Re: Is the four colour theory also valid on a closed surface ?

From: Eric Baird (eric_baird_at_compuserve.com)
Date: 10/12/04 

Date: Tue, 12 Oct 2004 23:11:08 +0000 (UTC)

On Mon, 11 Oct 2004 15:28:23 -0500, "Eugene Shubert"
<http://www.everythingimportant.org> wrote:

>"josefmatz" <josefmatz@arcor.de> wrote in message
>news:416adee1_1@news.arcor-ip.de...
>> I have heared, that the four colour problem is mathematically
>> approved for limited areas on a plane or a curved 2
>> dimensional surface.
>>
>> What i ask is: Is this also valid for a closed surface (e.g. the
>> complete surface of a ball ?)
>> Or are the necessary 5 or 6 colors ?

If you take a hollow rubber ball with a thin skin, with countries
drawn on it, and stick a pin in the /centre/ of one of the countries,
piercing the ball, you haven't altered the ball's colouring
properties.

You can then take that pinhole and stretch it out until the ball
becomes a cup, and then finally becomes a flat circular ***.

The original "exterior" side of the *** then becomes a
"special-case" flat map, where one country (the one that you pierced)
bounds the entire perimeter of the map ... but although the map is
"special", it's still "conventional", so four colours is still
guaranteed to be enough to colour it in.

So yes, the ball is also guaranteed to be colourable in four colours,
as a result of the "simple" 2D result.
 

>>
>> Thanks if you know the answer
>
>The topology of the surface does matter.
>The plane and sphere require a minimum of 4 colors, the Klein bottle,
>Möbius strip and Projective plane require a minimum of 6 colors and
>the torus needs 7. The place to ask questions like this is at
>sci.math.
>
>This newsgroup is for posting childishly simple yet baffling riddles
>like http://www.everythingimportant.org/relativity/special.pdf
>

I think that was supposed to say "maximum".

[Followups set to sci.math]

Regards
=Erk= (Eric Baird)
: " Whoot! "
: -- Tiny Clanger

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