
> 
> I have a question about the consistent of height function defined on a domino tiling. I always see papers claims that height function is defined consistently. But I am confused with the consistent. What does consistent mean?
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> > In lozenge tiling, we can define height function by if $u\to v$ is the positive direction, then
> >  $$h(v)-h(u)= \begin{cases} 1 & \text{if we follow an edge of a lozenge} 
> >  \\
> >  -2 & \text{if we cross a lozenge diagonally} \end{cases} $$
> > 
> > 
> > "This height function is defined consistently because the rules are consistent for a single lozenge and the definition extends consistently across unions." What does mean?
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> > 
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> Also, for domino tiling, we can always define the height function by starting a point and traverse "+1" if left black and "-1" if left white. The author of some papers claims that the proof of the consistency of the height function seen the paper of Thurston Conway's tiling groups. But how to prove that? 
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I feel like it is a little hard to show consistently for domino tiling. Could you give some hints? Thanks.



Proof of consistent of height function
Inconsistency would mean that the sum of these increments ("+1" if left black and "-1" if left white) along some closed cycle is nonzero. But every cycle can be decomposed as a sum of elementary "local" cycles, and for these you can directly verify that the sum along the cycle vanishes.


