Comments on: Can you get there from here?

Question: Can you get there from here?

thrako — Fri, 23 May 2008 17:45:18 -0800

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.

By: paulsc

paulsc — Fri, 23 May 2008 17:55:15 -0800

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

By: dixie flatline

dixie flatline — Fri, 23 May 2008 18:03:03 -0800

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.

By: thrako

thrako — Fri, 23 May 2008 18:03:28 -0800

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.

By: Horselover Fat

Horselover Fat — Fri, 23 May 2008 18:04:13 -0800

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.

By: Lazlo Hollyfeld

Lazlo Hollyfeld — Fri, 23 May 2008 18:08:43 -0800

Nauru is pretty close. But it does have a slight dent in the right side.

By: UrineSoakedRube

UrineSoakedRube — Fri, 23 May 2008 18:08:58 -0800

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.

By: monkeymadness

monkeymadness — Fri, 23 May 2008 18:09:00 -0800

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.

By: GuyZero

GuyZero — Fri, 23 May 2008 18:09:23 -0800

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.

By: Lazlo Hollyfeld

Lazlo Hollyfeld — Fri, 23 May 2008 18:09:46 -0800

Swaziland is close too.

By: whatzit

whatzit — Fri, 23 May 2008 18:11:33 -0800

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.)

By: GuyZero

GuyZero — Fri, 23 May 2008 18:11:34 -0800

Morocco is out - it's not even a star.

By: charlesv

charlesv — Fri, 23 May 2008 18:11:48 -0800

Here's an article on the subject. Can't see the details.

By: the christopher hundreds

the christopher hundreds — Fri, 23 May 2008 18:13:23 -0800

So would Colorado and Wyoming count if they were countries? I'm trying to understand what you're looking for.

By: GuyZero

GuyZero — Fri, 23 May 2008 18:14:27 -0800

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.

By: HotPatatta

HotPatatta — Fri, 23 May 2008 18:14:29 -0800

I'll email you the JSTOR article if you MeFi mail me your email address.

By: the christopher hundreds

the christopher hundreds — Fri, 23 May 2008 18:15:06 -0800

I didn't preview late enough... charlesv's link explains it.

By: HotPatatta

HotPatatta — Fri, 23 May 2008 18:16:20 -0800

The article has a bunch of technical gobbledygook but doesn't provide any specific examples of convex nations.

By: skybolt

skybolt — Fri, 23 May 2008 18:18:54 -0800

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.

By: UrineSoakedRube

UrineSoakedRube — Fri, 23 May 2008 18:19:02 -0800

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.

By: _dario

_dario — Fri, 23 May 2008 18:22:18 -0800

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.

By: thrako

thrako — Fri, 23 May 2008 18:22:32 -0800

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.

By: _dario

_dario — Fri, 23 May 2008 18:24:50 -0800

should have previewed, sorry.

By: mdn

mdn — Fri, 23 May 2008 18:25:41 -0800

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.

By: ormondsacker

ormondsacker — Fri, 23 May 2008 18:27:06 -0800

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.

By: whatzit

whatzit — Fri, 23 May 2008 18:27:11 -0800

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?

By: Flunkie

Flunkie — Fri, 23 May 2008 18:27:47 -0800

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.

By: Nelson

Nelson — Fri, 23 May 2008 18:28:26 -0800

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.

By: Flunkie

Flunkie — Fri, 23 May 2008 18:38:09 -0800

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

By: Flunkie

Flunkie — Fri, 23 May 2008 18:50:41 -0800

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.

By: Flunkie

Flunkie — Fri, 23 May 2008 18:52:10 -0800

Tekong Basore

By which I mean Tekong Besar. Apparently.

By: aubilenon

aubilenon — Fri, 23 May 2008 18:55:49 -0800

Nauru is very close. I believe it's territorial waters is (are?) convex - the non-convex parts are smoothed out by the 22km bubble.

By: ormondsacker

ormondsacker — Fri, 23 May 2008 18:59:01 -0800

aubilenon - yeah, I said that earlier.

Here is the territorial waters site. Here is convex Nauru (black border).

By: Flunkie

Flunkie — Fri, 23 May 2008 18:59:21 -0800

but besides that, it looks concave to me.

And by which, I mean "but besides that, it looks convex to me". Sheesh.

By: drscroogemcduck

drscroogemcduck — Fri, 23 May 2008 19:00:23 -0800

if you decrease the measuring precision some of curves making the shape concave will become straight lines.

By: TheOnlyCoolTim

TheOnlyCoolTim — Fri, 23 May 2008 19:14:59 -0800

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?

By: nebulawindphone

nebulawindphone — Fri, 23 May 2008 19:38:18 -0800

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.

By: Rumple

Rumple — Fri, 23 May 2008 19:54:12 -0800

I propose the Saudi-Iraqi Neutral Zone

By: Pater Aletheias

Pater Aletheias — Fri, 23 May 2008 20:12:36 -0800

What about Dominica?

By: jewzilla

jewzilla — Fri, 23 May 2008 21:04:44 -0800

when Tuvalu is completely underwater it'll meet your criteria. And if al gore is right then there may soon be many convex nations!

By: cowbellemoo

cowbellemoo — Fri, 23 May 2008 21:27:33 -0800

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!

By: paultopia

paultopia — Fri, 23 May 2008 21:33:47 -0800

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

I win. :-)

By: el_lupino

el_lupino — Fri, 23 May 2008 22:01:07 -0800

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.

By: Gyan

Gyan — Sat, 24 May 2008 00:15:26 -0800

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.

By: ctmf

ctmf — Sat, 24 May 2008 00:31:04 -0800

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.

By: tetranz

tetranz — Sat, 24 May 2008 04:21:10 -0800

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".

By: teg

teg — Sat, 24 May 2008 05:47:22 -0800

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.)

By: aihal

aihal — Sat, 24 May 2008 05:49:38 -0800

Sri Lanka must be pretty close...

By: teg

teg — Sat, 24 May 2008 05:50:13 -0800

(Meant to link to map of Saskatchewan.)

By: nax

nax — Sat, 24 May 2008 06:56:28 -0800

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)

By: TheOnlyCoolTim

TheOnlyCoolTim — Sat, 24 May 2008 08:31:29 -0800

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.

By: Johnny Assay

Johnny Assay — Sat, 24 May 2008 09:01:52 -0800

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.

By: TheOnlyCoolTim

TheOnlyCoolTim — Sat, 24 May 2008 12:24:41 -0800

I guess it depends how you consider the setup. I'd call the cardinal directions straight and Colorado a convex state.

By: hAndrew

hAndrew — Sun, 25 May 2008 10:09:37 -0800

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?

By: TheOnlyCoolTim

TheOnlyCoolTim — Sun, 25 May 2008 10:25:40 -0800

Excellent work, but they've either been discussed or aren't countries. Still, nice work.