The Vanishing Area Puzzle

Area doesn't change when you rotate or flip over a shape, yet that is what seems to happen here!
Each pattern is made with 16 pieces
In one picture they fit snugly, with no gaps.  The other has a white gap in the middle.  So the first picture has a way of using the pieces to cover more area.  But how can this be?  the area of the pieces are no different than before?

This puzzle is from an article by Martin Gardner in Scientific American, May 1961.  If you give up, try printing this page and cut out the pieces to manipulate. You can also look for clues at the source of the puzzle here

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