Math Is Fun Forum

Anthony.R.Brown

Banned

Registered: 2006-11-16

Posts: 516

Re: Mapping with only four colors-proof by Li, Ming and me

Would the Four Colour Problem Solution! still Stand in the Universe! with things Like Black Holes! where an Area' s Dimension is in More than one Place at the Same time!!

A.R.B

Zhylliolom

Real Member

Registered: 2005-09-05

Posts: 412

Re: Mapping with only four colors-proof by Li, Ming and me

Yes he admits the donut will be different. But he states that a sphere is more or less the same as a plane with infinite streches-you can assume the four infinite corners picked together as the other pole of the sphere. Projective Geometry, he argues.

Yes, the sphere is the one-point compactification of the plane; this doesn't change the coloring. A map on the surface of a torus may sufficiently be colored with 7 different colors. I can discuss how one determines the number of colors required for a map on a given closed surface, it is related to the genus of the surface (which is why the sphere and the plane have the same sufficient number of colors, but a torus is different), if you wish (this is a generalization of the four color theorem to any closed surface, not just the plane). Interestingly, the Klein bottle is the one exception to the rule: it is sufficiently colored with 6 colors, although in theory this should be 7 colors.

George,Y

Member

Registered: 2006-03-12

Posts: 1,379

Re: Mapping with only four colors-proof by Li, Ming and me

Seems tooooo hard.

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X'(y-Xβ)=0