Today (12th February) is the one-year anniversary of the first PeerJ papers! As Matt put it in an email this morning:

Hard to believe it’s been a year already. On the other hand, it’s also hard to believe that it’s only been a year. PeerJ is just such an established part of my worldview now.

That’s exactly right. PeerJ has so completely rewritten the rule-book (on price, speed and quality of service) that now when I’m thinking about new papers I’m going to write, the question I ask myself is no longer “Where shall I send this?” but “Is there any reason not to send it to PeerJ?”

dorsals-ab-composite

Dorsals A and B (probably D8 and D9) of NHM R5937, “The Archbishop”, a still-undescribed brachiosaurid sauropod from the Upper Jurassic Tendaguru Formation of Tanzania, which I will get done this year, and which is destined for PeerJ. Top row: dorsal view with anterior to the right. Bottom row, from left to right: left lateral, posterior, right lateral, anterior.

Yesterday in the comments of a post on The Scholarly Kitchen, Harvey Kane asked me “I am curious as to where you get the notion that publishing OA is less expensive and in some way “better” than the traditional model?” My reply was (in part):

My notion that OA publishing yields better results than traditional is rooted in the online-only nature of articles, which allows them to ignore arbitrary limits on word-count, number of figures, use of colour, etc., and to exploit online-only formats such as video, 3d models, CT-slice stacks, etc. In my own field of vertebrate palaeontology, it’s now routine to see in PLOS ONE descriptive articles that are many times more comprehensive than their equivalents in traditional journals — see for example the recent description of the frog Beelzebufo.

Of course there is nothing specific to open-access about this: there is no technical reason why an online-only subscription journal shouldn’t publish similarly detailed articles. But my experience so far has been that they don’t — perhaps because they are tied to the mindset that pages and illustrations are limited resources.

For Beelzebufo in PLOS ONE, read baby Parasaurolophus in PeerJ, which we described as “the world’s most open-access dinosaur“. This paper is 83 pages of technicolour goodness, plus all the 3d models you can eat. And the crazy thing is, this sort of detail in descriptive papers is not even exceptional any more — see for example the recent description of Canardia in PLOS ONE, or this analysis of croc respiration in PeerJ

Years ago, I said that in the Archbishop descriptions I wanted to raise the bar for quality of illustration. Well, I’ve taken so long over getting the Archbishop done that the bar has been raised, and now I’m scrambling to catch up. Certainly the illustrations even in our 2011 description of Brontomerus are starting to look a bit old-fashioned.

And of course, the truly astonishing thing about PeerJ is that it does this so very cheaply. Because I’m already a member (which cost me $99), the Archbishop description is going to be free to me to publish this year. (This year for sure!) If we also get our Barosaurus neck preprint published properly this year,then I’ll have to find $100 to upgrade my Basic membership to Enhanced. That’s cheap enough that it’s not even worth going through the hassle of trying to get Bristol to pay for me. And if I ever hit a year when I publish three or more papers, I’ll upgrade once more (for another $100) to the Investigator plan and then that’s it: I’m done paying PeerJ forever, however many papers I publish there. (Matt jumped straight to the all-you-can-eat plan, so he wouldn’t even have to think about it ever again.)

Barosaurus lentus holotype YPM 429, Vertebra Q (C?13). Top row: left ventrolateral view. Middle row, from left to right: anterior view, with ventral to the right; ventral view; posterior view, with ventral to the left. Bottom row: right lateral view, inverted. Inset shows diapophyseal facet on right side of vertebra, indicating that the cervical ribs were unfused in this individual despite its great size. Note the broad, flat prezygapophyseal facet visible in anterior view. (Taylor and Wedel 2013b: figure 6)


Barosaurus lentus holotype YPM 429, Vertebra Q (C?13). Top row: left ventrolateral view. Middle row, from left to right: anterior view, with ventral to the right; ventral view; posterior view, with ventral to the left. Bottom row: right lateral view, inverted. Inset shows diapophyseal facet on right side of vertebra, indicating that the cervical ribs were unfused in this individual despite its great size. Note the broad, flat prezygapophyseal facet visible in anterior view. (Taylor and Wedel 2013b: figure 6)

PeerJ’s pricing is making PLOS ONE’s $1350 APC look distinctly old-fashioned; and the $3000 charged by the legacy publishers (for a distinctly inferior product) is now frankly embarrassing. You might expect that as such low prices, PeerJ’s quality of service would suffer, but that’s not been our experience: editing, reviewing, typesetting and proofing for our neck-anatomy paper were all up there with the best we’ve received anywhere.

And it’s great to see that it’s not just minor researchers like Matt and me who are persuaded by PeerJ: they’ve now accumulated a frankly stellar list of 20 universities (so far) with institutional plans for researchers to publish there. When I say “stellar” I mean that the list includes Harvard, MIT, Cambridge, Berkeley, Stanford, Johns Hopkins, UCL, Carnegie Mellon, Duke … the list goes on.

We can only hope that the next year, and the next ten and twenty, are as successful for PeerJ as the first has been; and that other New Generation publishers will join it in pushing the field forward.

I leave the last word to Matt:

I’m getting Vicki a lifetime membership for Valentine’s Day. Because I’m a romantic.

She’s a lucky, lucky woman.

Giraffatitan skull photos

February 10, 2014

Giraffatitan skull left lateral

Let it never be said that we don’t take good care of our commenters. Heck, we’ll even degrade ourselves by blogging about theropods, if that’s what it takes to keep you all happy.

Giraffatitan skull left anterolateral

Derp dah durr

Today’s post is a response to this comment by Dean, asking for lateral view photos of the skull of Giraffatitan. Mike and I did get to spend some quality time with the T1 skull (a.k.a. “Old Toilet-Face”) when we were in Berlin in 2008.

Giraffatitan skull anterior

Unfortunately, most of our photos turned out not-so-hot. The room around the skull was not large, so we couldn’t get back very far from it. Hence our photos are plagued by perspective distortions.

Giraffatitan skull right anterolateral

Ah hurr hurr hurr

Also, we didn’t have a tripod along and the light level was fairly low, and the combination of handheld shots and long exposure times meant that most of the shots are at least a little blurry.

Giraffatitan skull right lateral

BUT. It was still a thrill to see that skull up close.

