TITLE: Four colour theorem in $3$ dimensions?
QUESTION [8 upvotes]: There is, of course the 4 colour theorem, which has been proven - every map can be coloured in just 4 colours.
However, has anything been examined in $3$ dimensions? By that, I mean how many different colours would I need to be able to colour in the parts of any cut up $3$D object?
Is it infinite, because any shape can have infinite sides?

REPLY [2 votes]: There are meaningful generalizations, if you consider surfaces like sphere, torus, Möbius band etc. as "3D objects". The minimum number of required colors for the mentioned surfaces is 4, 7 and 6, respectively.