A bit late but I think you're going to have points with infinite hexagons in any neighbourhood of them. Otherwise the set of hexagons would be finite because of compactness.
Which I think means those points can't really lie on the edge of or interior of a hexagon. But I can't rule out that they might lie on a corner, it's hard to reason about.
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But it _is_ possible to make a ball out of hexagons if you use infinitely many of them!
A bit late but I think you're going to have points with infinite hexagons in any neighbourhood of them. Otherwise the set of hexagons would be finite because of compactness.
Which I think means those points can't really lie on the edge of or interior of a hexagon. But I can't rule out that they might lie on a corner, it's hard to reason about.