The crazy thing about Giraffatitan is that the skull looks like it’s going to be pretty sweet when you see it from the side. Because you’re thinking it’s going to be kinda narrow, like a giraffe’s head. Then you get even a partial front view and suddenly the animal’s whole skull looks like a partially-deflated whoopie cushion (whereas in life it looked like a mostly-inflated whoopie cushion). And then you have to live with the knowledge that one of the most majestic animals that ever lived on Earth was afflicted with derpty-face. I’ll bet they went extinct from shame.

Giraffatitan skull dorsal oblique

Still, there is some cool anatomy to see here, especially the snout-troughs leading down from the external nares, and the neurovascular foramina on the maxillae.

And, crucially, brachiosaurs had the good taste to hide their freakish countenances 45 feet up, where they could be safely ignored by everyone other than pterosaurs and birds. This has not escaped the notice of exhibit designers:

Giraffatitan skeleton hmmm

Go here for the unmarked original.

Illustration talk slide 9

Illustration talk slide 10

Illustration talk slide 11

Illustration talk slide 12

That link in the first slide will take you here, and the rest of the series is here.

Following on from Matt’s post about the difficulty of photographing big specimens without distortion, I thought I’d have a play with our best Sauroposeidon C8 photo, which I think is this one:

sauroposeidon-c8-alone

(That’s been the basis for classic SV-POW! posts such as Your neck is pathetic and Darren’s new indeterminate Wealden maniraptoran is inadequate.)

I was motivated by Andy Farke’s comment:

Another–and perhaps more important–area where surface models excel is when you can remove colors on the original specimen that wash out relevant details…I bet this is probably the case for the example vertebra of Sauroposeidon. How many fossae and foramina just don’t show up well on the photos above?

Andy was talking about completely colourless 3d surface models, in which the 3d shape allows a render to make shadows that bring out the subtle shapes. But it made me wonder whether we could get anywhere just by washing out the most prevalent colour in the photo.

I started by doing a big, fat Gaussian blur on a duplicate layer — 500 pixels in each direction — and sampling the colour in the middle, to get a rough-and-ready average. (There may be a better way — please shout if you know one.) That average colour was#7e6b2f. I used it to run Colour To Alpha on another duplicate of the original layer, so that we’d be left with only residual colours. Here’s the result:

sauroposeidon-c8-alone-colour-completely-removed

I’m in two minds about this. It may be informative, but it sure is ugly. To compromise, I reinstated the original layer underneath this mostly-transparent one, and turned its opacity down to 75%. Here’s the result — a nice compromise:

sauroposeidon-c8-alone-colour-removed

Of course, there are endless other approaches you can take — that’s the blessing and the curse of image-editing programs like GIMP. For example, here’s what I got doing a simple Colours → Auto → White Balance:

sauroposeidon-c8-alone-whitebalanced

I’m not sure that isn’t the best of the bunch, in terms of informativeness.

I also tried something else — not amazingly successfully, but I think it’s worth seeing. Since the two photos that Matt showed in the previous post were evidently taken from somewhat different angles, I thought I’d have a go at compositing them into a red-cyan anaglyph. Because the variation in camera position is mostly dorsoventral rather than anteroposterior, the vert has to be pointed upwards for the two eyes to see the two versions from different horizontal points. Here’s the best I could do:

c8-anaglyph

I would say this is of some value; but it’s nowhere near as good as, for example, the anaglyph of Cervical S of the Archbishop. I could sit and look at that one all day. The problems with this one arise for three reasons.

First, I had to reduce both parts of the Sauroposeidon anaglyph to monochrome (since one was already in that form), so all colour information was lost.

Second, I had to scale the high-resolution picture to the same size as the lower-resolution one, throwing away more detail.

Finally, and most important, the two photos were not taken with the intention that they should be used to make an anaglyph. To work well, this has to be done with the images taken under the same lighting conditions, at the same distance from the specimen, from perspectives differing by about the distance between the pupils of the viewer, and with the camera-position difference being perfectly in the plane of the specimen. Needless to say, none of these conditions was met in this case, so it’s actually quite impressive that it works as well as it does.

We have a lot of options for illustrating specimens these days. Postage-stamp-sized greyscale photos really don’t cut it any more.

Here are two photos of what I infer to be C8 of OMNH 53062, the holotype of Sauroposeidon. The top one was taken by Mike during our visit to the OMNH in 2007. If you’re a regular you may recognize it from several older posts: 1, 2, 3. The bottom one was taken by Mike Callaghan, the former museum photographer at the OMNH, sometime in 1999 or 2000. I used it in Wedel et al. (2000) and Wedel and Cifelli (2005).

Sauroposeidon OMNH 53062 C8 photos compared

You’ll notice that the two photos are far from identical. In both cases, the photographers were up on ladders, as far above the vertebra as they could get, and there are still significant perspective effects. That’s just a fact of life when you’re taking photos of a vertebra that is 1.4 meters long, from anything lower than a helicopter. In Mike Taylor’s shot, the neural spine looms a little too large; in Mike Callaghan’s shot, the prezygapophysis looks a little too small, probably because it was curving off at the edge of the shot. So neither photograph is “right”; both distort the morphology of the specimen in different ways. Here’s how the two images stack up, with the outlines scaled to the same length:

Sauroposeidon OMNH 53062 C8 outlines compared

When I ran a draft of this post past Mike, he wrote (with permission to post):

I think the current draft misses an important point: the warning. We really can’t trust photos, however carefully taken, and however beautifully composited into TNFs*. You’re welcome to quote me as having said I’d have assumed the two C8s were different vertebrae. For that matter, I bet I could have worked up several taxonomically significant characters to distinguish them. Yikes.

* TNF = Taylor Normal Form, i.e., making multi-view photos like the ones here and here.

So the moral is, photos of big specimens almost always involve some distortion. This is clearly not ideal. But I have a plan for fixing it. I am hoping to get back to the OMNH this spring, and the next time I’m there, I’m going to take photos of this vertebra from a zillion angles and make a 3D model through photogrammetry. Happily, Heinrich Mallison has been producing a very helpful series of tutorials on that very topic over at dinosaurpaleo: 1, 2, 3, 4, with more on the way (I’ll update the links here later). Update: Don’t forget to check out Peter Falkingham’s (2012) paper in PE on making photogrammetric models with free software.

Armed with that model, it should be possible to produce a perspective-free lateral view image of the vertebra, to which all of the previous photos can be compared. I can’t use CT data because this vertebra has never been CTed; it’s too big to fit through a medical CT scanner, and probably too fragile to be packed up and shipped to an industrial CT machine like they used on Sue (not to mention that would require a significant chunk of money, which is probably not worth spending on a problem that can be solved in other ways).

