For reasons that I really can’t quite explain, I became determined to
answer the question “which state is the most square?”.
How did this question come up in the first place? My wife’s dad
asked my five year old son what he thought the most square state is
while he was looking at his state map. I figured it had to be one of
either Colorado or Wyoming, but my son (who loves states) immediately
answered “New Mexico”. New Mexico hadn’t really crossed my mind, but
the more I looked at it, the more I thought he might actually be
right!
Regardless, I decided I would try to make a small project out of it.
How do you define “most square”?
I don’t think there’s a right or wrong answer here; it’s going to be a
bit subjective. I ended up doing something fairly simple: I just
counted the fraction of the state’s area that overlaps with a square
of roughly the same volume.
To lay that out in a bit more detail:
Make a square that has roughly the same area as the state
Overlay the square on top of the state and measure how much of the area overlaps
Find the placement and orientation of the square that maximizes the overlap
Measure the overlap as a percentage of the total area
Methodology
First, I put each state on a lattice. Then, I scaled each state up or
down so that the number of lattice points that fell within each
state’s boundary was very close to 400 (+/- 2). In case that’s
confusing, here’s a picture of me scaling California up and down until
exactly 400 points fell within California’s state boundary:
Then, I found the position of the 20x20 square (so 400 points) that
overlaps with the most points inside the state. Here’s a picture of
me doing that for Vermont:
In Vermont’s case, the best square overlaps with 266 of the 400 points.
Then, I rotated each state in a bunch of different orientations to see
if there was an orientation where I could find an even better 20x20
square. For Vermont, here is the best one (it overlaps with 290
points instead of just 266 above):
For fun, here’s a picture of me rotating Texas at 10 different angles to see which is best:
So, in the end, for every state, I found the rotation that had the 20x20 square that overlapped with the most points inside the state’s (rotated) boundary.
Results!
First, the least square states (Hawaii and Florida):
I thought that the answer had to be either Colorado or Wyoming:
Here are the top 2. Notice the winner!
Full results
And for the curious, here is every state, both in the original
orientation as well as in the orientation that maximized the
"squareness".
