Unexpectedly Intriguing!
15 September 2023

Topology is a branch of mathematics that studies the properties of geometric objects that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending. In mapmaking, topology has to do with how features like points, lines, and polygons share geometry.

Now, what do you suppose might happen if those two points of view of what topology is were combined?

You don't have to wonder, because Topology Fact has taken a map of the lower 48 contiguous states of the U.S. and revisualized it. All state borders have been either stretched, twisted, crumpled, or otherwise bent as needed into rectangles, then positioned to indicate if they border each other or an ocean. Here's the result:

The topologist's map makes it easy to see some neat features that would be harder to see on a more traditional map. Such as:

• Maine is the only state that only borders just one other state.
• Four states, Maine, Rhode Island, South Carolina, and Washington only border two other states.
• Two states, Missouri and Tennessee, border eight other states each (and technically, each other).
• No state is more than four states away from an ocean.

The downside to all the stretching is that the topologist's map doesn't do well in depicting how large the states actually are with respect to each other. Texas has the largest area of all the states shown, but is tiny on the map. States like California (second-largest) and Arizona (fifth-largest) look almost identical in size, while Oregon is stretched to enormous proportion.

The map also suggests that both Arizona and New Mexico border an ocean. Both however are landlocked. That could be easily fixed however, if the rectangles for California, Arizona, New Mexico, and Texas were extended to the bottom edge of the map and the rectangle for Oregon shortened to indicate California has a beach.

The map also leaves open the question of how Alaska and Hawaii might be added to this kind of map. As created, they would be isolated islands separated from the rest. But then, that's exactly how wall maps of the United States in schoolrooms often display them.

If you want to take the map of the lower 48 states to another level of abstraction, mathmaticians can also turn to graph theory to revisualize how the boundaries of states are connected to each other. If nothing else, it makes it easier to count the number of borders for each state, but fully disconnects each from their geographic area.

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