Packing Explorations

I've run packing activities before on gridded paper. While it was passable, there were usually a lot of questions about whether objects were successfully packed or peeking out over the line. Here we'll look at an arts and crafts project that serves as a preamble to exploring packing problems.


Setup

In order to do this activity, you'll first need to make an adjustable square container. I've created a couple models based on photographers' adjustable viewing frames. For something quick to set up, here is a basic frame. For something easier to use that requires much more effort and care to put together, here is an advanced frame. For both, I recommend cardstock, but it is really more necessary for the advanced frame, for which you will also need glue or tape.

Aside from the frame, we'll want objects to pack. I've used \(1"\) as my unit because this plays nicely with commercially available \(1" \times 1"\) squares and \(1"\)-diameter disks. Here are some options:


Basic Frame

To create the basic frame, you need only cut out the two V-shaped pieces. You can move them relative to one another to create rectangles of various sizes by keeping the window's angles right. This can be done by choosing both a vertical and a horizontal length on the ruled V and then aligning the sides of the other V with both tics (inner and outer) of each measurement. Here is the basic frame set to \(3 \times 3\):


Advanced Frame

For the advanced frame, you will want to print all four pages. (Note: the last three pages are identical.) I have printed two pages in white and two in orange, all on cardstock. You don't need different colors, but the cardstock is highly recommended for thickness and rigidity.

After you cut everything out, you should have:

  • \(1\) vertical ruler

  • \(1\) horizontal ruler

  • \(11\) long L-shaped pieces

  • \(11\) long I-shaped pieces

  • \(9\) segmented \(3 \times 1\) rectangles (don't cut the dashed lines)

To assemble the frame

  1. First take three L's and three I's. Create a V by pairing an L and an I.

  2. Next add another layer by placing a second L and I on top, but mirrored.

  3. Then add a third layer, again mirroring.

  4. Apply a layer of glue between each layer or wind tape around all three in a few places to get a \(3\)-ply V:

  5. Repeat steps \(1-4\) two more times to end with three V's, total.

  6. Create a fourth V with only two layers using the remaining \(2\) L's and \(2\) I's.

  7. For the third layer, use the horizontal and vertical rulers, again gluing or taping it into a \(3\)-ply V.

  8. Take the nine segmented \(3 \times 1\) rectangles and fold them in thirds to get \(3\)-layer squares.

  9. Glue between layers so they lie flat.

  10. Take two unruled V's and three squares. Place three squares on one of the V's, one on its vertex and one at the end of each arm, gluing them into place.

  11. Glue the other V to these three squares so that one V is supported above the other by these three square pillars, creating two long slots between pillars.

  12. Take the third unruled V and thread each of its arms through the slots between pillars

  13. Place three squares on the vertex and arm ends of this V, gluing them into place

  14. Glue the ruled V to these three pillars, linking the two V-complexes together.

  15. Flip the structure over and glue the last three squares to this new V. (These will serve as feet, keeping it level with the other layer.)

Now that the frame is done, we can use it to explore packings, like in the image below. The thickness of the frame allows it to actually referee whether sufficiently thick objects are inside it or not.


Challenges

The challenges we'll look at will revolve around two types of questions:

  • Given \(n\) \(1\)-diameter disks, what is the smallest square that can contain them with no overlap? How do the circles need to be arranged?

  • Given \(n\) \(1 \times 1\) squares, what is the smallest square that can contain them with no overlap? How do the squares need to be arranged?

I tend to work mostly with circles, since the differences between arrangements for different values of \(n\) tend to be starker and more surprising. After seeing this with circles, the squares can be counterintuitive in the other direction.

My default way to lead this activity is by having the students explore a quantity \(n\) and work together to shave down their record. For example, the group could start with \(7\) circles and see what they can do. Once they are pretty sure they have done the best they can, increase the number of disks by \(1\).

