July 29, 2014
I wrote this paper in 2003-4, and it was rejected without review when I submitted it back then. (For, I think, specious reasons, but that’s a whole nother discussion. Forget I mentioned it.)
I haven’t touched the manuscript since then (except to single-space it for submission as a preprint). It’s ten years old. That’s a problem because it’s an analysis of a database of dinosaur diversity, and as everyone knows, the rate of recognising new dinosaurs has gone through the roof. That’s the reason I never made any attempt to update and resubmit it: dinosaur diversity is a fast-moving target, and each time through the submit-reject cycle takes long enough for the data to be outdated.
So much for the history. Now the question: how should I cite this paper? Specifically, what date should I give it? If I cite it as from 2004, it will give the misleading impression that the paper has been available for ten years; but if I cite it as from 2014, it will imply that it’s been worked on at some point in the last ten years. Both approaches seem misleading to me.
At the moment, I am citing it as “Taylor (2014 for 2004)”, which seems to more or less capture what’s meant, but I don’t know whether it’s an established convention. Is there an established convention?
Releated: where in mv publications list should it appear? At present I am sorting it under 2014, since that’s when it came out; but should it be under 2004, when it was written? I guess publication date is the one to go far — after all, it’s not unusual even now for papers to spend a year or more in press, and it’s the later (publication) date that’s cited.
Help me out. How should this be done?
- Taylor, Michael P. 2014 (written in 2004). A survey of dinosaur diversity by clade, age, place of discovery and year of description. PeerJ PrePrints 2:e434v1. doi: 10.7287/peerj.preprints.434v1
New (but very old) preprint: A survey of dinosaur diversity by clade, age, place of discovery and year of description
July 11, 2014
Today, available for the first time, you can read my 2004 paper A survey of dinosaur diversity by clade, age, place of discovery and year of description. It’s freely available (CC By 4.0) as a PeerJ Preprint. It’s one of those papers that does exactly what it says on the tin — you should be able to find some interesting patterns in the diversity of your own favourite dinosaur group.
“But Mike”, you say, “you wrote this thing ten years ago?”
Yes. It’s actually the first scientific paper I ever wrote (bar some scraps of computer science) beginning in 2003. It’s so old that all the illustrations are grey-scale. I submitted it to Acta Palaeontologica Polonica way back on on 24 October 2004 (three double-spaced hard-copies in the post!) , but it was rejected without review. I was subsequently able to publish a greatly truncated version (Taylor 2006) in the proceedings of the 2006 Symposium on Mesozoic Terrestrial Ecosystems, but that was only one tenth the length of the full manuscript — much potentially valuable information was lost.
My finally posting this comes (as so many things seem to) from a conversation with Matt. Off work sick, he’d been amusing himself by re-reading old SV-POW! posts (yes, we do this). He was struck by my exhortation in Tutorial 14: “do not ever give a conference talk without immediately transcribing your slides into a manuscript”. He bemoaned how bad he’s been at following that advice, and I had to admit I’ve done no better, listing a sequence of old my SVPCA talks that have still never been published as papers.
The oldest of these was my 2004 presentation on dinosaur diversity. Commenting on this, I wrote in email: “OK, I got the MTE four-pager out of this, but the talk was distilled from a 40ish-page manuscript that was never published and never will be.” Quick as a flash, Matt replied:
If I had written this and sent it to you, you’d tell me to put it online and blog about how I went from idea to long paper to talk to short paper, to illuminate the process of science.
And of course he was right — hence this preprint.
I will never update this manuscript, as it’s based on a now wildly outdated database and I have too much else happening. (For one thing, I really ought to get around to finishing up the paper based on my 2005 SVPCA talk!) So in a sense it’s odd to call it a “pre-print” — it’s not pre anything.
Despite the data being well out of date, this manuscript still contains much that is (I think) of interest, and my sense is that the ratios of taxon counts, if not the absolute numbers, are still pretty accurate.
I don’t expect ever to submit a version of this to a journal, so this can be considered the final and definitive version.
