First I thought I'd better check to make sure Thompson hadn't “cooked the books” by misrepresenting their shapes. I googled up some images of Polyprion (a.k.a Atlantic wreckfish) and Scorpaena; he appears to have done them justice, though perhaps exaggerated the differences a bit:

Of course, we already knew that you could morph shapes as disparate as Professor McGonagall and a cat into each other, so the "success" of this little experiment tells us nothing of interest scientifically. Furthermore, the program does not present us with an explicit grid describing the deformation, so we cannot even tell if the deformation looks like Thompson’s.

To address this I tried another approach, which was to morph the grid and carry the fish along for the ride; that also produced plausible looking results:

What this exercise did clarify for me is that a critical point of Thompson’s transformation images is that the deformations be in some sense simple, or regular. In mathematical terms this means it should be possible to describe them with few parameters (numbers). If I allow every point of interest in image A to move to an arbitrary new position in image B, I could describe the transformation with 2*n numbers (x,y displacements of each point), where n is the number of points (called fiducial points in the trade). For a fish, I could probably get by with n on the order of 10 or so fiducial points, so with 20 numbers I could get as cubist as I wanted. To the extent that the deformation is regular, I should be able to get by with many fewer numbers than that. I can describe a scaling transformation with just one number, a scale factor; for a two-dimensional rotation or shear I might need 4.

How many numbers do I need to describe Thompson's transformations? Closer to 1 or to 20? Thompson’s diagrams seem to be saying

*look how regular Nature’s transformations are*! But is it true? Certainly growth is fairly regular, unless you have teenage children. It is not a simple zoom (or dilation or scaling) transformation, because some aspects of the organism scale according to different powers of the overall size. For example, the weight of an organism is proportional to its volume, and so scales as the cube of its overall size, but the compression strength of bone is proportional to its cross-sectional area, and so scales as the square of the linear dimensions. This means that if you simply scale a cat to the size of a lion, its bones will be too thin for its weight, and they will fracture. To compensate, bones have to get thicker faster as the body grows. This general principle is called allometric scaling, and has been known for a long time.

Thompson’s transformations are not so much about growth, however, but about form; not the ontogenic transformations but the phylogenetic. Are the deformations between related species describable with few parameters, and if so, is that interesting? Maybe all this time these transformations have been some kind of conjurer’s trick, a clever iconic image with no real significance. How many parameters would I need to describe the warping of Polyprion into Scorpaena? The horizontal lines get squished towards the back; one number could probably describe the extent of the squishing. Thompson curves the verticals, which would burn extra parameters, since curves are mathematically more expensive to describe than lines. However, I didn't curve them, since curves are also more expensive to do in Free Morphing, which only gives you lines to work with; you would need to approximate a curve with a series of lines, which gets tedious rather quickly. However my lines seemed to work almost as well as Thompson's curves. So maybe you could turn Polyprion into Scorpaena with a few numbers. Some of Thompson's other transformations are more complex, and start to seem like special pleading. Given that he lets himself choose from a rather open-ended family of tranformations, you would also technically require one or two parameters to specify the type of transformation; it starts to look like an exercise in minimum length encoding or Kolmogorov complexity, i.e., the shortest way to describe something complicated.

If we accept the point that many morphological transformations between species are simple in the sense of being describable with a small number of parameters, is this biologically interesting? After all, Nature is not using a warping algorithm or specifying parameters. Or is She? The transcription factors mentioned in the last post, the regulators of genes, bind with a strength determined by the sequence of the DNA they are binding to. Change the sequence, change the binding strength, change how hard this regulator turns that gene on or off, and therefore change, or modulate, all the downstream effects of that gene. If that gene says grow the front, or the back, of the critter this much, then you change the shape. So maybe there's something here after all...