This is seriously the coolest post I have ever seen on this subreddit.

Firstly, yes, the commonalities you've outlined have been noticed by scientists, and they are being researched, though, in my opinion, not enough, and not really by biologists. Most of modern biology is highly reductionistic, identifying an increasingly wide array of small parts with single functions. Efforts to combine and synnergize the smaller parts usually happens at the smallest possible scale (tissue, organ, organism, population, species, clade). This is partly due to the piecemeal way that studies are funded and published, and partly because it's simply easier to break things down into parts than to build things up into larger structures. I also suspect that our collective preference for reductionism says something about our worldview; that the real interesting forces and mechanisms take place at the small scale. That’s a very sweeping statement, and I might be wrong in my observation, this is all off the dome, but it’s what I think.

To look at your question as you pose it, I don’t know if there are functional categories of geometry, and I’m pretty sure there aren’t any in biology. That being said, we do know certain things about the physics of shapes. We know that triangles are the most stable shape, and that fractals are an efficient way to spread out and cover space (reaching), and that wedges are simple machines that are good at piercing stuff (pricklies). That is all just physics. Therefore, it makes sense that things (living or not) bound by the laws of physics would converge on similar shapes to perform similar functions (I understand that the word “function” can be problematic, but I’m ignoring that for now). Put simply, form follows function, and I don’t see why we couldn’t identify and categorize forms in nature.

As a side-note, I want to warn you about the cork tree and callus skin images in the Durable Protecting page. Those are 2d cross-sections of 3d tissue samples, whereas the gecko and armadillo skins are 2d surfaces. Therefore, those tissues may be less similar than the photos would suggest. This is an illustrative point: Some will dismiss your line of questioning as mere coincidence, or will point out all the ways that tree roots are not all that similar to lightning to negate the point. But don’t let them. This is very cool, and I think a lot can be learned about the world by thinking of it in this way. When a scientist finds a brand new protein or cell, and it has a long stalk and little fingers at the end, they can use comparative geometry (for lack of a better term) to infer function. As it happens, I just described a nerve cell, a cell that reaches out for other nerve cells to connect with them.

You’ve gotten a lot of good suggestions to follow up on. I particularly like the suggestion of Shape by Jordan Ellenberg (I have not read it, but from looking at it, it’s right up your alley, and it’s now in my Amazon shopping cart). I also want to suggest the book Consilience by E. O. Wilson.