## Animations of polyominoes folding from one level to two

In 2009 I explored techniques for folding a polyomino that is one level thick to the same figure that is two levels thick. This resulted in an article,
"Folding Polyominoes from One Level to Two", by Greg N. Frederickson,
College Mathematics Journal, vol. 42, no. 4 (2011), pp. 265-274.
To give a better idea how the polyominoes actually fold, I have created a large diagram of the folding dissection in Figure 3 which you can print, cut out, and fold.

Four challenges were left to readers. Here are their solutions.

I have also produced a series of animations.
(Note: The animations range in size up to 2.9 Megabytes. Please do not attempt to download them if you have a slow connection.)
Here is the animation (0.8 Mb) of my canonical folding dissection of a Greek Cross from one level to two, which appears in Figure 3 of the article. This folding assumes that the four pieces adjacent to the star HV-piece are rounded, although to be honest I didn't shave them for this animation. The reason why is that it's impossible (for me) to see any interpenetration of pieces when the animation is run at this speed. If we were to slow the animation down sufficiently, then we would have a chance to see it, but at the risk of being so slow as to put everyone to sleep!
Here is the animation (1.4 Mb) of my folding dissection of a Greek Cross from one level to two, which appears in Figure 4 of the article. This folding is not rounded and is consequently more complex. As discussed in the article, I split the star HV-piece into two half-stars. After I fold the pieces adjacent to the two half-stars by 90 degrees, I fold one of the half-stars by 90 degrees relative to the other. This then allows me to complete the folding (through the remaining 90 degrees) of the pieces adjacent to the half-stars. Once this is accomplished, I can fold the one half-star back 90 degrees relative to the other. Finally, I fold the eight isosceles right triangles into their final position.
Here is the animation (2.9 Mb) of my folding dissection of a Cross of Lorraine from one level to two, which appears in Figure 5 of the article. This folding is not rounded and is thus substantially more complex. There are two star HV-pieces that are now each split into half-stars. The folding with respect to each pair of half-stars is similar to what we saw before, except that now these two sequences of motions must be interleaved. It's complicated by the fact that a number of the pieces are longer in length than pieces in the preceding Greek Cross folding. I must admit that it took me a number of tries to order the individual foldings so that no piece interpenetrated another.
Here is the animation (0.8 Mb) of my special folding dissection of an F-pentomino from one level to two, which appears in Figure 13 of the article. This folding is rounded. The nonconvex green piece causes the problem, when it folds so that its concave notch surrounds the roof of a green house-shaped piece. It is the side of the roof furthest from the axis of rotation that will obstruct the movement.
Here is the animation (2.2 Mb) of my hybrid folding dissection of a quasi-well-formed polyomino from one level to two, which appears in Figures 17 and 18 of the article. This folding is rounded, because I use a star HV-piece. You can see where the four pieces attached to the star rotate 90 degrees early in the animation. Once most of the other pieces have rotated into position, these four pieces simultaneously rotate the final 90 degrees. The only remaining folds are for the isosceles right triangles and then the final rotation from a view of the "red level" to a view of the "green level".
For additional background material, see:
Piano-Hinged Dissections: Time to Fold!, by Greg N. Frederickson,
A K Peters, 2006.

Text and animations are copyright 2009-2010 by Greg Frederickson
and may not be copied, electronically or otherwise,
without his express written permission.

Last updated November 27, 2011.