Can you get there from here?

May 23, 2008 5:45 PM

Are there any convex nations?

A convex nation is one where: given two locations in the country the straight line connecting the two points lies entirely within the country (and not in the ocean). I can't find any countries that meet this definition.

If we add territorial waters, how many nations are convex? I can't find good maps that show territorial waters, so I don't know which nations have "dents" in their waters.

posted by thrako to law & government (54 comments total)

Bolivia, by your definition. It's landlocked. There are several landlocked countries.
posted by paulsc at 5:55 PM on May 23, 2008

I don't think Bolivia works. For example, consider the area near the border with Paraguay. e.g. if you picked Tarija and Puerto Aguirre as the two points, the line between them would run through Paraguay, violating the condition.
posted by dixie flatline at 6:03 PM on May 23, 2008

Bolivia, by your definition. It's landlocked. There are several landlocked countries.

Thanks for the response, but I'm looking for something else. I want the straight line between any two points in the nation to lie entirely within the nation. For Bolivia the straight line between Tarija and Puerto Aguirre crosses Paraguay.
posted by thrako at 6:03 PM on May 23, 2008

Well, the rivers that define so many borders lack the mathematical precision you would like for this. Including territorial waters probably gets you a lot of island nations, like Iceland. Papua New Guinea and Egypt look pretty close. Interesting question.
posted by Horselover Fat at 6:04 PM on May 23, 2008

Nauru is pretty close. But it does have a slight dent in the right side.
posted by Lazlo Hollyfeld at 6:08 PM on May 23, 2008

The extent of territorial waters is out to 22 km (or 12 nautical miles) from the coast. So if you go to wikipedia's list of island nations, and look for the smallest ones, like Antigua and Barbuda, those should work.
posted by UrineSoakedRube at 6:08 PM on May 23, 2008

This is a good question. I don't think there are any. From a cursory look I found Morocco is close. Can you can change your question slightly to ask which are geometrically stars, i.e. where there is a 'center' from which a line drawn to any other point in the region is entirely in the region? It's a much weaker condition, still a cool mathematical property of regions, and you'll probably get some hits.
posted by monkeymadness at 6:09 PM on May 23, 2008

Interesting question. Egypt, no, just barely not. Because there are so many curvy borders I would expect this to be pretty rare if not impossible. I think Colorado and Wyoming are your best bets. You essentially need a country with no borders defined by natural features if you assume that all natural features are concave.
posted by GuyZero at 6:09 PM on May 23, 2008

Swaziland is close too.
posted by Lazlo Hollyfeld at 6:09 PM on May 23, 2008

paulsc: entirely within the country - not just ocean. If you look at the CIA map, a straight line from Tarija to Puerto Aguirre will cross through Paraguay, thus leaving Bolivia.

I have to say, this seems a doubtful prospect. First, you need a country generally shaped with no, um, undercuts? There's a math-word for this that I don't remember.

Due to the vagaries of coastlines, it's doubtful-to-none that there is an ocean-bordering country that fits this quality. Also, no islands are allowed within this country's domain, or it's over the water. Though the physical boundaries (rivers, mountains) that are often used as borders have the same squirrely shapes, let's consider them anyway (here). But no.

Since territorial waters are generally defined by their distance from permanent above-water land, they will generally "bubble" around the existing outline. Except where countries are less than the minimum distance (22km) from shoreline, in which case the border goes along the median. This may be an out somewhere...

Seems any most likely convex place would have an "artificial" rather than naturally-shaped border, like Wyoming (except the pesky corner near Yellowstone) or Colorado (this may be a winner, except it's not a country).

(Also, there's some interesting discussions on this very topic if you do some googling. I especially like this one, which ventures into determining which country is the most convex.)
posted by whatzit at 6:11 PM on May 23, 2008

Morocco is out - it's not even a star.
posted by GuyZero at 6:11 PM on May 23, 2008

Here's an article on the subject. Can't see the details.
posted by charlesv at 6:11 PM on May 23, 2008

So would Colorado and Wyoming count if they were countries? I'm trying to understand what you're looking for.
posted by the christopher hundreds at 6:13 PM on May 23, 2008

I,A convex nation is one where: given two locations in the country the straight line connecting the two points lies entirely within the country (and not in the ocean)

Another definition would be that you could walk around the entire border of the country and you would only ever turn in one direction, either left or right.
posted by GuyZero at 6:14 PM on May 23, 2008

I'll email you the JSTOR article if you MeFi mail me your email address.
posted by HotPatatta at 6:14 PM on May 23, 2008

I didn't preview late enough... charlesv's link explains it.
posted by the christopher hundreds at 6:15 PM on May 23, 2008

The article has a bunch of technical gobbledygook but doesn't provide any specific examples of convex nations.
posted by HotPatatta at 6:16 PM on May 23, 2008

Hmm, I think Colorado and Wyoming, like Christopher 100's said, would do it. So where are there square nations? The only places where modern borders (ie, 20th century) were created are in the middle east and Africa, hence all those straight lines. A rectangle or trapezoid should do it for you. how about Egypt and Sudan? Or Nigeria? I don't have the map in front of me so i can't tell, but I bet Africa would be a safe bet. Jordan is out as it has that jug handle in the east.
posted by skybolt at 6:18 PM on May 23, 2008

