Wednesday, August 19, 2020

[odgwyppg] Density of visible points on a square versus hexagonal lattice

Consider standing at one point on a square grid of points (lattice).  Grid points are separated by unit distance.  How many points can you see closer than a given distance?  This is an orchard visibility problem: points block from view other points directly behind them.

How does the density of visible points compare to that of the equilateral triangle lattice (points separated by unit distance)?  On one hand, the triangular grid has more points because hexagonal close packing has greater packing density than square.  On the other hand, the triangular lattice's greater 6-fold symmetry compared to the square lattice's 4-fold symmetry causes more points to be obscured.

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