[Math] Number of rectangles to cover a circle

areacircles

After searching around I found this is similiar to the Gauss Circle but different enough (for me anyway) that it doesn't translate well.

I have a circle, radius of 9 that I need to completely cover with rectangles 4 x 8. What is the minimum number of whole rectangles required.
My own calculations concluded it's between 8 and 11. I found this by setting the maximum number of panels equal to (diameter of circle / panel height)x(diameter of circle/panel width) to obtain 10.125 rounded up to 11
I do think I could safely use 10 due to waste.
I then found the minimum by setting the number of panels equal to (area of circle/area of panel). This gave me an answer of 7.95, rounded up to 8.

Is there a better way to do this?

Best Answer

It is possible to cover the circle by $11$ rectangles.

A cover of 11 rectangles

We can construct the $11$ rectangles by following procedure.

  • Center the circle of radius $9$ at origin.
  • Start covering the circle with rectangle $C_0 = [5,9] \times [-4,4]$ (the red one).
  • Rotate $C_0$ with respect to origin for angles $\frac{2k}{7} k \pi $ for $k = 1,\ldots, 6$. This gives us $6$ new rectangles $C_1, C_2, \ldots, C_6$ (the gray ones).
  • Making copies of $C_0$, $C_2$ and $C_5$ and shift them inwards for a radial displacement $4$. This give us $3$ rectangles $C'_0$, $C'_2$ and $C'_5$ (the green ones). To make the covering work, one need to shift $C'_2$ and $C'_5$ a little bit tangentially too.
  • What's remain on the circle can be covered by the rectangle $[-7,1] \times [-2,2]$ (the blue one).

According to Erich's packing center, the current best known covering of circle by $18$ unit squares has radius $r \approx 2.116$. Since $2.116 \le 2.25 = \frac{9}{4}$, this means there is no known covering of our circle with $9$ rectangles. This leaves us with the question whether we can reduce the number of rectangles from $11$ to $10$.

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