So, photogrammetry to the rescue, or am I just deluding myself? Let me know what you think in the comments.

Finally, I should mention that the idea of superseding photographs with 3D photogrammetric models is not original. I got religion last week while I was having beers with Martin Sander and he was showing me some of the models he’s made. He said that going forward, he was going to forbid his students to illustrate their specimens only with photographs; as far as he was concerned, now that 3D models could be cheaply and easily produced by just about everyone, they should be the new standard. Inspiring stuff–now I must go do likewise.

Some previous posts on Sauroposeidon:

References

“Look at all the things you’ve done for me
Opened up my eyes,
Taught me how to see,
Notice every tree.”

So sings Dot in Move On, the climactic number of Stephen Sondheim’s Pulitzer Prize-winning music Sunday in the Park with George, which on the surface is about the post-impressionist painter Georges Seurat, but turns out to be a study of obsession and creativity.

xx

Un dimanche après-midi à l’Île de la Grande Jatte – 1884 [A Sunday Afternoon on the Island of La Grande Jatte – 1884]

“Taught me how to see”? What kind of talk is that? One the surface, it seems silly — we all know how to see. We do it constantly, without thinking. Yet it’s something that artists talk about all the time. And anyone who’s sat down and seriously tried to paint or draw something will have some understanding of what the phrase means. We have such strong implicit ideas of what things look like that we tend to reproduce what we “know” is there rather than what’s actually there. Like I said, we see without thinking.

In fact, the psychology of perception is complicated and sophisticated, and the brain does an extraordinary amount of filtering of the visual signals we get, to save us the bother of having to consciously process way too much data. This is a whole scientific field of its own, and I’m going to avoid saying very much about it for fear of making a fool of myself — as scientists so often do when wandering outside their own field. But I think it’s fair to say that we all have a tendency to see what we expect to see.

xx

Phylogeny of Sauropoda, strict consensus of most parsimonious trees according to Wilson (2002:fig. 13a)

In the case of sauropods, this tendency has meant that we’ve all been startlingly bad at seeing pneumaticity in the caudal vertebrae of sauropods. Because the literature has trained us to assume it’s not there. For example, in the two competing sauropod phylogenies that dominated the 2000s, both Wilson (2002) and Upchurch et al. (2004) scored caudal pneumaticity as very rare: Wilson’s character 119, “Anterior caudal centra, pneumatopores (pleurocoels)”, was scored 1 only for Diplodocus and Barosaurus; and  Upchurch et al. (2004:286) wrote that “A few taxa (Barosaurus, Diplodocus, and Neuquensaurus) have pleurocoel-like openings in the lateral surfaces of the cranial [caudal] centra that lead into complex internal chambers”. That’s all.

And that’s part of the reason that every year since World War II, a million people have walked right past the awesome mounted brachiosaur in the Museum Für Naturkunde Berlin without noticing that it has pneumatic caudals. After all, we all knew that brachiosaur caudals were apneumatic.

But in my 2005 Progressive Palaeontology talk about upper limits on the mass of land animals estimated through the articular area of limb-bone cartilage, I included this slide that shows how much bigger the acetabulum of Giraffatitan is than the femoral head that it houses:

Screenshot from 2014-01-24 17:30:30

And looking at that picture made me wonder: those dark areas on the sides of the first few caudals (other than the first, which is a very obvious plaster model) certainly look pneumatic.

Then a few years later, I was invited to give a talk at the Museum Für Naturkunde Berlin itself, on the subject “Brachiosaurus brancai is not Brachiosaurus“. (This of course was drawn from the work that became my subsequent paper on that subject, Taylor 2009) And as I was going through my photos to prepare the slides of that talk, I thought to myself: darn it, yes, it does have pneumatic caudals!

So I threw this slide into the talk, just in passing:

Screenshot from 2014-01-24 17:32:06

Those photos were pretty persuasive; and a closer examination of the specimen on that same trip was to prove conclusive.

Meanwhile …

Earlier in 2009, I’d been in Providence, Rhode Island, with my Index Data colleagues. I’d managed to carve a day out of the schedule to hope along the coast to the Yale Peabody Museum in New Haven, Connecticut. My main goal was to examine the cervicals of the mounted Apatosaurus (= “Brontosaurus“) excelsus holotype (although it was also on that same trip that I first saw the Barosaurus holotype material that we’ve subsequently published a preprint on).

The Brontosaurus cervicals turned out to be useless, being completely encased in plaster “improvements” so that you can’t tell what’s real and what’s not. hopefully one day they’ll get the funding they want to take that baby down off its scaffold and re-prep the material.

But since I had the privilege of spending quality time with such an iconic specimen, it would have been churlish not to look at the rest of it. And lo and behold, what did I see when I looked at the tail but more pneumaticity that we thought we knew wasn’t there!

Wedel and Taylor (2013b: Figure 10).

An isolated pneumatic fossa is present on the right side of caudal vertebra 13 in Apatosaurus excelsus holotype YPM 1980. The front of the vertebra and the fossa are reconstructed, but enough of the original fossil is visible to show that the feature is genuine. (Wedel and Taylor 2013b: Figure 10).

What does this mean? Do other Giraffatitan and Apatosaurus specimens have pneumatic tails? How pervasive is the pneumaticity? What are the palaeobiological implications?

Stay tuned! All will be revealed in Matt’s next post (or, if you can’t wait, in our recent PLOS ONE paper, Wedel and Taylor 2013b)!

References

[This is part 4 in an ongoing series on our recent PLOS ONE paper on sauropod neck cartilage. See also part 1, part 2, and part 3.]

Big Bend Vanessa 182 small

Weird stuff on the ground, Big Bend, 2007.

Here’s a frequently-reproduced quote from Darwin:

About thirty years ago there was much talk that geologists ought only to observe and not theorise; and I well remember some one saying that at this rate a man might as well go into a gravel-pit and count the pebbles and describe the colours. How odd it is that anyone should not see that all observation must be for or against some view if it is to be of any service!

It’s from a letter to Henry Fawcett, dated September 18, 1861, and you can read the whole thing here.

I’ve known this quote for ages, having been introduced to it at Berkeley–a copy used to be taped to the door of the Padian Lab, and may still be. It’s come back to haunt me recently, though. An even stronger version would run something like, “If you don’t know what you’re looking for, you won’t make the observation in the first place!”

