Why are folded membranes an advantage?

Because folded membranes offer a greater volume at a very minimal increase in size. A good example where a folded membrane was an advantage was our own brain. The 'outter lining' of our brain is created out of folded parts that have increased the volume of our brain and yet led us to retain almost the same size through thousands of years. 

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Diagrams for sure 67(8? ; 4