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Images of Math

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Re: Images of Math

#21

Post by Someone »

JadeNB wrote: post #18
Someone wrote: post #17

Earlier quoted context omitted.

"You never need more than four colors to color every country on a map a different color from its neighbors" Is that true in today's geography? I know of one almost [1] counterexample (of that specific statement, not of the four-color theorem): the North Sea, Belgium, the French Republic, Germany and the Kingdom of the Netherlands all border each other, so you need five different colors to color them on a map. [1] Alm…

> [1] Almost because the Kingdom of the Netherlands isn't a country. Neither is the North Sea! Nonetheless, this is a neat example. > Back to my original question: does anybody know of a valid counterexample for the statement on countries? A standard counterexample to the hypotheses of the 4-colour theorem (though not to the conclusion, as consulting a map easily verifies) is Michigan, which is not connected. The 4-c…

According to http://en.m.wikipedia.org/wiki/Quadripoint, there currently isn't any quadripoint between countries, although it is awfully close between Namibia, Botswana, Zambia and Zimbabwe.

Trijunctions are very common. Luxembourg has 3 of them, Germany (if I count correctly) 7.

Re: Images of Math

#22

Post by dedward »

JadeNB wrote: post #18
Someone wrote: post #17

Earlier quoted context omitted.

"You never need more than four colors to color every country on a map a different color from its neighbors" Is that true in today's geography? I know of one almost [1] counterexample (of that specific statement, not of the four-color theorem): the North Sea, Belgium, the French Republic, Germany and the Kingdom of the Netherlands all border each other, so you need five different colors to color them on a map. [1] Alm…

> [1] Almost because the Kingdom of the Netherlands isn't a country. Neither is the North Sea! Nonetheless, this is a neat example. > Back to my original question: does anybody know of a valid counterexample for the statement on countries? A standard counterexample to the hypotheses of the 4-colour theorem (though not to the conclusion, as consulting a map easily verifies) is Michigan, which is not connected. The 4-c…

I was under the impression corners don't matter... it's about borders, not corners. Draw any kind of map you like, you only need four colours.

Re: Images of Math

#23

Post by JadeNB »

dedward wrote: post #22
JadeNB wrote: post #18

Earlier quoted context omitted.

> [1] Almost because the Kingdom of the Netherlands isn't a country. Neither is the North Sea! Nonetheless, this is a neat example. > Back to my original question: does anybody know of a valid counterexample for the statement on countries? A standard counterexample to the hypotheses of the 4-colour theorem (though not to the conclusion, as consulting a map easily verifies) is Michigan, which is not connected. The 4-c…

I was under the impression corners don't matter... it's about borders, not corners. Draw any kind of map you like, you only need four colours.

> I was under the impression corners don't matter... it's about borders, not corners. Draw any kind of map you like, you only need four colours.

Corners matter (or do not matter, depending on how you like to phrase it!) only in the slightly non-intuitive sense that countries that meet only at a corner must be declared not to 'meet' at all for purposes of the theorem. Otherwise, you can have four regions meeting only at a corner (say by sub-dividing a square), which therefore use 4 colours, and a 'moat' surrounding them all, which therefore requires a 5th colour.

Re: Images of Math

#24

Post by JadeNB »

Someone wrote: post #21
JadeNB wrote: post #18

Earlier quoted context omitted.

> [1] Almost because the Kingdom of the Netherlands isn't a country. Neither is the North Sea! Nonetheless, this is a neat example. > Back to my original question: does anybody know of a valid counterexample for the statement on countries? A standard counterexample to the hypotheses of the 4-colour theorem (though not to the conclusion, as consulting a map easily verifies) is Michigan, which is not connected. The 4-c…

According to http://en.m.wikipedia.org/wiki/Quadripoint , there currently isn't any quadripoint between countries, although it is awfully close between Namibia, Botswana, Zambia and Zimbabwe. Trijunctions are very common. Luxembourg has 3 of them, Germany (if I count correctly) 7.

Wow—regardless of its use to answer the question, that's a new word for me, and a fascinating article. (Who would have expected such a debate to lead to literal shots fired? Well, I guess any student of cartography and history, but not me.) Thanks!

Re: Images of Math

#25

Post by Someone »

JadeNB wrote: post #23
dedward wrote: post #22

Earlier quoted context omitted.

I was under the impression corners don't matter... it's about borders, not corners. Draw any kind of map you like, you only need four colours.

> I was under the impression corners don't matter... it's about borders, not corners. Draw any kind of map you like, you only need four colours. Corners matter (or do not matter, depending on how you like to phrase it!) only in the slightly non-intuitive sense that countries that meet only at a corner must be declared not to 'meet' at all for purposes of the theorem. Otherwise, you can have four regions meeting only…

Or just n>4 regions meeting in a corner. Allowing 'meeting in a single point' turns the theorem into the 'infinite color theorem', which would be uninteresting.

Re: Images of Math

#26

Post by dedward »

Someone wrote: post #25
JadeNB wrote: post #23

Earlier quoted context omitted.