OLYMPUS DIGITAL CAMERA

Kent Sanders looking at scans of BYU 12613, a posterior cervical of either Kaatedocus or an anomalously small Diplodocus, at the University of Utah in May, 2008.

For example: I started CT scanning sauropod vertebrae with Rich Cifelli and Kent Sanders back in January, 1998. Back then, I was interested in pneumaticity, so that’s what I looked for, and that’s what I found–work which culminated in Wedel et al. (2000) and Wedel (2003). It wasn’t until earlier this year that I wondered if it would be possible to determine the spacing of articulated vertebrae from CT scans. So everything I’m going to show you, I technically saw 15 years ago, but only in the sense of “it crossed my visual field.” None of it registered at the time, because I wasn’t looking for it.

A corollary I can’t help noting in passing: one of the under-appreciated benefits of expanding your knowledge base is that it allows you to actually make more observations. Many aspects of nature only appear noteworthy once you have a framework in which to see them.

OLYMPUS DIGITAL CAMERA

BYI 12613 going through a CT scanner at the University of Utah medical center. We were filming for the “Megasaurus” episode of Jurassic CSI. That shoot was crazy fun.

So anyway, the very first specimen we scanned way back when was the most anterior of the three plaster jackets that contain the four cervical vertebrae that make up OMNH 53062, which was destined to become the holotype of Sauroposeidon. I’ve written about the taphonomy of that specimen here, and you can read more about how it was excavated in Wedel and Cifelli (2005). We scanned that jacket first because, although the partial vertebrae it contains are by far the most incomplete of the four, the jacket is a lot smaller and lighter than the other two (which weigh hundreds of pounds apiece). Right away we saw internal chambers in the vertebrae, and that led to all of the pneumaticity work mentioned above.

Sauroposeidon C5 cross section Wedel 2007b fig 14

Internal structure of a cervical vertebra of Sauroposeidon, OMNH 53062. A, parts of two vertebrae from the middle of the neck. The field crew that dug up the bones cut though one of them to divide the specimen into manageable pieces. B, cross section of C6 in posterior view at the level of the break, traced from a CT image and photographs of the broken end. The left side of the specimen was facing up in the field and the bone on that side is badly weathered. Over most of the broken surface the internal structure is covered by plaster or too damaged to trace, but it is cleanly exposed on the upper right side (outlined). C, the internal structure of that part of the vertebra, traced from a photograph. The arrows indicate the thickness of the bone at several points, as measured with a pair of digital calipers. The camellae are filled with sandstone. Wedel (2007: fig. 14).

Happily for me, that first jacket contains not only the posterior two-thirds of the first vertebra (possibly C5), but also the front end of the second vertebra. Whoever decided to plow through the second vertebra to divide the specimen into manageable chunks in the field made a savvy choice. Way back in 2004 I realized that the cut edge of the second vertebra was not obscured by plaster, and therefore the internal structure could be seen and measured directly, which is a lot cleaner than relying on the artifact-heavy CT scans. (The CT scans are noisy because the hospital machines we had access to start to pant a bit when asked to punch x-rays through specimens this large and dense.) A figure derived from that work made it into a couple of papers and this post, and appears again above.

But that’s pneumaticity, which this post is allegedly not about. The cut through the second vertebra was also smart because it left the intervertebral joint intact.

Figure 11. Fifth and partial sixth cervical vertebrae of Sauroposeidon. Photograph and x-ray scout image of C5 and the anterior portion of C6 of Sauroposeidon OMNH 53062 in right lateral view. The anterior third of C5 eroded away before the vertebra was collected. C6 was deliberately cut through in the field to break the multi-meter specimen into manageable pieces for jacketing (see [37] for details). Note that the silhouettes of the cotyle of C5 and the condyle of C6 are visible in the x-ray.

Fifth and partial sixth cervical vertebrae of Sauroposeidon.
Photograph and x-ray scout image of C5 and the anterior portion of C6 of Sauroposeidon OMNH 53062 in right lateral view. The anterior third of C5 eroded away before the vertebra was collected. C6 was deliberately cut through in the field to break the multi-meter specimen into manageable pieces for jacketing (see Wedel and Cifelli 2005 for details). Note that the silhouettes of the cotyle of C5 and the condyle of C6 are visible in the x-ray. Taylor and Wedel (2013: figure 11).

Here are a photo of the jacket and a lateral scout x-ray. The weird rectangles toward the left and right ends of the x-ray are boards built into the bottom of the jacket to strengthen it.

Figure 12. CT slices from fifth cervical vertebrae of Sauroposeidon. X-ray scout image and three posterior-view CT slices through the C5/C6 intervertebral joint in Sauroposeidon OMNH 53062. In the bottom half of figure, structures from C6 are traced in red and those from C5 are traced in blue. Note that the condyle of C6 is centered in the cotyle of C5 and that the right zygapophyses are in articulation.

CT slices from fifth cervical vertebrae of Sauroposeidon.
X-ray scout image and three posterior-view CT slices through the C5/C6 intervertebral joint in Sauroposeidon OMNH 53062. In the bottom half of figure, structures from C6 are traced in red and those from C5 are traced in blue. Note that the condyle of C6 is centered in the cotyle of C5 and that the right zygapophyses are in articulation. Taylor and Wedel (2013: figure 12).

And here’s a closeup of the C5/C6 joint, with the relevant radiographs and tracing. The exciting thing here is that the condyle is centered almost perfectly in the cotyle, and the zygapophyses are in articulation. Together with the lack of disarticulation in the cervical rib bundle (read more about that here and in Wedel et al. 2000), these things suggest to us that the vertebrae are spaced pretty much as they were in life. If so, then the spacing between the vertebrae now tells us the thickness of the soft tissue that separated the vertebrae in life.

I should point out here that we can’t prove that the spacing between the vertebrae is still the same as it was in life. But if some mysterious force moved them closer together or farther apart, it did so (1) without  decentering the condyle of C6 within the cotyle of C5, (2) without moving the one surviving zygapophyseal joint out of contact, and (3) without disarticulating the cervical ribs. The cervical ribs were each over 3 meters long in life and they formed vertically-stacked bundles on either side below the vertebrae; that’s a lot of stuff to move just through any hypothetical contraction or expansion of the intervertebral soft tissues after death. In fact, I would not be surprised if the intervertebral soft tissues did contract or expand after death–but I don’t think they moved the vertebrae, which are comparatively immense. The cartilage probably pulled away from the bone as it rotted, allowing sediment in. Certainly every nook and cranny of the specimen is packed with fine-grained sandstone now.

