Consider a lattice of the edges of the cubic honeycomb. Look along certain angles (rational slopes) and see holes that go all the way through. What are some large holes and what is the diameter of a cylinder that would fit through them? (The orthogonal holes are obvious.) Also consider other lattices of edges, for example the diamond lattice or other honeycombs.
Consider a much more general problem of space consisting of empty and solid regions, and a line segment within an empty region. For each point on the line segment, draw on the perpendicular plane the largest circle centered at that point that does not enter a solid region. The union of these circles forms a tube of varying diameter. Returning to the cubical lattice, what is the shape of the tube for a diagonal line segment connecting two vertices? It tapers to a point on either end.
We can further generalize by asking for the largest ellipse, no longer necessarily centered on the point on the line segment, but containing that point.
Two dimensions? Four?
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