> I was under the impression corners don't matter... it's about borders, not corners. Draw any kind of map you like, you only need four colours. Corners matter (or do not matter, depending on how you like to phrase it!) only in the slightly non-intuitive sense that countries that meet only at a corner must be declared not to 'meet' at all for purposes of the theorem. Otherwise, you can have four regions meeting only…

Or just n>4 regions meeting in a corner. Allowing 'meeting in a single point' turns the theorem into the 'infinite color theorem', which would be uninteresting.

Right.

The point I'm making is that this isn't simply that "there isn't anything in our current geo-political makeup where enough countries meet to disprove the theorem"... the theorem holds up under any theoretical design.

Re: Images of Math

#27

Post by JadeNB »

Someone wrote: post #17
jordigh wrote: post #3

You never need more than four colors to color every country on a map a different color from its neighbours. This was proved in the 20th century — but nobody knows why it is true. Heh. I suppose we've proved it, but this author doesn't accept this proof as an explanation for why it's true.

"You never need more than four colors to color every country on a map a different color from its neighbors" Is that true in today's geography? I know of one almost [1] counterexample (of that specific statement, not of the four-color theorem): the North Sea, Belgium, the French Republic, Germany and the Kingdom of the Netherlands all border each other, so you need five different colors to color them on a map. [1] Alm…

As a geographical idiot who has just consulted a map, I am puzzled by:

> I know of one almost [1] counterexample (of that specific statement, not of the four-color theorem): the North Sea, Belgium, the French Republic, Germany and the Kingdom of the Netherlands all border each other, so you need five different colors to color them on a map.

On Google Maps, Belgium seems to interpose entirely between France and the Netherlands. Am I deceived by appearances, or is there some subtle geopolitical point?

Re: Images of Math

#28

Post by JadeNB »

dedward wrote: post #26
Someone wrote: post #25

Earlier quoted context omitted.

Or just n>4 regions meeting in a corner. Allowing 'meeting in a single point' turns the theorem into the 'infinite color theorem', which would be uninteresting.

Right. The point I'm making is that this isn't simply that "there isn't anything in our current geo-political makeup where enough countries meet to disprove the theorem"... the theorem holds up under any theoretical design.

> The point I'm making is that this isn't simply that "there isn't anything in our current geo-political makeup where enough countries meet to disprove the theorem"... the theorem holds up under any theoretical design.

No, it doesn't! Of course, as a true theorem, its conclusion holds whenever its hypotheses do—but there are hypotheses, not all of which are satisfied by any theoretical (or, perhaps more importantly, real-world) design.

One is that meeting at a corner is not counted as 'meeting', which you may fairly object is just a definition rather than a hypothesis; but another, which is a genuine hypothesis rather than just a matter of definition, is that the countries / regions must be connected [0]. (For a real-world example where this hypothesis is not satisfied—although the conclusion still is—on the level of states, see Michigan; and, on the level of countries, see the US and Alaska. I don't know if it is possible to make all countries connected by building imaginary land bridges across non-territorial waters, but I doubt it.)

EDIT: As Someone pointed out at the top of this thread (http://news.ycombinator.com/item?id=8801082 and then, in response to my confusion, http://news.ycombinator.com/item?id=8804876), this disconnectedness can actually create an issue; the fact that it technically isn't an issue for the particular case that he or she mentions is just because the regions involved aren't technically 'countries', rather than because no pathological arrangement of countries can break the 4-colour theorem.

EDIT: [0] Probably I should say 'open and connected', to avoid the pathologies a malicious topologist could cook up.

Re: Images of Math

#29

Post by Someone »

JadeNB wrote: post #27
Someone wrote: post #17

Earlier quoted context omitted.

"You never need more than four colors to color every country on a map a different color from its neighbors" Is that true in today's geography? I know of one almost [1] counterexample (of that specific statement, not of the four-color theorem): the North Sea, Belgium, the French Republic, Germany and the Kingdom of the Netherlands all border each other, so you need five different colors to color them on a map. [1] Alm…

As a geographical idiot who has just consulted a map, I am puzzled by: > I know of one almost [1] counterexample (of that specific statement, not of the four-color theorem): the North Sea, Belgium, the French Republic, Germany and the Kingdom of the Netherlands all border each other, so you need five different colors to color them on a map. On Google Maps, Belgium seems to interpose entirely between France and the Ne…

Read my post again, especially the part after Confused?

The kingdom of the Netherlands borders the French Republic in the Caribbean.

Re: Images of Math

#30

Post by JadeNB »

Someone wrote: post #29
JadeNB wrote: post #27

Earlier quoted context omitted.

As a geographical idiot who has just consulted a map, I am puzzled by: > I know of one almost [1] counterexample (of that specific statement, not of the four-color theorem): the North Sea, Belgium, the French Republic, Germany and the Kingdom of the Netherlands all border each other, so you need five different colors to color them on a map. On Google Maps, Belgium seems to interpose entirely between France and the Ne…

Read my post again, especially the part after Confused? The kingdom of the Netherlands borders the French Republic in the Caribbean.

Ah—I knew (because the 4-colour theorem holds!) that there must be a meeting at a corner or a disconnection, but I couldn't see either! Thanks for clarifying.
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