0

There is a series of circles with the same diameter in a rectangle large enough, so that the circle can cover every point in the rectangle (each point can be covered two or more times), and the number of circles is as small as possible. So is the figure formed by the line connecting the centers of the circles a regular hexagon? How to prove or simply explain it?

I know that hexagons have the highest space utilization, but don't know if it can be generalized to this conclusion.

Alpha Beta
  • 159
  • See https://math.stackexchange.com/questions/216211/what-is-the-optimal-solution-for-covering-a-rectangle-with-circles and https://stackoverflow.com/questions/7716460/fully-cover-a-rectangle-with-minimum-amount-of-fixed-radius-circles If answers on these questions are true, then for case when rectangle dimensions are much greater than circles radius hexagonal pattern is optimal. – Ivan Kaznacheyeu Jun 07 '22 at 12:08
  • @above Thank you! But the answer in the first site mentions that there is no optimal solution to this problem, and neither site gives a proof. Is this problem an open problem? You say "is optimal" but I can't find the answer from these sites. Is there some explanations I have overlooked? – Alpha Beta Jun 07 '22 at 15:01

0 Answers0