Modelling the folding of paper into three dimensions using affine transformations

We model the folding of ordinary paper via piecewise isometries R-2-->R-3. The collection of crease lines and vertices in the unfolded paper is called the crease pattern. Our results generalize the previously known necessity conditions from the more restrictive case of folding paper flat (into R-2); if the crease pattern is foldable, then the product (in a non-intuitive order) of the associated rotational matrices is the identity matrix. This condition holds locally in a multiple vertex crease pattern and can be adapted to a global condition. Sufficiency conditions are significantly harder, and are not known except in the two-dimensional single-vertex case. (C) 2002 Elsevier Science Inc. All rights reserved.