- Taylor, Michael P. 2006. Dinosaur diversity analysed by clade, age, place and year of description. pp. 134-138 in Paul M. Barrett and Susan E. Evans (eds.), Ninth international symposium on Mesozoic terrestrial ecosystems and biota, Manchester, UK. Cambridge Publications. Natural History Museum, London, UK. 187 pp.
- Taylor, Michael P. 2014 (written in 2004). A survey of dinosaur diversity by clade, age, place of discovery and year of description. PeerJ PrePrints 2:e434v1. doi:10.7287/peerj.preprints.434v1
I think it’s fair to say that this “bifurcation heat-map”, from Wedel and Taylor (2013a: figure 9), has been one of the best-received illustrations that we’ve prepared:
Back when the paper came out, Matt rashly said “Stand by for a post by Mike explaining how it came it be” — a post which has not materialised. Until now!
This illustration was (apart from some minor tweaking) produced by a program that I wrote for that purpose, snappily named “vcd2svg“. That name is because it converts a vertebral column description (VCD) into a scalable vector graphics (SVG) file, which you can look at with a web-browser or load into an image editor for further processing.
The vertebral column description is in a format designed for this purpose, and I think it’s fairly intuitive. Here, for example, is the fragment describing the first three lines of the figure above:
Taxon: Apatosaurus louisae
Specimen: CM 3018
Taxon: Apatosaurus parvus
Specimen: UWGM 155556/CM 563
Taxon: Apatosaurus ajax
Specimen: NMST-PV 20375
Basically, you draw little ASCII pictures of the vertebral column. Other directives in the file explain how to draw the various glyphs represented by (in this case) “Y”, “V”, “u”, and “n”.
It’s pretty flexible. We used the same program to generate the right-hand side (though not the phylogenetic tree) of Wedel and Taylor (2013b: figure 2):
The reason I mention this is because I released the software today under the GNU General Public Licence v3.0, which is kind of like CC By-SA. It’s free for anyone to download, use, modify and redistribute either verbatim or in modified form, subject only to attribution and the requirement that the same licence be used for modified versions.
vcd2svg is written in Perl, and implemented in part by the SVG::VCD module, which is included in the package. It’s available as a CPAN module and on GitHub. There’s documentation of the command-line vcd2svg program, and of the VCD file format. Also included in the distribution are two documented examples: the bifurcation heat-map and the caudal pneumaticity diagram.
Folks, please use it! And feel free to contribute, too: as the change-log notes, there’s work still to be done, and I’ll be happy to take pull requests from those of you who are programmers. And whether you’re a programmer or not, if you find a bug, or want a new feature, feel free to file an issue.
A final thought: in academia, you don’t really get credit for writing software. So to convert the work that went into this release into some kind of coin, I’ll probably have to write a short paper describing it, and let that stand as a proxy for the actual program. Hopefully people will cite that paper when they generate a figure using the software, the way we all reflexively cite Swofford every time we use PAUP*.
Update (12 April 2014)
- Swofford, D. L. 2002. PAUP*: phylogenetic analysis using parsimony (* and other methods). Sinauer Associates, Sunderland, MA.
- Wedel, M.J., and Taylor, M.P. 2013. Neural spine bifurcation in sauropod dinosaurs of the Morrison Formation: ontogenetic and phylogenetic implications. Palarch’s Journal of Vertebrate Palaeontology 10(1): 1-34. ISSN 1567-2158.
- Wedel, Mathew J., and Michael P. Taylor. 2013. Caudal pneumaticity and pneumatic hiatuses in the sauropod dinosaurs Giraffatitan and Apatosaurus.PLOS ONE 8(10):e78213. 14 pages. doi:10.1371/journal.pone.0078213 [PDF]
“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.“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.
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:
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:
Those photos were pretty persuasive; and a closer examination of the specimen on that same trip was to prove conclusive.
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!
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)!
- Taylor, Michael P. 2009. A re-evaluation of Brachiosaurus altithorax Riggs 1903 (Dinosauria, Sauropoda) and its generic separation from Giraffatitan brancai (Janensch 1914). Journal of Vertebrate Paleontology 29(3):787-806.
