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Mathematics
Linear and Abstract Algebra
Question regarding fundamental region of a lattice
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[QUOTE="Office_Shredder, post: 6861022, member: 53426"] Given that you have the right volume, if the lattice shifts don't intersect they have to cover the whole space. Consider an enormous sphere, so that effects around the surface don't matter, then you have an equal number of non intersecting volumes using this region vs a region you know is a fundamental unit (all the lattice shifts in the sphere) which both have the same volume, and the latter leaves no empty space, so the former can't leave empty space either. Suppose ##z\in x+P\cap y+P## for ##x,y## lattice points and z any arbitrary point. Then ##z=x+p_1=y+p_2## for ##p_1,p_2\in P##. Hence ##x-y=p_2-p_1##. ##x-y## is another lattice point, so a contradiction can be generated by showing you can't get a lattice point by adding two elements of the region (I've been a little sloppy here, you can add two elements from the surface of the region and get a lattice point, the real issue is if you can add two interior points. I'll let you think it over) [/QUOTE]
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Linear and Abstract Algebra
Question regarding fundamental region of a lattice
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