Actually, these two shortcomings kind of cancel each other out. The sheets of tissue that fold, invaginate and turn inside out in a developing embryo do indeed grow, and it is that very growth that provides part of the motive force which, in origami, is supplied by the human hand. If you wanted to make an origami model with a sheet of living tissue instead of dead paper, there is at least a sporting chance that, if the sheet were to grow in just the right way, not uniformly but faster in some parts of the sheet than in others, this might automatically cause the sheet to assume a certain shape - and even fold or invaginate or turn inside out in a certain way - without the need for hands to do the stretching and folding, and without the need for any global plan, but only local rules. And actually it's more than just a sporting chance, because it really happens. Let's call it 'auto-origami'. How does auto-origami work in practice, in embryology? It works because what happens in the real embryo, when a sheet of tissue grows, is that cells divide. And differential growth of the different parts of the sheet of tissue is achieved by the cells, in each part of the sheet, dividing at a rate determined by local rules. So, by a roundabout route, we return to the fundamental importance of bottom-up local rules as opposed to top-down global rules. It is a whole series of (far more complicated) versions of this simple principle that actually go on in the early stages of embryonic development.
Here's how the origami goes in the early stages of vertebrate development. The single fertilized egg cell divides to make two cells. Then the two divide to make four. And so on, with the number of cells rapidly doubling and redoubling. At this stage there is no growth, no inflation. The original volume of the fertilized egg is literally divided, as in slicing a cake, and we end up with a spherical ball of cells which is the same size as the original egg. It's not a solid ball but a hollow one, and it is called the blastula. The next stage, gastrulation, is the subject of a famous bon mot by Lewis Wolpert: 'It is not birth, marriage, or death, but gastrulation, which is truly the most important time in your life.'
Gastrulation is a kind of microcosmic earthquake which sweeps over the blastula's surface and revolutionizes its entire form. The tissues of the embryo become massively reorganized. Gastrulation typically involves a denting of the hollow ball that is the blastula, so that it becomes two-layered with an opening to the outside world (see the computer simulation on p. 231). The outer layer of this 'gastrula' is called the ectoderm, the inner layer is the endoderm, and there are also some cells thrown into the space between the ectoderm and endoderm, which are called mesoderm. Each of these three primordial layers is destined to make major parts of the body. For example, the outer skin and nervous system come from the ectoderm; the guts and other internal organs come from the endoderm; and the mesoderm furnishes muscle and bone.
The next stage in the embryo's origami is called neurulation. The diagram on the right shows a cross-section through the middle of the back of a neurulating amphibian embryo (it could be either a frog or a salamander). The black circle is the 'notochord', a stiffening rod that acts as a precursor of the backbone. The notochord is diagnostic of the phylum Chordata, to which we and all vertebrates belong (although we, like most modern vertebrates, have it only when we are embryos). In neurulation, as in gastrulation, invagination is much in evidence. You remember I said that the nervous system comes from ectoderm. Well, here's how. A section of ectoderm invaginates (progressively backwards along the body like a zip fastener), rolls itself up into a tube, and is pinched off where the sides of the tube 'zip up' so that it ends up running the length of the body between the outer layer and the notochord. That tube is destined to become the spinal cord, the main nerve trunk of the body. The front end of it swells up and becomes the brain. And all the rest of the nerves are derived, by subsequent cell divisions, from this primordial tube.*
I don't want to get into the details of either gastrulation or neurulation, except to say that they are wonderful, and that the metaphor of origami holds up pretty well for both of them. I am concerned with the general principles by which embryos become more complicated through inflating origami. Below is one of the things that sheets of cells are observed to do during the course of embryonic development, for example during gastrulation. You can easily see how this invagination could be a useful move in inflating origami, and it does indeed play a major role in both gastrulation and neurulation.
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