Anyway, barring actual preserved cartilage, this is a best-case scenario for trying to infer intervertebral spacing in a fossil. If articulation of the centra, zygs, and cervical ribs doesn’t indicate legitimate geometry, nothing ever will. So if we’re going to use the fossils to help settle this at all, we’re never going to have a better place to start.

Figure 14. Geometry of opisthocoelous intervertebral joints. Hypothetical models of the geometry of an opisthocoelous intervertebral joint compared with the actual morphology of the C5/C6 joint in Sauroposeidon OMNH 53062. A. Model in which the condyle and cotyle are concentric and the radial thickness of the intervertebral cartilage is constant. B. Model in which the condyle and cotyle have the same geometry, but the condyle is displaced posteriorly so the anteroposterior thickness of the intervertebral cartilage is constant. C. the C5/C6 joint in Sauroposeidon in right lateral view, traced from the x-ray scout image (see Figure 12); dorsal is to the left. Except for one area in the ventral half of the cotyle, the anteroposterior separation between the C5 cotyle and C6 condyle is remarkably uniform. All of the arrows in part C are 52 mm long.

Geometry of opisthocoelous intervertebral joints.
Hypothetical models of the geometry of an opisthocoelous intervertebral joint compared with the actual morphology of the C5/C6 joint in Sauroposeidon OMNH 53062. A. Model in which the condyle and cotyle are concentric and the radial thickness of the intervertebral cartilage is constant. B. Model in which the condyle and cotyle have the same geometry, but the condyle is displaced posteriorly so the anteroposterior thickness of the intervertebral cartilage is constant. C. the C5/C6 joint in Sauroposeidon in right lateral view, traced from the x-ray scout image (see Figure 12); dorsal is to the left. Except for one area in the ventral half of the cotyle, the anteroposterior separation between the C5 cotyle and C6 condyle is remarkably uniform. All of the arrows in part C are 52 mm long. Taylor and Wedel (2013: figure 14).

So, by now, you know I’m a doofus. I have been thinking about this problem literally for years and the data I needed to address it was sitting on my hard drive the entire time. One of the things I pondered during those lost years is what the best shape for a concave-to-convex intervertebral joint might be. Would the best spacing be radially constant (A in the figure above), or antero-posteriorly constant (B), or some other, more complicated arrangement? The answer in this case surprised me–although the condyle is a lot smaller in diameter than the cotyle, the anteroposterior separation between them in almost constant, as you can see in part C of the above figure.

Figure 13. Joint between sixth and seventh cervicals vertebrae of Sauroposeidon. X-ray scout image of the C6/C7 intervertebral joint in Sauroposeidon OMNH 53062, in right lateral view. The silhouette of the condyle is traced in blue and the cotyle in red. The scale on the right is marked off in centimeters, although the numbers next to each mark are in millimeters.

Joint between sixth and seventh cervicals vertebrae of Sauroposeidon.
X-ray scout image of the C6/C7 intervertebral joint in Sauroposeidon OMNH 53062, in right lateral view. The silhouette of the condyle is traced in blue and the cotyle in red. The scale on the right is marked off in centimeters, although the numbers next to each mark are in millimeters. Taylor and Wedel (2013: figure 13).

Don’t get too worked up about that, though, because the next joint is very different! Here’s the C6/C7 joint, again in a lateral scout x-ray, with the ends of the bones highlighted. Here the condyle is almost as big in diameter as the cotyle, but it is weirdly flat. This isn’t a result of overzealous prep–most of the condyle is still covered in matrix, and I only found its actual extent by looking at the x-ray. This is flatter than most anterior dorsal vertebrae of Apatosaurus–I’ve never seen a sauropod cervical with such a flat condyle. Has anyone else?

The condyle of C6 is a bit flatter than expected, too–certainly a lot flatter than the cervical condyles in Giraffatitan and the BYU Brachiosaurus vertebrae. As we said in the paper,

It is tempting to speculate that the flattened condyles and nearly constant thickness of the intervertebral cartilage are adaptations to bearing weight, which must have been an important consideration in a cervical series more than 11 meters long, no matter how lightly built.

Anyway, obviously here the anteroposterior distance between condyle and cotyle could not have been uniform because they are such different shapes. Wacky. The zygs are missing, so they’re no help, and clearly the condyle is not centered in the cotyle. Whether this posture was attainable in life is debatable; I’ve seen some pretty weird stuff. In any case, we didn’t use this joint for estimating cartilage thickness because we had no reason to trust the results.

Figure 15. First and second dorsal vertebrae of Apatosaurus CM 3390. Articulated first and second dorsal vertebrae of Apatosaurus CM 3390. A. Digital model showing the two vertebrae in articulation, in left lateral (top) and ventral (bottom) views. B-G. Representative slices illustrating the cross-sectional anatomy of the specimen, all in posterior view. B. Slice 25. C. Slice 31. D. Slice 33. E. Slice 37. F. Slice 46. G. Slice 61. Orthogonal gaps are highlighted where the margins of the condyle and cotyle are parallel to each other and at right angles to the plane of the CT slice. 'Zygs' is short for 'zygapophyses', and NCS denotes the neurocentral synchondroses.

First and second dorsal vertebrae of Apatosaurus CM 3390.
Articulated first and second dorsal vertebrae of Apatosaurus CM 3390. A. Digital model showing the two vertebrae in articulation, in left lateral (top) and ventral (bottom) views. B-G. Representative slices illustrating the cross-sectional anatomy of the specimen, all in posterior view. B. Slice 25. C. Slice 31. D. Slice 33. E. Slice 37. F. Slice 46. G. Slice 61. Orthogonal gaps are highlighted where the margins of the condyle and cotyle are parallel to each other and at right angles to the plane of the CT slice. ‘Zygs’ is short for ‘zygapophyses’, and NCS denotes the neurocentral synchondroses. Taylor and Wedel (2013: figure 15).

Kent Sanders and I had also scanned several of the smaller sauropod vertebrae from the Carnegie collection (basically, the ones that would fit in the trunk of my car for the drive back to Oklahoma). Crucially, we’d scanned a couple of sets of articulated vertebrae, CM 3390 and CM 11339, both from juvenile individuals of Apatosaurus. In both cases, the condyles and cotyles are concentric (that’s what the ‘orthogonal gaps’ are all about in the above figure) and the zygs are in articulation, just as in Sauroposeidon. These are dorsals, so we don’t have any cervical ribs here to provide a third line of evidence that the articulation is legit, but all of the evidence that we do have is at least consistent with that interpretation.