A convex nation is one where: given two locations in the country the straight line connecting the two points lies entirely within the country (and not in the ocean)

He's expanding that to include ones where: given two locations in the country, the straight line connecting the two points lies entirely within the country or its territorial waters.
posted by UrineSoakedRube at 6:19 PM on May 23, 2008

According to the OP's definition of "convex", a landlocked nation might be convex, but notice "within the country", which implies not only "on dry land" (almost any landlocked nation would then be "convex") but "inside the nation's boundaries".
I feel almost any landlocked nation is to be excluded, together with any nation with a coastline long enough to have a bay, sound, fjord or other "concave features". Some possible candidates might be small independent islands such as Nauru (which has a shallow bay anyway). Considering territorial waters might widen your list a little.
posted by _dario at 6:22 PM on May 23, 2008

Oh I totally forget the other part of the question - if there is no convex nation what is the smallest group of nations that is convex?

Thanks for all the answers.
posted by thrako at 6:22 PM on May 23, 2008

should have previewed, sorry.
posted by _dario at 6:24 PM on May 23, 2008

It depends how detailed you want to get - ethiopia and afghanistan look more convex but there are certainly parts where it wouldn't work. Even among US states it looks like only the rectangles (WY & CO) would pass your test, and they aren't technically convex, just not concave anywhere. I think the problem is that real borders are often quite messy and zig zaggy... even tiny places like kuwait and vatican city seem to fail. Borders that were all convex would probably have to be imposed from without, rather than naturally arise between two areas.

It is funny though that so many ancient cities used to have round borders with the old wall - it was just that beyond them was unclaimed territory, not the other guy's house.
posted by mdn at 6:25 PM on May 23, 2008

What if you add in the territorial waters of Nauru? The curve in the island itself is negated by the straight-line naval border with Kiribati. Looks convex from here.
posted by ormondsacker at 6:27 PM on May 23, 2008

if there is no convex nation what is the smallest group of nations that is convex?
Well, if you accept the shorelines-suck axiom, you're looking for a cluster of landlocked countries. None of those clusters fit the bill either. Can I make you feel better by offering another non-country, Saskatchewan?
posted by whatzit at 6:27 PM on May 23, 2008

Sealand, sort of. If you don't consider its inherent three-dimensionality (i.e. it's impossible to get from one tower straight to the other; you have to travel up to the platform first). And, uh, if you consider it a country.
posted by Flunkie at 6:27 PM on May 23, 2008 [1 favorite has favorites]

Vatican City is awfully close. San Marino not so much, I think the borders follow the mountain tops. If you ignore the harbours Monaco is also close.
posted by Nelson at 6:28 PM on May 23, 2008

If you think that Monaco is close^*, then consider Belize, which is arguably closer, and a whole hell of a lot larger.

*: which I don't, but hey, whatever
posted by Flunkie at 6:38 PM on May 23, 2008

Singapore is really, really close. Looks like the lines for some parts of the island of Tekong Basore to parts of the area around the town (or, uh, whatever) of Sembawang just barely graze into Malaysia, but besides that, it looks concave to me.
posted by Flunkie at 6:50 PM on May 23, 2008

Tekong Basore

By which I mean Tekong Besar. Apparently.
posted by Flunkie at 6:52 PM on May 23, 2008

Nauru is very close. I believe it's territorial waters is (are?) convex - the non-convex parts are smoothed out by the 22km bubble.
posted by aubilenon at 6:55 PM on May 23, 2008

aubilenon - yeah, I said that earlier.

Here is the territorial waters site. Here is convex Nauru (black border).
posted by ormondsacker at 6:59 PM on May 23, 2008

but besides that, it looks concave to me.

And by which, I mean "but besides that, it looks convex to me". Sheesh.
posted by Flunkie at 6:59 PM on May 23, 2008

if you decrease the measuring precision some of curves making the shape concave will become straight lines.
posted by drscroogemcduck at 7:00 PM on May 23, 2008

The JSTOR article does let us know how we might stretch and deform the currently existing borders to create a world that's similar, but with convex countries. Thrako could potentially implement his question.

Wikipedia on landlocked countries lists contiguous groups of landlocked nations, and none of them work.

I checked certain less-recognized, but still serious business countries and recognized microstates like Andorra and Liechtenstein, and they're also out.

Territorial waters are generally supposed to follow the indentations of the coastline, but countries are allowed to claim less and apparently try to claim more. According to the article the Philippines' territorial waters are rectangular but over the 12 mile limit. However, the two territorial water maps for that country that Google Images comes up with show the boundary not to be completely rectangular or convex. (And on preview, this additional map)

I'm thinking you might just be left with the trivial islands?
posted by TheOnlyCoolTim at 7:14 PM on May 23, 2008

The Turtle Mountain Chippewa in North Dakota, the Kickapoo of Kansas, and the Prairie Band Potawatomi (also in Kansas) all appear to have reservations that are perfectly rectangular, and hence convex. There may be more I haven't spotted yet — the maps here give tribal land outlines if you want to keep hunting.