Another way to go about it is to take the current record-holding arrangements found by mathematicians and create puzzles out of those. Eckard Specht's Packomania has a list of current records. Based on current records and rounding up to take into account the precision of our ruling, here are a few challenges:

  1. Pack \(1\) disks into a \(1 \times 1\) square.

  2. Pack \(2\) disks into a \(1.8 \times 1.8\) square.

  3. Pack \(3\) disks into a \(2 \times 2\) square. (Can you make the square slightly smaller than \(2 \times 2\)?)

  4. Pack \(4\) disks into a \(2 \times 2\) square.

  5. Pack \(4\) disks into a \(2.5 \times 2.5\) square.

  6. Pack \(6\) disks into a \(2.7 \times 2.7\) square.

  7. Pack \(7\) disks into a \(2.9 \times 2.9\) square.

  8. Pack \(8\) disks into a \(3 \times 3\) square. (Can you make the square slightly smaller than \(3 \times 3\)?)

  9. Pack \(9\) disks into a \(3 \times 3\) square.

  10. Pack \(10\) disks into a \(3.4 \times 3.4\) square.

  11. Pack \(11\) disks into a \(3.6 \times 3.6\) square. (Can you make the square slightly smaller than \(3.6 \times 3.6\)?)

  12. Pack \(12\) disks into a \(3.6 \times 3.6\) square.

  13. Pack \(13\) disks into a \(3.8 \times 3.8\) square.

  14. Pack \(14\) disks into a \(3.9 \times 3.9\) square.

  15. Pack \(15\) disks into a \(4 \times 4\) square. (Can you make the square slightly smaller than \(4 \times 4\)?)

  16. Pack \(16\) disks into a \(4 \times 4\) square.

  17. Pack \(17\) disks into a \(4.3 \times 4.3\) square.

  18. Pack \(18\) disks into a \(4.4 \times 4.4\) square.

  19. Pack \(19\) disks into a \(4.5 \times 4.5\) square. (Can you make the square slightly smaller than \(4.5 \times 4.5\)?)

  20. Pack \(20\) disks into a \(4.5 \times 4.5\) square.


Notes

  • A lot of the packings can have some pretty counterintuitive properties. For example:

    • if \(n = k^2\), then it would seem the best you could do for \(n\) disks is a \(k \times k\) square, lining the disks in a rectangular grid. This stops being true for \(n = 49 = 7^2\), however!

    • You might also expect that if a disk is loose (i.e., can be slid around), then the packing could be made more efficient. The record holder for \(n = 7\) has been proven optimal, however, and it has a loose disk! Check out the record holders at Packomania.

  • One way to think of packing efficiency is packing density. If packnig disks in a square, for example, one could take the total area of the disks and divide it by the area of the square to get the fraction of the square covered. A higher fraction covered is a denser packing. How dense can a packing be? You might note that if three disks mutually kiss, they leave a triangle-ish shaped gap. If we're trying to pack a plane with disks, the best we can do is a hexagonal packing that only leaves those sorts of gaps. When packing in a square, there will similarly be a gap in a corner if a disk kisses two edges of the square or a gap along an edge if two disks and an edge mutually kiss. There are always going to be gaps of certain sizes or larger, but how many of them do you need?


Extra Challenges

  • Besides circles and squares, another interesting object to try to pack into a square is an equilateral triangle. It turns out most pattern block sets have shapes with edge lengths of \(1"\), so they would play nicely with our frames.

  • While we've focused on packing different shapes inside a square, you can also change the bounding shape. For example, you could try packing \(n\) circles inside a circle. Or \(n\) squares inside an equilateral triangle. It would not be too challenging to design an adjustable equilateral triangle frame. (Make a V with two ruled arms meeting at an angle of \(60^\circ\) and then use a third moving arm to change the three side lengths.) For a circle, it's a little more challenging. I haven't tried it, but you could make a ruled adjustable cylinder and devise your rulings to read off the diameter instead of the circumference. (Instead of ruling it in \(0.1\)-inch increments, rule it in \(\pi/10\)-inch increments but label them as though they were \(0.1\)-inch increments.) Making sure it stays circular could be an issue, though.


This post was sponsored by the Julia Robinson Mathematics Festival

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