- Upchurch, Paul, Paul M. Barrett and Peter Dodson. 2004. Sauropoda. pp. 259-322 in D. B. Weishampel, P. Dodson and H. Osmólska (eds.), The Dinosauria, 2nd edition. University of California Press, Berkeley and Los Angeles. 861 pp.
- Wedel, Mathew J., and Michael P. Taylor. 2013. Caudal pneumaticity and pneumatic hiatuses in the sauropod dinosaurs Giraffatitan and Apatosaurus. PLOS ONE 8(10):e78213. 14 pages. doi:10.1371/journal.pone.0078213
- Wilson, J.A. 2002. Sauropod dinosaur phylogeny: critique and cladistic analysis. Zoological Journal of the Linnean Society 136:217-276.
December 13, 2013
It’s now widely understood among researchers that the impact factor (IF) is a statistically illiterate measure of the quality of a paper. Unfortunately, it’s not yet universally understood among administrators, who in many places continue to judge authors on the impact factors of the journals they publish in. They presumably do this on the assumption that impact factor is a proxy for, or predictor of, citation count, which is turn is assumed to correlate with influence.
As shown by Lozano et al. (2012), the correlation between IF and citations is in fact very weak — r2 is about 0.2 — and has been progressively weakening since the dawn of the Internet era and the consequent decoupling of papers from the physical journal that they appear in. This is a counter-intuitive finding: given that the impact factor is calculated from citation counts you’d expect it to correlate much more strongly. But the enormous skew of citation rates towards a few big winners renders the average used by the IF meaningless.
To bring this home, I plotted my own personal impact-factor/citation-count graph. I used Google Scholar’s citation counts of my articles, which recognises 17 of my papers; then I looked up the impact factors of the venues they appeared in, plotted citation count against impact factor, and calculated a best-fit line through my data-points. Here’s the result (taken from a slide in my Berlin 11 satellite conference talk):
I was delighted to see that the regression slope is actually negative: in my case at least, the higher the impact factor of the venue I publish in, the fewer citations I get.
There are a few things worth unpacking on that graph.
First, note the proud cluster on the left margin: publications in venues with impact factor zero (i.e. no impact factor at all). These include papers in new journals like PeerJ, in perfectly respectable established journals like PaleoBios, edited-volume chapters, papers in conference proceedings, and an arXiv preprint.
My most-cited paper, by some distance, is Head and neck posture in sauropod dinosaurs inferred from extant animals (Taylor et al. 2009, a collaboration between all three SV-POW!sketeers). That appeared in Acta Palaeontologia Polonica, a very well-respected journal in the palaeontology community but which has a modest impact factor of 1.58.
My next most-cited paper, the Brachiosaurus revision (Taylor 2009), is in the Journal of Vertebrate Palaeontology — unquestionably the flagship journal of our discipline, despite its also unspectacular impact factor of 2.21. (For what it’s worth, I seem to recall it was about half that when my paper came out.)
In fact, none of my publications have appeared in venues with an impact factor greater than 2.21, with one trifling exception. That is what Andy Farke, Matt and I ironically refer to as our Nature monograph (Farke et al. 2009). It’s a 250-word letter to the editor on the subject of the Open Dinosaur Project. (It’ a subject that we now find profoundly embarrassing given how dreadfully slowly the project has progressed.)
Google Scholar says that our Nature note has been cited just once. But the truth is even better: that one citation is in fact from an in-prep manuscript that Google has dug up prematurely — one that we ourselves put on Google Docs, as part of the slooow progress of the Open Dinosaur Project. Remove that, and our Nature note has been cited exactly zero times. I am very proud of that record, and will try to preserve it by persuading Andy and Matt to remove the citation from the in-prep paper before we submit. (And please, folks: don’t spoil my record by citing it in your own work!)
