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2D shape can be folded in various ways. For example, trapezoid can have its sides (with the acute angles) folded, so that it will effectively become rectangle – with the difference, that part of the rectangle's interior will have "thickness" doubled (maybe also tripled, if the two opposite sides will overlap inside of the rectangle).

How to model such folds?

Areas should be preserved. In the trapezoid -> rectangle example, after summing areas of different parts of the rectangle – with the overlapping doubled/tripled "thickness" parts counted two or three times respectively – the final area should be equal to area of the initial trapezoid.

The goal is to examine properties like: maximum width or diameter – of the folded shape – with and without respecting the doubled/tripled/etc. thickness.

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Will not folding any shape to its interior reduce (or leave the same) the diameter and the width? And the folds could always be scrunched into the interior, so effectively any reduction of the original shape is achievable this way. –  Joseph O'Rourke Nov 22 '12 at 13:47

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