In a rectangle that's 2 by 200 units long, it's trivial to draw 400 non-overlapping unit-diameter circles.
But in the same rectangle, can you draw 401 circles?
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Posted 14 July 2013 - 08:35 AM
Vidi vici veni.
Posted 14 July 2013 - 03:59 PM
I'm pretty sure you can fit way more than 401 circles in the rectangle.
what would your array be?
Sorry. There was an error in my thinking. Is Bonanova right? I think so. I can't possibly think of another way of doing this.
Posted 15 July 2013 - 05:28 AM
my idea is to zigzag..but its too long
Then a group of three top row circles packed against the top of the box. Then three more down, then three more up. Do that 66+ times, to the end of the box. It seems like that permits the top row to be compacted in length. That may be enough to fit two extra (199 = 2 = 201) top row balls.
It has to be the case that two extra top row circles appear.
That won't happen unless at some point (at least in two cases) a top row circle is directly above a bottom row circle. So the top row circles must move up and down in groups.
Start with a group of three top row circles nestled down in triangular packing position.
Then a group of three top row circles packed against the top of the box.
Then three more down, then three more up. Do that 66+ times, to the end of the box.
It seems like that permits the top row to be compacted in length.
That may be enough to fit two extra (199 = 2 = 201) top row balls.
Vidi vici veni.
Posted 15 July 2013 - 06:06 AM
I don't feel I have a very solid proof. But: a rectangular packing clearly won't work, a hexagonal packing where circles are touching in lines parallel to the long axis of the box will never allow you to make a third row of circles (and I don't think it would allow enough free space on one side to somehow squeeze in another circle because rectangular packing is as squeezed into a corner as you can get), and the most efficient way of packing I can envision that will make a third row of circles is to take a slice of a hexagonal packing like so:
Where the circles directly above and directly below each other are touching. In that case, each "repeating unit" of three circles would be separated from the previous one by sqrt(3) ~= 1.73, and you could only fit 115 of them in the 200 unit long rectangle to get about 345 circles in.
Posted 15 July 2013 - 07:35 AM
Consider the drawing below.
The top (red) row of circles jog up and down in groups of three.
In 20 circles distance, the row has compacted about 1/7 diameter.
After 200 circles, the compaction would be of the order of 1.5 diameters.
See green circle and dimension lines on the right end.
201 circles should fit across the top row.
Vidi vici veni.
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