So, here’s an interesting thing: in CM 3390, above, the first dorsal is cranked up pretty sharply compared to the next one, but the condyle is still centered in the cotyle and the zygs are in articulation. Now, the vertebrae have obviously been sheared by taphonomic deformation, but that seems to have affected both vertebrae to the same extent, and it’s hard to imagine some kind of taphonomic pressure moving one vertebra around relative to the next. So I think it’s at least plausible that this range of motion was achievable in life. Using various views and landmarks, we estimate the degree of extension here somewhere between 31 and 36 degrees. That’s a lot more than the ~6 degrees estimated by Stevens and Parrish (1999, 2005). And, as we mentioned in the paper, it nicely reinforces the point made by Upchurch (2000), that flexibility in the anterior dorsals should be taken into account in estimating neck posture and ROM.

Figure 16. Dorsal vertebrae of Apatosaurus CM 11339. Articulated middle or posterior dorsal vertebrae of Apatosaurus CM 11339. A. X-ray scout image showing the two vertebrae in articulation, in left lateral view. B–D. Slices 39, 43 and and 70 in posterior view, showing the most anterior appearance of the condyles and cotyles.

Dorsal vertebrae of Apatosaurus CM 11339.
Articulated middle or posterior dorsal vertebrae of Apatosaurus CM 11339. A. X-ray scout image showing the two vertebrae in articulation, in left lateral view. B–D. Slices 39, 43 and and 70 in posterior view, showing the most anterior appearance of the condyles and cotyles. Taylor and Wedel (2013: figure 16).

Here’s our last specimen, CM 11339. No big surprises here, although if you ever had a hard time visualizing how hyposphenes and hypantra fit together, you can see them in articulation in parts C and D (near the top of the specimen). Once again, by paging through slices we were able to estimate the separation between the vertebrae. Incidentally, the condyle IS centered in the cotyle here, it just doesn’t look that way because the CT slice is at an angle to the joint–see the lateral scout in part A of the figure to see what I mean.

So, what did we find? In Sauroposeidon the spacing between C5 and C6 is 52mm. That’s pretty darn thick in absolute terms–a shade over two inches–but really thin in relative terms–only a little over 4% of the length of each vertebra. In both of the juvenile Apatosaurus specimens, the spacing between the vertebrae was about 14mm (give or take a few because of the inherent thickness of the slices; see the paper for details on these uncertainties).

Now, here’s an interesting thing: we can try to estimate the intervertebral spacing in an adult Apatosaurus in two ways–by scaling up from the juvenile apatosaurus, or by scaling sideways from Sauroposeidon (since a big Apatosaurus was in the same ballpark, size-wise)–and we get similar answers either way.

Scaling sideways from Sauroposeidon (I’m too lazy to write anymore so I’m just copying and pasting from  the paper):

Centrum shape is conventionally quantified by Elongation Index (EI), which is defined as the total centrum length divided by the dorsoventral height of the posterior articular surface. Sauroposeidon has proportionally very long vertebrae: the EI of C6 is 6.1. If instead it were 3, as in the mid-cervicals of Apatosaurus, the centrum length would be 600 mm. That 600 mm minus 67 mm for the cotyle would give a functional length of 533 mm, not 1153, and 52 mm of cartilage would account for 9.8% of the length of that segment.

Scaling up from the juveniles: juvenile sauropods have proportionally short cervicals (Wedel et al. 2000). The scanned vertebrae are anterior dorsals with an EI of about 1.5. Mid-cervical vertebrae of this specimen would have EIs about 2, so the same thickness of cartilage would give 12mm of cartilage and 80mm of bone per segment, or 15% cartilage per segment. Over ontogeny the mid-cervicals telescoped to achieve EIs of 2.3–3.3. Assuming the cartilage did not also telescope in length (i.e., didn’t get any thicker than it got taller or wider), the ratio of cartilage to bone would be 12:120 (120 from 80*1.5), so the cartilage would account for 10% of the length of the segment–almost exactly what we got from the based-on-Sauroposeidon estimate. So either we got lucky here with our tiny sample size and truckloads of assumptions, or–just maybe–we discovered a Thing. At least we can say that the intervertebral spacing in the Apatosaurus and Sauroposeidon vertebrae is about the same, once the effects of scaling and EI are removed.

Finally, we’re aware that our sample size here is tiny and heavily skewed toward juveniles. That’s because we were just collecting targets of opportunity. Finding sauropod vertebrae that will fit through a medical-grade CT scanner is not easy, and it’s just pure dumb luck that Kent Sanders and I had gotten scans of even this many articulated vertebrae way back when, since at the time we were on the hunt for pneumaticity, not intervertebral joints or their soft tissues. As Mike has said before, we don’t think of this paper as the last word on anything. It is, explicitly, exploratory. Hopefully in a few years we’ll be buried in new data on in-vivo intervertebral spacing in both extant and extinct animals. If and when that avalanche comes, we’ll just be happy to have tossed a snowball.

References

I just found out — thanks to a tweet from abertonykus — that this exists:

sauropod_vertebra_picture_adventure__by_classicalguy-d6ssfil

That’s me on top of the Giraffatitan, Matt to the right, and Darren swinging from its wattle.

It’s the work of classicalguy on Deviant Art. He provides a poem and some brief commentary along with the original. There also one for the Tetrapod Zoology podcats, and one for Tom Holtz.

Last time, we looked at how including intervertebral cartilage changes the neutral pose of a neck – or, more specifically, of the sequence of cervical vertebrae. The key finding (which is inexplicably missing from the actual paper, Taylor and Wedel 2013c) is that adding cartilage of thickness x between vertebrae whose zygapophyses are height y above the mid-height of the centra elevates the joint’s neutral posture by x/y radians.

Figure 14. Geometry of opisthocoelous intervertebral joints. Hypothetical models of the geometry of an opisthocoelous intervertebral joint compared with the actual morphology of the C5/C6 joint in Sauroposeidon OMNH 53062. A. Model in which the condyle and cotyle are concentric and the radial thickness of the intervertebral cartilage is constant. B. Model in which the condyle and cotyle have the same geometry, but the condyle is displaced posteriorly so the anteropos- terior thickness of the intervertebral cartilage is constant. C. the C5/C6 joint in Sauroposeidon in right lateral view, traced from the x-ray scout image (see Figure 12); dorsal is to the left. Except for one area in the ventral half of the cotyle, the anteroposterior separation between the C5 cotyle and C6 condyle is remarkably uniform. All of the arrows in part C are 52 mm long.

