But, the problem is that paper cannot be a fish. Fish are made of flesh and bone, not fibrous sheets. The fiber cannot become flesh, no matter how much it looks like it. And so subtlety can go a long way toward representing an object. The traditional crane is a good example of a model that reminds people of a crane, but not because if it's realism or detail, but because of it's simplicity and grace. By holding to more subtle techniques, there is often less paper to tuck away and hide, less thickness in that last fold.
It is awe-inspiring to see when complexity and simplicity come together. Where every inch of the paper is exactly where it needs to be. There is no bulk to hide, and no extra appendages just because there was an extra flap. The nature of paper forces the folder to think about the final arrangement of the paper in such a way that a painter or carver is unconcerned.