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Fun stuff, especially if you're a U.S. geography nerd (states with the largest coastlines, states that border the most states, all that good but mostly arbitrar
by joshuahedlund 13y ago
Fun stuff, especially if you're a U.S. geography nerd (states with the largest coastlines, states that border the most states, all that good but mostly arbitrary stuff)
Pedantic, I'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).
- thejteam 13y agoThat was what I thought it would be at first as well. My definition of "most" concave: pick two points at random in the state. What is the probability the segment connecting them leaves the state? In this case the hard part is defining what to do about water. As an example, I live in Maryland. It is easily the most concave if you don't treat the Chesapeake Bay as part of the state. If you do, not quite as concave. See maps in the link for the shape of MD if you are not familiar with it.
- rm999 13y agoIf you're going to include water I think Hawaii may have Maryland beat. Maybe even Alaska?
- thejteam 13y agoI forgot about islands... But I don't really think of Hawaii as "concave". So I'm going to refine my definition to only allow the points to be chosen in connected areas.
- pavel_lishin 13y ago> So I'm going to refine my definition to only allow the points to be chosen in connected areas. Are two areas considered connected if they cross a river, once?
- howeman 13y agoI was going to go with "Divide the land area of a state by the area of its convex hull. Which is the lowest percentage?"
- mjw 13y agoYeah that was what I was expecting they'd do also. Maybe you'd want to compute the convex hull on a per-connected-component basis, though, to avoid allowing small islands to have undue influence. Either that, or say that any area of sea within its convex hull gets treated as part of the state's area. This approach differs in that it special-cases sea over other kinds of land.
- lessnonymous 13y agoWould you not get the mean height above sea level of all points along the border and divide that by the mean height above sea level of the entire state (which I guess is really an average of some sampling of points)?
- mjw 13y agoFor that suggestion I was just considering it a two-dimensional problem with a "sea or not sea" distinction, not using heights above sea level.
- tzs 13y agoMy first thought was to go with the length of the state's boundary that is on the convex hull divided by the length of the whole boundary. So...we've got at least 3 ways so far to assign a measure of convexity/concavity to a state, and they give quite different results. For instance, your method and the method in the comment you responded to both would assign a low concavity to a state that is almost a square, except that the boundary has a high frequency, low amplitude sawtooth pattern imposed on it. Mine would score that has a very high concavity.
- thejteam 13y ago
- mjw 13y agoMaths challenge: I wonder if you could prove this "probability of a random line segment violating convexity" definition equivalent to something given in terms of a ratio of different areas like the area to convex hull area suggestion below.
- TallGuyShort 13y agoI thought it was going to look at which state had the biggest average drop in elevation from its border, which in hindsight is silly since the entire surface of the earth is pretty convex...
- baddox 13y agoOn the scale of most US states, the convexness of the surface of the Earth is fairly minimal.
- stephengillie 13y agoA state with a mountain range would be fairly convex by comparison. An entirely flat state, like Kansas, would probably have the lowest relative center to edge ratio.
- jlgreco 13y agoI was thinking of concavity as a state which had the most distance outside the state between two points inside the state, somehow proportional to size or area. Florida seems particularly concave to me (it's practically shaped like a banana), Massachusetts slightly less so.
- klodolph 13y agoI think this metric might weight Tennessee higher than one would intuitively desire. Perhaps: 1. Draw a line segment between two connected points in the state, such that only the endpoints of the line are contained in the state. 2a. Draw a line segment perpendicular to this one, such that one end touches the first line, the other end touches the border of the state, and only one endpoint is contained within the state. The length of the largest such line is the "concavity" of the state. 2b. Alternatively, measure the area contained between line #1 and the state border. The largest such area is the "concavity" of the state. 2c. Alternatively, measure the ratio of the area in #2b to the length of the corresponding line in #1.
- eru 13y agoDoesn't the length of the coastline depend on the scale you measure at?