# Count Bacula, part 5: Indescribable geometries

It does go on, doesn’t it? How many parts could there be? I could add a few more, but this post might be the last for a bit. (also, posted a bit late today – apologies)

I wrote my dissertation in a maze of then-current scientific controversy – how do you make phylogenies, how do you use them, how do evolutionary processes relate to the results, what is homology, what is adaptation, how did the thrice-damned bats evolve anyway, and more. One area I had to get educated in was the then-exploding realm of morphometrics.

I used the images below in previous baculum posts but here I’m trying to show how seriously three-dimensional these little bones are. It might amuse you to know that each type I’ve represented below is more extreme in other species of megabats, so that a more dish-shaped version of the one on the left is the spitting image of a Romulan starship in the original Star Trek, and the more extended-knob version of the one on the right is ditto for a Klingon one.

My first task, and pretty much my Master’s degree, was to capture the shapes mathematically. This was twenty-five years ago, mind. No quick-and-simple scanning, no 3-D printing, hell, not even compatible IBM/Mac systems. We thought “PDF” was simply a detail of physical printing. Emailing a single Mb crashed whole computers.

3-D was out. You’d need a zillion dollars of grant to pay for the scanning/digitizing hardware, computer power, and programs for that. I was forced to work with the 2-D dorsal (top-down) view only.

Landmark-to-landmark measuring was out. The bones were so different that choosing such things would be imposing more order than I was discovering. I became knowledgeable of a thing called elliptical Fourier analysis, an algorithm you use on curvature that takes a shape rendered in X-Y coordinates, and goes point-by-point around it, creating summary measures (“harmonics”) of the different ways the curves go. Since I wanted mathematically independent trends to analyze, I then subjected the harmonics to another beast called principal-components analysis, generating things called eigenvectors – each one, itself, supposed to capture a distinctive shape feature. Bear in mind I had to write my own programs in BASIC to turn the output from the first analysis (which was a basement DIY by a morphometrician) readable by SAS, which back then had no canned analyses and you had to write programs for it in its own language.

As Leigh Brackett put it in The Reavers of Skaith,

The hellish thing about the ritual was that it worked.

Once I realized how well what I’d designed captured complex shapes, I gnashed my teeth that X-Y-Z coordinates weren’t feasible, and I still do. The elliptical Fourier analysis would have nailed the overall shapes to the wall.

All of that was laborious but ultimately, merely clever. The real intellectual task remained: to deliver a rigorous account of how bacula have evolved throughout the history of megabats. I’d been struck that no one had actually done this for any mammal, as researchers had been either hand-wavy about what the bone “did” or was “for,” or focused solely on whether it was “good” for generating phylogenies, or both.

And the problem appears in precisely the image to the right: that you can’t compare things mathematically as if they’re historically and causally independent, if they’re not. Yes, you lack a tail and a chimp lacks a tail, but that’s one evolutionary instance (shared with the other apes too), not two or more.

About a decade before my study, Joseph Felsenstein had suggested a method he called independent contrasts. Its math isn’t actually heinous, but it is necessarily highly precise and rigorous – it’s a matter of filtering out previously-occurring changes from the changes you’re comparing at any one point. Hard to believe today, but back then, most faculty were insufficiently skilled at systematics to understand this method let alone to promote it among their grad students, and luckily, my advisor was an exception. Also luckily, the program Comparative Analysis by Independent Contrasts (CAIC) was published by Andy Purvis and Andrew Rambaut just in time, in 1995 – in fact, I used a pre-publication version – and I daresay I was one of its most ambitious early adopters.

A lot of the people using the technique weren’t being very interesting, to my way of thinking. More size-based allometry … like the world needed more of that. It didn’t interest me when a linear relationship without the contrasts was validated by finding the same one with the contrasts (typically what happened with size). I was interested in the “hey wait a minute” outcomes:

• Traditional analysis = association, contrast analysis = scattered
• Traditional = scattered, contrast analysis = association

When this happens, it means a non-linear and non-intuitive outcome doesn’t mean “random” or “nothing,” but rather that some biology of note is occurring.

The first step was what all good scientists do: corpse-collecting; in my case, a more complete panoply of penises across megabat species, so I became quite the ghoul of museum collections, finding the organs, rendering them transparent, and staining the bones. Then came all the hallucinatory math, and finally, I applied the character mapping technique I described in Count Bacula, part 3: The case of the missing baculum to the contrast values to see what had happened to the shapes, for whom, and in what order.

From the myriad of results, I’ll pick just this one. The letter codes indicate positions in the phylogeny, suffice to say that the higher up the diagram, the more letters (as you can see here). The point is that the evolutionary changes represented by open circles are small in magnitude – and they always precede the changes represented by closed circles, which are much bigger.

In other words, there are small but important changes that occur in megabat bacula, without which no bigger changes occur. This is related to another point of mine, visible in that “shape by shape” graph above, that there are evidently “impossible” bacular shapes in these animals – theoretically and mathematically feasible, but historically, past a hard morphometric limit that is almost certainly a matter of developmental constraint.

The implications aren’t minor. Consider this: diversity of anatomical shapes display a shifting mosaic of options among closely-related taxa, not one-time unique events which battle it out in some kind of one-will-survive contest. Evolutionary change is all about who currently has what, and what cannot physically happen, than it is about a triumph or special status of what happens in one or another instance.