familiar outlines of the US states emerge from thousands of boundary changes to their underlying counties over the last few hundred years.  Did you know Virginia once spanned from the Atlantic Ocean to the Mississippi River?


We’re developing a new web-based tool for visualizing these historic boundary changes and it’s nearly ready for prime time.  We’ll announce the beta release here soon.

So with the knowledge that US state boundaries have already been subject to drastic change over time, let’s have some fun with geographic information systems to visualize drastic mathematically-induced changes to those familiar US state boundaries.

For our experiment, let’s keep all our current state capital cities right where they are since they are laden with the necessary infrastructure of government.  But we’ll move the state boundary lines Voronoi-style so anywhere you travel in each of our new states you’ll be closer to the state capital than any other state capital.  In other words, when you’re standing anywhere inside our newly outlined Virginia, you will always be closer to the Virginia state capital, Richmond, than any other state capital.  That seems very efficient.  Let’s have a look.

US States with capitals

Here’s that familiar grade-school wall map of the lower 48 US states and their capital cities.   Now let’s tweak the map with GIS software to reconfigure the states, Voronoi-style.

US Voronoi states with capitals

Wow, what a difference Voronoi makes.

Let’s measure just how much the states have changed in our new layout.   In absolute terms, Utah and New Mexico make the biggest land grabs while Texas and California lose the most real estate.  But as a percentage of their current area, Rhode Island is the big winner ballooning in size by over 240% while Massachusetts shrinks 60%.

To visualize the state-by-state changes, Todd Burks from neighboring Clemons Library overlayed the two maps.

Intrigued?  Read more about Voronoi and Thiessen polygon GIS techniques.

As a former Geographic Information Systems Specialist for the Scholars' Lab, Kelly worked with faculty, staff, and students to visualize, analyze, create, and manage geographic data. He earned a master's degree in Geographic Information Science from Indiana University Purdue University Indianapolis. His research interests include applied geography and cartography as an art form.


  1. What would be really interesting imho: see if the existing boundaries can be approximated by Voronoi regions of a different set of “capitals”.

    And then see how much the fit can be improved by allowing each state two capitals.

  2. I did this (by hand) a few years ago for the northeastern states, and am delighted to see that it has been now done by computer. A further question: suppose the states are simultaneously allowed to relocate their capitals (within present state boundaries) to maximize the area captured by the Voronoi map; what would the redrawn map then look like?

  3. I like Interpunct’s suggestion. Certainly more practical. But more needs to be clearly shown about state capitols. What factors chiefly determined their choice? Center of population, or wealth, or power, or transportation? Waiting to see what SL comes up with in their geography play.

  4. Am I correct this uses geography as the primary variable and not existing population densities? A more plausible representation would not be where any given person would be closest to a state capital, but where the most people would be closest. It would be similar to how the population center of America moved west over time.

  5. Edit: the 38-state plan guy’s name was C. Etzel Pearcy.

  6. Very interesting work! It looks surprisingly similar to the 38-state plan made 40+ years ago by UCLA geographer Limbaugh (without any computer power whatsoever).