The original thread is here:

https://news.povray.org/povray.advanced-users/thread/%3Cweb.5a6fdd3c85506b565cafe28e0%40news.povray.org%3E/

7 years later, with a bit more experience with dot products, matrices, and all
manner of other things:

Placing a heavy weight on the top corner of a box, and having the vertical edge
partially collapse in an accordian fashion would likely require subdividing the
edge into numerous segments, and then rotating the vertices into place,
maintaining a constant linear "arc length" of the linear spline.

From there, one would have to determine suitable vertices in the two box faces

Theorem.  That would introduce new perturbations in the sheet face, and the
vertices connected to the crumpled edge would have to be displaced.

Those perturbations and displacements would need to be processed by some sort of
algorithm that worked out the cascade of displacements throughout the rest of
the box, ensuring that the lengths and areas stayed constant.

https://people.math.ethz.ch/~acannas/Student_Papers/BSc_Theses/2024_bsc_kunz_mathematics_of_origami.pdf

https://community.wolfram.com/groups/-/m/t/148287

Although I was a graduate student in organic chemistry at the time, when I first
discovered POV-Ray I was very interested in sheet metal development, and I would
say that the books and software concerning making 3D shapes out of flat sheets
of metal ("development") would be an invaluable resource as well.

Doing the reverse: starting with a 3D mesh and unfolding it into a flat sheet
(if possible) would be an amazing accomplishment - it's how professional
packages uv-unwrap meshes.

- BE

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