Figure 14. Geometry of opisthocoelous intervertebral joints. Hypothetical models of the geometry of an opisthocoelous intervertebral joint compared with the actual morphology of the C5/C6 joint in Sauroposeidon OMNH 53062. A. Model in which the condyle and cotyle are concentric and the radial thickness of the intervertebral cartilage is constant. B. Model in which the condyle and cotyle have the same geometry, but the condyle is displaced posteriorly so the anteroposterior thickness of the intervertebral cartilage is constant. C. the C5/C6 joint in Sauroposeidon in right lateral view, traced from the x-ray scout image (see Figure 12); dorsal is to the left. Except for one area in the ventral half of the cotyle, the anteroposterior separation between the C5 cotyle and C6 condyle is remarkably uniform. All of the arrows in part C are 52 mm long.

But how thick was the intervertebral cartilage in sauropods?

At the moment, no-one really knows. As Kent Stevens (2013) points out in his contribution to the PLOS ONE sauropod gigantism collection:

Determining the ONP of a sauropod’s cervical vertebral column given only its bones requires is necessarily speculative since the cartilage, and thus the intervertebral spacing, is unknown.

Part of the our goal in our own PLOS collection paper (Taylor and Wedel 2013c) was to take some very tentative first steps towards estimating the cartilage thickness. To do this, we used two approaches. First, we looked at CT scans of articulated vertebrae; and second, we measured the cartilage thickness in a selection of extant animals and thought about what we could extrapolate.

Since the CT scans were Matt’s domain, I’m going to pass over those for now, in the hope that he’ll blog about that part of the paper. Here, I want to look at the extant-animal survey.

Figure 18. Cartilage in the neck of a rhea. Joint between cervicals 11 (left) and 10 (right) of a rhea, sagittally bisected. Left half of neck in medial view. The thin layers of cartilage lining the C11 condyle and C10 cotyle are clearly visible.

Figure 18. Cartilage in the neck of a rhea. Joint between cervicals 11 (left) and 10 (right) of a rhea, sagittally bisected. Left half of neck in medial view. The thin layers of cartilage lining the C11 condyle and C10 cotyle are clearly visible.

The first thing to say is that our survey is inadequate in many ways. We worked with the specimens we could get hold of, in the state we had them. This means that:

  • we have a very arbitrary selection of different animals,
  • they are at different ontogenetic stages, and
  • their cartilage thickness was measured by a variety of methods.

Our goal was not at all to reach anything like a definitive answer, but just to get the question properly asked, and so hopefully to catalyse much a more detailed survey.

With that proviso out of the way, here are our main results (from Table 4 of the paper, though here I have removed the sauropod CT-scan rows since we’ll be writing about those separately).

Taxon Thickness Reference Notes
Turkey 4.56% This study Difference in measurements of intact neck and articulated sequence of cleaned, degreased and dried vertebrae.
Ostrich 6.30% Cobley et al. (2013) Difference in measurements of individual vertebrae with and without cartilage.
Rhea 2.59% This study Measurement of in situ cartilage in bisected neck.
Alligator 14.90% This study Measurement of in situ cartilage from photograph of cross section.
Horse 6.90% This study Measurement of in situ cartilage from photograph of cross section.
Camel 13.00% This study Crude measurement from condyle margin to cotyle lip of lateral-view X-ray. This is an interim measurement, which we hope to improve on when we obtain better images.
Dog 17.00% This study Measurement of intervertebral gaps in lateral-view X-ray, uncorrected for likely concavity of cotyles.
Giraffe 24.00% This study Difference in measurement of intact neck and closely articulated sequence of cleaned vertebrae. Young juvenile specimen.
Muraenosaurus 14.00% Evans (1993) Measurement of in situ cartilage in fossils.
Cryptoclidus 20.00% Evans (1993) Measurement of in situ cartilage in fossils.

We’ve expressed the measurements as a ratio between cartilage thickness and the length of the bone itself — that is, cartilage/bone. Another way to think of this is that the percentage is a correction factor which you need to add onto bone length to get whole-segment length. Note that this is not the same ratio as the proportion of total segment length that consists of cartilage: that would be (cartilage thickness + bone length) / bone length.

(We also tossed in some measurements of plesiosaur neck cartilage that Mark Evans made way back when. Get that thing properly published, Mark!)

Even this small survey throws up some interesting points.

First, there is a huge range of proportional cartilage thicknesses: almost an order of magnitude from the 2.59% of the Rhea up to the 24% of the juvenile giraffe — or, even if you discard that because of its ontogenetic stage, up to 17% for the dog. And note that the 17% for the dog is probably an under-estimate, since we were working from an X-ray that doesn’t show the concavity of the vertebral cotyles.

Figure 22. Dog neck in X-ray. Neck of a dog (dachsund), in X-ray, with the seven cervical vertebrae indicated. This photo has been used with permission from the Cuyahoga Falls Veterinary Clinic.

Figure 22. Dog neck in X-ray. Neck of a dog (dachsund), in X-ray, with the seven cervical vertebrae indicated. This photo has been used with permission from the Cuyahoga Falls Veterinary Clinic.

(Two reviewers expressed scepticism that this is the usual condition for dogs, but this X-ray is consistent with those of other dogs illustrated in the veterinary literature.)

The second thing to note is that the cartilage measurements for birds (average 4.5%) are are much lower than those of crocodilians (14.9%) or mammals (15.2%). What does this mean? Among these groups, sauropods are most closely related to birds; but birds and crocs form the extant phylogenetic bracket, so we can’t tell from phylogeny alone whether to expect them to more closely approach the avian or crocodilian condition. Furthermore, in being opisthocoelous (condyle in front, cotyle at the back) sauropod cervicals most closely resemble those of mammals in gross structure — and they have the thickest cartilage of all.

The third thing to note is that there is considerable variation within groups. Although the cartilage is proportionally thin for all three birds, it’s more than twice as thick in the ostrich as in the rhea (although some of this could be due to the different measurement methods used for these two birds). More interestingly, among mammals the cartilage is twice as thick in camels as in horses. In the horse, the condyles are deeply inserted into the cotyles of the preceding vertebrae; but in camels, they don’t reach even the lip of the cotyle. This should worry us, as horse and camel cervicals are grossly similar, and no osteological correlates have been identified that would allow us to determine from the bones alone how very different the cartilage is between these two mammals. So it seems possible that there were similarly dramatic differences in the neck-cartilage thickness of different sauropods.

