A piece of fabric shaped like a regular hexagon with unit-length sides is given, with vertices A...F and center O. Let a fold correspond to folding some fabric along a straight line, without stretching. For example, bringing A to E and B to D by folding along line FC is a fold. Let a uniform folding be a series of folds giving a result with uniform fabric thickness. For example, if a uniform-folding result is 3 layers thick somewhere, it is 3 layers thick everywhere.
Find a three-fold uniform folding giving a triangle.
Find a four-fold uniform folding giving a
six-fold uniform folding giving a square, or prove that none exists.
Is there a five-fold uniform folding that gives a square?