What's the Most Concave State in the U.S.? Using R to Solve a Geography Puzzle

Fun stuff, especially if you&#x27;re a U.S. geography nerd (states with the largest coastlines, states that border the most states, all that good but mostly arbitrary stuff)<p>Pedantic, I&#x27;m sure, but from the title before I clicked on it I was trying to think of the state whose shape might independently be considered the most concave (though that may be much harder to define). This version of concavity depends largely on the shapes of the states around it (e.g. if Nevada split into 6 horizontal states, suddenly California would be the winner).

Any idea whether this depends on the projection used to get the map in the first place? I mean, the &quot;straight lines&quot; are really curves that lie in the boundary of the Earth&#x27;s surface, unless I&#x27;m missing something major.

Aaaaannnd now I&#x27;m playing FTL again. Lining up those 5-room beam strikes is just too much fun!<p>On topic, though, this is pretty cool. Rivers and coastlines seem to be the best way to get appropriately jagged borders. It&#x27;s interesting to look at states across the map from east to west and see the shapes get simpler and more geometric over time.

Cancavity has a simple definition, its measure is: area of the region divided by the area of its convex hull. Most concave is value nearest to zero. You can&#x27;t just make up definitions.

It is an odd definition of concave. It would make more sense to me to require that the line joining two points on the state&#x27;s border does not cross in and out of the original state when determining the number of other states it crosses... This doesn&#x27;t really capture concave as a geometric concept either but more aligned with the idea that it is a local property.

I would have thought concave meant in the Z dimension. Find the state with the highest elevations on any two sides with the lowest relative elevations in between.

Pretty cool. But why is this posted on RapGenius?

This reminded me that I want to finish reading the book &#x27;How the States Got Their Shapes.&#x27; <a href="http://amzn.to/16zBxBo" rel="nofollow">http:&#x2F;&#x2F;amzn.to&#x2F;16zBxBo</a> There are some crazy reasons some states have their strange borders.

It appears that the author makes a mistake in his attempts to simplify the problem, because although he is correct that he only needs to look at points on the edges, he goes on to suggest that he is looking only at corners of the polygon, and not at any of the (infinite number of) points between the corners.

Interesting. But an odd definition of concavity. Surely a better definition would be the state whose area relative to its convex hull is smallest. I&#x27;d guess Hawaii or Florida off the top of my head.

It would be neat if the algorithm found the most illustrative solution for each case (e.g. maximizing minimal distance in each state). Especially the Arkansas one is obviously the edge case.

Is this a clever way of recruiting data engineers?