Note: I said that no osteological correlates have been identified. That doesn’t mean they don’t exist. One thing I would love to see is a serious attempt to analyse cartilage thickness across a broad range of mammals, and to examine the corresponding dry bones to see whether in fact there are correlates that could be informative in this respect. One lesson that Matt and I have learned over and over again is that there’s often plenty of data in places that are out in the open, but where no-one’s thought to look.

Next time: more on searching for osteological correlates of cartilage. Then, measurements of sauropod-neck cartilage from CT scans, and likely implications for cartilage thickness in life.

References

As I mentioned a few days ago, Matt and I have a couple of papers in the new PLOS ONE Sauropod Gigantism collection. We were each lead author on one and second author on the other, so for convenience’s sake we’ll refer to them as my paper (Taylor and Wedel 2013c on neck cartilage) and Matt’s paper (Wedel and Taylor 2013b on caudal pneumaticity.)

Mine is very simple in concept (although it ended up at 17 pages and 23 figures). It’s all about addressing one of the overlooked variables in reconstructing the postures of the necks of sauropods (and indeed of all tetrapods). That is, the spacing between consecutive vertebrae, and the effect this has on “neutral pose”.

The concept of “neutral pose” goes back to the DinoMorph work of Stevens and Parrish (1999). They defined it (p. 799) as follows: “We determined the neutral poses for each animal, wherein the paired articular facets of the postzygapophyses of each cervical vertebra were centered over the facets of the prezygapophyses of its caudally adjacent counterpart.”

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Taylor and Wedel (2013c: Figure 3). Articulated sauropod vertebrae. Representative mid-cervical vertebra of Giraffatitan brancai, articulating with its neighbours. The condyle (ball) on the front of each vertebra’s centrum fits into the cotyle (socket) at the back of the preceding one, and the prezygapophyses articulate with the preceding vertebra’s postzygapophyses. These vertebrae are in Osteological Neutral Pose, because the pre- and postzygapophyseal facets overlap fully.

One of the more fundamental flaws in Stevens and Parrish (1999) is the assumption that animals habitually rest their necks in neutral pose — an assumption that is unsupported by evidence and, as it turns out, false (Vidal et al. 1986, Taylor et al. 2009). But let’s leave that aside for the moment, and consider what neutral pose actually represents.

The fact that there is even such a thing as neutral articulation between two consecutive vertebrae is due to there being three points of contact between those vertebra: as with the legs of a tripod, three points is the minimum number you need to fix an object in three-dimensional space. Two of these points are at the zygapophyses, as noted in the original definition above. The third point is the articulation between the centra.

The centrum has been curiously overlooked in discussions of neutral pose, but needless to say its length is crucial in establishing what is neutral. In the image above, if the centrum was longer, then the angle between the consecutive vertebrae would need to be raised in order to keep the zygapophyses articulated.

And of course it was longer in life, because of the cartilage in between the consecutive centra. (The use of the more specific term “osteological neutral pose” goes some way to recognising that tissues other than bone have been overlooked, but the problem has not really been addressed or even properly acknowledged in published works before our paper.)

xx

Taylor and Wedel (2013c: Figure 5). Intervertebral gaps in camel necks. Head and neck of dromedary camels. Top: UMZC H.14191, in right lateral view, posed well below habitual posture, with apparently disarticulated C3/C4 and C4/C5 joints. Photograph taken of a public exhibit at University Museum of Zoology, Cambridge, UK. Bottom: OUMNH 17427, in left lateral view, reversed for consistency with Cambridge specimen. Photograph taken of a public exhibit at Oxford University Museum of Natural History, UK. Inset: detail of C4 of the Oxford specimen, showing articulations with C3 and C5. The centra are separated by thick pads of artificial ‘‘cartilage’’ to preserve spacing as in life.

You simply can’t ignore cartilage when modelling neck postures and expect to get anything resembling a meaningful result. That is, presumably, the reason why the habitual posture of rabbits in life exceeds the most extended posture we were able to obtain when manipulating dry vertebrae of a hare: compare Vidal et al. (1986: fig. 4) with Taylor et al. (2009: fig. 1).

How big is the effect? That depends on the thickness of the cartilage and the height of the zygapophyses above the center of rotation. Here is an illustration that we should have put in the paper, but which inexplicably neither of us thought of:

figNEW-angle-at-zygs

Influence of intervertebral cartilage on vertebral articulation angle. Consider the posterior vertebra (black) as fixed. The blue vertebra represents neutral pose of the preceding vertebra with centra abutting and zygapophyseal facets maximally overlapped. The red vertebra indicates neutral pose once intervertebral cartilage is added between the vertebra (where else?) The green lines show the angle by which the more anterior vertebra must be inclined in order to accommodate the cartilage, and the magenta line shows the height of the zygapophyseal articulation above the center of rotation between the two vertebrae.

Here’s some elementary trigonometry. Suppose the intervertebral cartilage is x distance thick at mid-height of the centra, and that the height of the zygs above this mid-height point (the magenta line) is y. The triangle between the middle of the condyle of the posterior vertebra, the middle of the cotyle of the anterior one and the zygapophyseal articulation is near enough a right-angled triangle as makes no odds.

Consider the angle θ between the green lines. Sin(θ) = opposite/hypotenuse = x/y, and by similarity, the additional angle of inclination of the anterior vertebra is also θ.

But for small angles (and this is generally a small angle), sin(θ) ≈ θ. So the additional inclination in radians = cartilage thickness divided by zygapophyseal height. For example, in vertebrae where the zygs are 23 cm above the mid-height of the centra, adding 4 cm of intervertebral cartilage adds about 4/23 = 0.174 radians = 10 degrees of extra inclination. (That’s pretty similar to the angle in the illustration above. Eyeballing the cartilage thickness and zyg height in the illustration suggests that 23:4 ratio is about right, which is a nice sanity-check of this method.)

millionaire-stupid-contestant4

At this point, I am cursing my own stupidity for not putting this diagram, and the very simple calculation, into the paper. I guess that can happen when something is written in a hurry (which to be honest this paper was). The formula is so simple — and accurate enough within tolerances of inevitable measurement error — that we really should have used it all over the place. I guess that will have to go in a followup now.

Anyway — next time, we’ll address this important related question: how thick, in fact, was the cartilage between the cervicals of sauropods?

References

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