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Just today, I was making tortilla chips, and I began to wonder, what is the most efficient way to pack circular quarters onto the plane?

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This sort of circle packing is most efficient for circles, but what if quarter circles are used instead of whole circles, is it still this method?

Does anyone have an answer for this? What is the most efficient way to pack quarter circles onto the plane?

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    $\begingroup$ I think you mean "in a square of side $2d$" (diameter $d$) rather than "in the plane". $\endgroup$ – Ethan Bolker Feb 6 '16 at 23:47
  • $\begingroup$ I'm actually not sure... Would the answers be different? $\endgroup$ – Noah Harris Feb 6 '16 at 23:49
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    $\begingroup$ Probably. The best circle packing for the whole plane (or for a part of the plane that's large relative to the size of the circles) is the one based on hexagons, not squares. $\endgroup$ – Ethan Bolker Feb 6 '16 at 23:53
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I expect, but am not about to try to prove, that the optimal packing would be something like the following:

enter image description here

Of course, pushing each to be as tight as possible. (pardon my bad mspaint skills)

In fact, a similar packing argument allowed the ancient egyptians to derive the formula $\pi r^2$ for the area of the circle, noting that as you cut the circle into a large number of segments and alternate between pointed up and pointed down, the shape resembles a rectangle of height $r$ and width $\pi r$.

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