2012.10.30 Another map experiment: Voronoi states

Not letting another calendar year slide past without a single POIB post, unlike 2010! Here is an experiment with the geography of the U.S. states along similar lines as last year's border states. Rather than each new state comprising all points closer to a given state boundary than to any other state boundary, these states comprise all points closer to a given state capital than to any other state capital. That's right: this is a Voronoi decomposition of the Lower 48 using the state capitals as seeds. Because there's a state for every state capital, we can consider these new states reshaped versions of the existing lower 48 states. Click the map for a double-wide version, click again for the smaller version:

Some observations:

• Look around the map for a while, and you'll find a ton of places where a pair of Voronoi states border each other, such that their real counterparts don't border each other; to take the top left example, Washington and Montana. The converse is also necessarily common. At first while I was constructing the Voronoi boundaries, this didn't seem right to me, but it makes sense when you realize the Voronoi lines have nothing to do with the real state lines. (Except that the real state lines restrict the positions of the state capitals to the interior of their states, but that's not enough of a constraint to make a difference here.) I wonder if it's possible to pick a point for each state such that each Voronoi state using those points as seeds borders exactly the states that its real counterpart does.
• Note how the boundary between new Utah (Vutah? not great) and new Arizona (Avorona; Vorozona; an embarrassment of riches) is nearly the same as the real states' border, except shifted over. This means that the locations of Salt Lake City and Phoenix are nearly mirrors of each other across the state line, which in turn means that: (1) Salt Lake City is pretty much due north of Phoenix; (2) they're equidistant from the state line; (3) the distance of each from the state line is half the distance to the other city as the crow flies, if a crow has reason to fly from Phoenix to Salt Lake City or, unimaginably, vice versa. This is a pretty neat spatial relationship to uncover; I certainly never noticed it before. This seems to be the only case where a Voronoi border is nearly the same as a real border.
• Note that while Missouri and Tennessee border eight states each IRL (including each other), Voronnessee also borders eight but Voronissouri only borders five. West Vorginia also now borders eight states, tying with Voronnessee for the maximum. Voronnessee is also the center of a rather pleasing rosette of states.
• In many cases the area of a Voronoi state is similar to that of its real counterpart, but there are a few significant divergences: California (Calivornia) and Oregon (Voregon), for instance, get lobotomized, while Utah (Utonoi? not much better) bloats. Other states have their centers of gravity shift quite a bit, e.g. Georgia (Vorgia).

Here's a boiled-down explanation of how I constructed the map:

1. Start with the US map with state capitals in Illustrator.
2. Draw the Delaunay triangulation of the state capitals.
3. For each line drawn, rotate it 90° about its center by shift+dragging the line via its rotation cursor (which shows up when hovering the cursor just offset from any corner of the line's selection box).
4. Trim the lines at every place where three close-together lines (a badly inexact description) intersect, extending lines first where needed. Keep the part of the line that connects to another three-line intersection. Obviously.

That's it, basically! It sounds pretty arcane maybe when explained that way, but one gets the hang of it quickly with practice. It's fun! Trust me.

Bonus: here are the names of all the Voronoi states, in alphabetical order of their real counterparts: Vorobama, Vorlaska, Vorozona, Vorkansas, Calivornia, Vorolado, Voronnecticut, Delavore, Vlorida, Vorgia, Havoii, Voidoho, Villonoi, Vorondiana, Voronoiowa, Voronansas, Vorontucky, Vorisiana, Voronoine, Voryland, Voronassachusetts, Voronichigan, Voronnesota, Voronississippi, Voronissouri, Vorontana, Voronebraska, Nevorda, Noi Vorshire, Noi Vorsey, Noi Vorxico, Noi Vork, Vorth Carolina, Vorth Dakota, Vorhio, Vorklahoma, Voregon, Pennsylvornia, Vorde Island, South Caronoina, South Davorta, Voronnessee, Vorxas, Utonoi, Vormont, Vorginia, Vorshington, West Vorginia, Vorsconsin, Voronoyoming. Whew!