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My question is:

How to prove that a regular hexagon covers the euclidian plane without overlapping and gaps?

I think it would be the same as proofing the case that an equilateral triangle is covering the plane without overlapping and gaps. I would love to get some literature about it. Thanks alot!

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All internal angles of an hexagon are $2\pi/3$. Hence you can put 3 of them together to "cover" the whole angle $3\cdot 2\pi/3=2\pi$. Now keep doing the same at all vertices of all hexagons (those that are already there and those that you keep attaching).

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