All three seem to be recognized as distinct tribes by the BIA. Depending on your take on tribal sovereignty, their land might count.
posted by nebulawindphone at 7:38 PM on May 23, 2008

I propose the Saudi-Iraqi Neutral Zone
posted by Rumple at 7:54 PM on May 23, 2008

What about Dominica?
posted by Pater Aletheias at 8:12 PM on May 23, 2008

when Tuvalu is completely underwater it'll meet your criteria. And if al gore is right then there may soon be many convex nations!
posted by jewzilla at 9:04 PM on May 23, 2008

How about the French claim to Antarctica?

Territory, yeah, but that wedge is pretty cool. One research station scecedes from France and you've got your answer. Maybe you can email them and ask as a favor. And look at what it does to Australia's claim!
posted by cowbellemoo at 9:27 PM on May 23, 2008

SEALAND!!! (Assuming you don't count the helipad as part of the "country's" territory.) It's a rectangle!

I win. :-)
posted by paultopia at 9:33 PM on May 23, 2008

If you throw in five miles of territorial waters, Taiwan?

I expect the Chinese government will not be happy with my suggestion that Taiwan is a country, but come on, it's a topology question.
posted by el_lupino at 10:01 PM on May 23, 2008

The answer will be limited by the threshold of violation you are willing to accept. Trivially, there are no ideal straight lines, so even in,say, Colorado or Sealand, you should still expect a concave stretch within a border but whose curvature is slight enough to seem a straight line on a map.
posted by Gyan at 12:15 AM on May 24, 2008

what is the smallest group of nations that is convex?

It looks like Orange Free State and the little unlabeled one next to it are convex when taken together.
posted by ctmf at 12:31 AM on May 24, 2008

Maybe I'm the only one here who's a bit slow but it took me a while to understand the question. Where the question says "given two locations", I think it should say "given any two locations".
posted by tetranz at 4:21 AM on May 24, 2008

Can I make you feel better by offering another non-country, Saskatchewan?

Actually, Saskatchewan wouldn't meet the definition, as the Saskatchewan-Manitoba border has a subtle stair-step pattern. (I believe this is to deal with the inconsistency of the line of longitude and the mile-based grid system used for roads and farms.)
posted by teg at 5:47 AM on May 24, 2008

Sri Lanka must be pretty close...
posted by aihal at 5:49 AM on May 24, 2008

(Meant to link to map of Saskatchewan.)
posted by teg at 5:50 AM on May 24, 2008

The Vatican? (Can't tell if the line from the Audience Hall to the tip of St. Peter's goes through its own territory or into Italy)
posted by nax at 6:56 AM on May 24, 2008

Trivially, there are no ideal straight lines, so even in,say, Colorado or Sealand, you should still expect a concave stretch within a border but whose curvature is slight enough to seem a straight line on a map.

Colorado is bounded by lines of latitude and longitude, which are straight for the curved surface of the Earth.
posted by TheOnlyCoolTim at 8:31 AM on May 24, 2008

Colorado is bounded by lines of latitude and longitude, which are straight for the curved surface of the Earth.

Lines of longitude are straight, since they're segments of great circles. Lines of latitude are not, except for the equator. So even Colorado has a slight concavity: a great-circle route connecting the northeasternmost and northwesternmost corners of the state will run a few miles into Nebraska & Wyoming.

Also, even pace this objection, Saskatchewan doesn't work — you can see from the map that the border with Manitoba is kind of zig-zaggy. This has to do with the way the land was surveyed when it was being settled, and the fact that a one mile-by-one mile section of land will span more lines of longitude as you go north.
posted by Johnny Assay at 9:01 AM on May 24, 2008

I guess it depends how you consider the setup. I'd call the cardinal directions straight and Colorado a convex state.
posted by TheOnlyCoolTim at 12:24 PM on May 24, 2008

Mathematica contains cartographic data for countries, so it can be quickly used to rule out most countries. The data is only of a certain finite resolution, so it rules in some countries that we know are not convex, such as Vatican City.

Here are the countries it fails to rule out:

toSpherical[{x1_, x2_}] := {Cos[x1] Sin[x2], Sin[x1] Sin[x2], Cos[x2]}
test[s_] :=
With[{dat =
Most[toSpherical /@ (\[Pi]/180
CountryData[s, "FullCoordinates"][[1]])]},
Length[CountryData[s, "FullCoordinates"]] == 1 &&
Equal @@
Thread[Norm /@ (dat +
Cross @@@ Partition[dat - RotateRight[dat], 2, 1, {1, 1}]) >=
1]
]
Select[CountryData[], test]

{"ChristmasIsland", "Liechtenstein", "Macau", "Montserrat", "Nauru", "SanMarino", "VaticanCity"}

Have all these already been ruled out?
posted by hAndrew at 10:09 AM on May 25, 2008 [3 favorites has favorites]

Excellent work, but they've either been discussed or aren't countries. Still, nice work.
posted by TheOnlyCoolTim at 10:25 AM on May 25, 2008