What does all this mean? Admittedly, not much. It’s anecdote rather than data, and I’m posting it more because it amuses me than because it’s particularly persuasive. In fact if you remove the anomalous data point that is our Nature monograph, the slope becomes positive — although it’s basically meaningless, given that all my publications cluster in the 0–2.21 range. But then that’s the point: pretty much any data based on impact factors is meaningless.
- Farke, Andrew A., Michael P. Taylor and Mathew J. Wedel. 2009. Sharing: public databases combat mistrust and secrecy. Nature 461:1053.
- Lozano, George A., Vincent Larivière and Yves Gingras. 2012. The weakening relationship between the impact factor and papers’ citations in the digital age. Journal of the American Society for Information Science and Technology 63(11):2140-2145. doi:10.1002/asi.22731 [arXiv preprint]
- Taylor, Michael P. 2009. A re-evaluation of Brachiosaurus altithorax Riggs 1903 (Dinosauria, Sauropoda) and its generic separation from Giraffatitan brancai (Janensch 1914). Journal of Vertebrae Paleontology 29(3):787-806.
- Taylor, Michael P., Mathew J. Wedel and Darren Naish. 2009. Head and neck posture in sauropod dinosaurs inferred from extant animals. Acta Palaeontologica Polonica 54(2):213-230.
November 21, 2013
Well, folks, I’m back from Berlin. And what an extraordinary couple of days it was. There were in fact three days of open-access talks, though I was only able to be there for the first two. Day one was the satellite conference, aimed at early-career researchers; days two and three were the much larger main conference, attended mostly by heavy hitters: senior librarians, university administrators, a sprinkling of politicians, and of course some researchers and publishers.
It was my privilege to speak at both satellite and main conferences. This post is really just to advertise those talks. Why am I doing this? Because I’m convinced that they’re by far the most important talks I’ve ever given. It’s great fun to talk about Barosaurus at SVPCA, or about intervertebral cartilage in Bonn, but if someone says to me that that work doesn’t really matter in a cosmic sense, I’ll be hard put to find reasons why they’re wrong. But open access has profound and immediate consequences for health, industry, education, third-world development, and more fields than I can list.
So here are the talks.
Satellite meeting talk
First up, at the satellite conference, my subject was: Towards universal Open Access: what we can do about it, and who should do it. My goal here was to help researchers see what practical steps they can take right now towards the open-access goal that we all aspire to. I covered six areas:
- Publish our own work open access (whether Gold or Green)
- Review for Open Access journals
- Edit for Open Access journals
- Advocate Open Access policies
- Deprecate journal rank
- Talk about Open Access
Along the way, we talked about the open-access citation advantage, the (mostly non-) problem of article processing charges, the complete non-problem of “predatory open-access publishers”, the acceptable length of Green-OA embargoes (zero), the SV-POW! decision tree, publishers’ lack of control over what you do before you sign the copyright transfer, the inability of impact factor to predict citation count (post to come), the childishness of evaluating individuals by journal rank, and the knotty problem of who should take responsibility for fixing our current broken system.
Here are a few tweets that went out as I was giving this talk: “a blistering, fantastic presentation“, “Can we get a twitter round of applause … Absolutely BRILLIANT presentation“, “TOTALLY BRILLIANT“, “This is why we HAVE to record these conferences. Not recording that presentation would be a crime“, “It was AWESOME!“, and finally my favourites: “making you not just know #openaccess , but feel it” and “Mike’s talks at the #Berlin11 conference was 1of the most emotional 1’s I have ever seen!“
I actually don’t know whether it’s going to be possible for people who missed the live stream to watch this talk. That was the plan, but I heard a rumour that the recording went wrong. If a video does becomes available, I’ll let you know. In the mean time, you can at least get the slides [PowerPoint or exported PDF]. They are CC By.
Main meeting talk
In the main conference, I used my slot to remind us all that Open Access is about sharing, unity and sanity, not about money. Because I was addressing a more senior audience that necessarily has to think more about practicalities, finances, ways and means, I wanted to take the opportunity to remember that those are not the issues that gave birth to Open Access; rather, it started out as an unabashedly idealistic movement (as reading any of the three great declarations will show you). I don’t want us to walk away from that high-ground and be reduced to thinking only about practicalities, important though they are.
