Which organelles inside the cell have membranes with many convolutions and why?
One such organelle is the endoplasmic reticulum. This organelle is a highly folded membrane. The many folds result in more surface area for ribosomes to attach and thus allowing protein synthesis to occur at a more efficient rate.
Another organelle that consists of convolutions is the mitochondria. This organelle has two membranes one of which is folded inside the other. The folded inner membrane create what is known as cristae and is the site for a portion of the cellular respiration process. The folds in this organelle also provide more surface area for this chemical process to occur.