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Packing Pennies in the Plane |
For suppose
In the diagram to the right, this means that the points
in the yellow rhombus are never in the Voronoi cell
of the third disc.
5. The hexagonally circumscribed circle
Circumscribe a disc with
a regular hexagon,
and circumscribe the hexagon with a circle.
This gives what I call the hexagonally
circumscribed circle of the original disc.
It is a concentric circle whose radius
is
We shall need also another property of these circumscribed circles.
Suppose two of them intersect, but that the discs
themselves do not intersect. Then that intersection
can only intersect the Voronoi cell of
a third disc if the three discs are mutually touching
in the configuration of
discs in a hexagonal packing.
the
hexagonally circumscribed circles.
As a consequence, points in the yellow rhombus
never lie in
the Voronoi cell of a third disc.
5. The hexagonally circumscibed circle