----- Original Message -----
Sent: Sunday,
December 07, 2003 12:34 AM
Subject: Re:
[BoundaryPoint] Re: Four Color Maps
Four colors will always
suffice if none of the regions are fragmented.
When you created that purple swath of E, you fragmented both C and
E. When you make a fragment, you are actually limiting
the color choices for both it and its parent region to something
less than four, because both must have the same color. That
complicates things. Perhaps you could scatter enough odd
fragments such that six, seven, or even more colors would be
required.
The Four-Color Map Theorem was
suggested when Francis Guthrie discovered in 1852 that a map of the
English counties could be made with only four colors. The question
was asked if four colors would always suffice. Mathematicians
conjectured that they would, provided that no regions were
fragmented. Finding a proof for the theorem became a celebrated
mathematical problem that seemed to defy resolution. In the 1970's,
mathematicians and computers at the University of Illinois finally
devised a proof that ran to thousands of pages. Because the proof was so
voluminous and impossible to double-check manually, some still doubted.
Finally, in the 1990's, a new relatively simpler proof was devised at
Georgia Tech. It is now widely accepted that it's impossible to
design a map without fragmented regions on either a plane or a sphere
that requires more than four colors. Maps on the surfaces of
some bizarre three-dimensional geometric shapes have been shown to
require more colors.
Of course, this is a
mathematical exercise--not readily adaptable to the real world, with it
fragments, enclaves, condominia, neutral zones, conflicting claims,
etc.
Lowell G. McManus
Leesville,
Louisiana, USA
Yes, but the question originally raised was related to an Azerbaijani
enclave. BTW, I didn't see when I posted my map that Eric had already
beat me to it. I also didn't see I fragmented C until you point it
out. Not hard to see that any number of colors can be required if an
arbitrary number of enclaves are allowed. If you have N countries and
each country has an enclave directly inside of another country, N colors would
be required.
----- Original Message -----
Sent: Saturday, December 06, 2003 11:01
PM
Subject: Re: [BoundaryPoint] Re: Four
Color Maps
----- Original Message -----
Sent: Saturday, December 06, 2003 1:32
PM
Subject: [BoundaryPoint] Re: Four Color
Maps
> well i know this has been a perennially
interesting question eric
> but how about first showing us a
theoretical example of a topology
> requiring a 5th color
>
& then at least we will know what to seek on your behalf in
reality
>
> otherwise i think we may only continue to peck away
at this huge
> offering
> & finally shrug our
shoulders
> even if we had the geographical knowledge to apply to
it
Here is my stab at it. Hopefully, one day
the below picture will hang in the Louvre. Start with A, B and
C. Color them each a seperate color. Now, surround by D.
It must be a fourth color. Now make E, F and G which must be red,
yellow and blue, though not necessarily in that order. Now suppose E
claims a (purple) swath of land cutting two of the boundaries AC and
CB. Come to think of it, it is actually unnecessary for F and G to
exist. EFG could be one big red ring on the outside. Then to
make that swath, it would have to be a new color, so EFG must now be that
color. So the new map would show A, B, C and D, as is and change my
EFG outside to just one E and make it purple.
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