Publishers and their associates often say — rightly, as far as they go — that “Scientific and technical publishing is a business“. But no-one goes into it because of they money they can make. Everyone involved in doing or publishing research surely got into that business because their eyes were on a higher prize. So the burden of my talk was that publishing research is a mission; that far from “getting rid of the idealists“, we should cherish them; and that we should encourage rather than curb our own idealistic tendencies.
Perhaps the most satisfying part of the whole conference was giving this talk — you might almost call it a preach — and watching the nodding agreement spread across the audience. Folks, we’re about a great work. Let’s not forget that. Let’s not sell ourselves short.
The main session was unfortunately not livecast, and to the best of my knowledge, there were never any plans to record it. But as with my satellite talk, you can at least get the slides [PowerPoint or exported PDF]. They are CC By.
Since I made the slides available for download immediately after the talks (three days ago for the satellite meeting, two days ago for the main meeting), I’ve been surprised and delighted to see the download numbers — currently standing at 641 for the satallite talk and 939 for the main talk. The tweet announcing the main talk has also been retweeted 34 times and favourited 26 times. I hope that shows that I struck a chord.
I have an informal invitation to deliver the main-session talk next year to an Italian university, which I’ll be pleased to do once we’re able to sort out the details. I’m not sure whether more invitations are likely to be forthcoming, but I’ll mention them here if they do materialise.
I’d like to finish by thanking my employer, Index Data. As most of you know, I am not a career academic: I work on sauropods in my spare time (and advocate open access in my spare spare time), earning my living as a computer programmer. By the time the invitations to speak at the Berlin conferences came in, I’d already booked up my annual leave allowance, so I had to ask for permission to take unpaid days for the conference. Instead, Index Data gave me two more paid days — because they, like me, believe in the importance of open access.
This is all the more laudable since, if anything, universal open access will harm our business. A significant part of what we build is authentication mechanisms to allow people (legitimate) access to paywalled resources. Once everything is open, no-one will need to pay us to do that. It’s greatly to Index Data’s credit that, despite this, they want to help us push on towards a goal that will benefit society as a whole.
- Taylor, Michael P. Monday 18 November 2013. Towards universal Open Access: what we can do about it, and who should do it. Berlin 11 Satellite Conference for students and early-career researchers. [Slides PPT] [Slides PDF]
- Taylor, Michael P. Tuesday 19 November 2013. Open Access is about sharing, unity and sanity, not about money. Berlin 11 Open Access Conference: 10th Anniversary of the Berlin Declaration. [Slides PPT] [Slides PDF]
November 3, 2013
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.”
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.)
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:
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.)
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?
- Stevens, Kent A., and J. Michael Parrish. 1999. Neck Posture and Feeding Habits of Two Jurassic Sauropod Dinosaurs. Science 284:798-800.
- Taylor, Michael P., and Matthew J. Wedel. 2013c. The effect of intervertebral cartilage on neutral posture and range of motion in the necks of sauropod dinosaurs. PLOS ONE 8(10): e78214. 17 pages. doi:10.1371/journal.pone.0078214 [PDF]
- Taylor, Michael P., Mathew J. Wedel and Darren Naish. 2009. Head and neck posture in sauropod dinosaurs inferred from extant animals. Acta Palaeontologica Polonica 54(2):213-230. [PDF]
- Vidal, Pierre Paul, Werner Graf and Alain Berthoz. 1986. The orientation of the cervical vertebral column in unrestrained awake animals. Experimental Brain Research, 61:549-559. doi:10.1007/BF00237580
- Wedel, Mathew J., and Michael P. Taylor. 2013. Caudal pneumaticity and pneumatic hiatuses in the sauropod dinosaurs Giraffatitan and Apatosaurus. PLOS ONE 8(10):e78213. 14 pages. doi:10.1371/journal.pone.0078213 [PDF]