2-by-n Disk-Packing Paradox

The top box shows that it is easy to place unit diameter disks in a rectangle, namely by fitting two disks at a time side by side.
At first glance this seems to be the best possible for any .
It is therefore surprising that if is sufficiently large more disks can be packed.
The lower rectangle shows how a rectangle can be filled with 329 unit disks.
There are seven disks at each end and 105 sets of three disks in the middle.

SNAPSHOTS

  • [Snapshot]
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DETAILS

The packing using triples of disks in the second box is more efficient:
While for every two additional disks the top arrangement needs 1 unit length more, the arrangement with sets of two triples in the second rectangle uses only 0.99399 unit length per two disks.
Controls:
Use the "window" slider to view any part of the rectangles.
Click the "details" button to display the -coordinate of the centers of the disks and other information.
We suggest using the "window" slider only when the "details" option is switched off.
Construction of disk arrangement:
There are seven disks at each end and 105 sets of three disks in the middle. Start by placing the 105 triples into the center of the rectangle.
All disks that seem to touch in the diagram indeed touch each other, with the exception of disk 1 and disk 2, which do not touch each other. In detail:
Place 6 against the top and 8.
Place 7 against 9 and the bottom.
Place 5 next to 6 and 7.
Place 4 against 5 and 6.
Place 3 against the bottom and 5.
Place 2 against 3 and 4
Place 1 against 2 and 3.
Place reflections of all these to the right of the 105 center disks.
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Related Curriculum Standards

US Common Core State Standards, Mathematics