More Stackfolding dissections

In the late summer and fall of 2007, I became interested in folding dissections that have only one assemblage. To mimic a dissection such as two equal squares to one, I would have the smaller square be in a "tower" twice as high as that of the larger square. I wrote the following article,

"Unfolding an 8-high Square, and Other New Wrinkles", by Greg N. Frederickson,
which was the basis for my talk at the Eighth Gathering for Gardner, in March 2008.

I posted a sampling of my animations on a webpage with stackfolding animations.

When I was invited to give a talk at the DIMACS Workshop on Algorithmic Mathematical Art: Special Cases and Their Applications, May 11-13, 2009, I looked to extend that first paper in specifically algorithmic directions.

In particular, I studied how to fold an m-high stack of equilateral triangles to an n-high stack of equilateral triangles, for m and n being different natural numbers. I also studied how to fold any member of a class of "well-formed polyominoes" from a 1-high figure to the corresponding 2-high version. Below are several animations for the triangles. For the polyominoes, I shot some video, but I'm having trouble getting the appropriate aspect ratio for the triangles and the well-formed polyominoes.

A 1-high triangle to a 4-high triangle. Since each piece has non-zero thickness and pieces are hinged in a manner consistent with real hinges, we are forced to split two of the triangles so that we can actually do the hinging.

A 1-high triangle to a 9-high triangle. You can begin to see how one would handle 1-high to n²-high, handling each row of the 1-high triangle in turn.

A 4-high triangle to a 3-high triangle. Not so algorithmic, but a pretty stackfolding dissection nonetheless.

A 4-high hexagon to a 3-high hexagon. (This is a large file: 8 Mb.) This is also not so algorithmic.

A 2-high 10-pointed star to a 4-high 5-pointed star. (This file is even larger: 11 Mb.) The hinging looks chaotic but is not.

For additional background material, see:

Piano-Hinged Dissections: Time to Fold!, by Greg N. Frederickson,
A K Peters, 2006.