There’s this guy, you see, who knitted his way to a solution to an infamous problem in Roman history. This might be a bit premature (since academic journals haven’t weighed in yet), but I am persuaded that the mystery of the ancient Roman dodecahedrons has been solved. And why I’m persuaded affords a handy example for teaching how Bayesian reasoning works in making good historical inferences. [Update: This case likewise shows how Bayesian reasoning can incorporate new facts so as to change what’s likely: experts in the comments to this article subsequently persuaded me that a full accounting of the facts in my Bayesian model does not get as positive a result for this thesis as I had initially thought.]
I suppose I should begin by explaining what a “mysterious ancient Roman dodecahedron” is. It’s not just any dodecahedron from ancient Rome (I’ll show you an unrelated example shortly), but a very peculiarly consistent oddity that no one has been able to explain (mainly because no writing survives mentioning it). It’s a common object. Some hundred or so have been found, originating in the 2nd century A.D. and spanning a couple of centuries afterward. But only in France and northern and eastern Europe. It’s weird looking. And has peculiar features. Some are of stone manufacture, but most are cast bronze.
Some typical examples (one from Wikipedia, another from the Birmingham Musem) are shown to the right. Each is a twelve-sided hollow object, the sides generally symmetrical (an isohedron, so it looks a little like a twelve-sided die, something old-school role-playing-gamers will recognize), but every side has a circular hole in it, and the holes are different sizes, but the pattern of sizes (the sequence and arrangement) is the same on every object, even though the size of the object (and thus size of the holes) varies considerably, from kind of tiny (one and a half inches total diameter) to about the size of what would have then been a large adult fist (a little over four inches). The holes also sometimes have a sequence of parallel carved rings around them (sort of like gutters or guidelines in the face of the object), but many do not, so these appear to be a decorative flourish (a typical accent found in Roman tech of the time, where common utilitarian objects can be prettied up with some artsy flourishes like that).
But importantly, every corner of these objects has a solid knob sticking out of it, a bollard narrower at its base than at its tip (many of these just look like attached spheres), for twenty knobs in all. This most of all prevents the twelve-sided die analogy from quite being right, that plus the fact that the holes being of different size means each face has a different weight. They also aren’t inscribed with anything…a fact that is far more crucial to determining their purpose than you might at first think.
Just search “Roman dedocahedron” in Google Images and you’ll find dozens of examples. And yet…
What’s It For?
No one really knows. Many theories have been proposed, often argued in elaborate fashion. Wikipedia has an entry on them (but to avoid spoilers, don’t read that just yet) and lists the hypotheses so far explored as “candlestick holders; dice; survey instruments; devices for determining the optimal sowing date for winter grain; that they were used to calibrate water pipes; and army standard bases.”
The main problem with most of these hypotheses is that they all predict the same thing. For example, the “survey instrument” and “solar calendar” theories are based on coincidences of the angles produced by the holes in these dodecahedrons, but those coincidences would be there no matter what the object was used for–as already exemplified by the fact that the same angles lead to both proposed functions, perspectival surveying and calendrics. Despite those having nothing to do with each other.
Whatever function the objects had, they would always have these features. In Bayesian terms, that means the probability of these objects having these features if they weren’t built for either purpose is virtually 100%. Which renders that fact useless as evidence. What counts as evidence is a fact that is improbable on alternative theories, not a fact that is entirely expected on just about any theory you try to propose (atheists will find that complaint familiar…when we expect an outcome anyway, whether there be a God or not, it can’t count as evidence for God).
Before I survey the proposed theories, if you want a delightful surprise before I spoil it, take a moment and watch the video embedded here (sorry, there’s no transcript or narration, so if you are visually impaired, just keep reading along here, I’ll fill you in). That’s the correct solution. I’m fairly certain. You won’t start to realize what it is until more than halfway through the video. But once you’ve enjoyed that surprise, let’s see what theories that solution is competing with…
Apart from what I already mentioned, the surveying-instrument theory (defended by Amelia Carolina Sparavigna of the Department of Applied Science and Technology at the Politecnico di Torino in Italy) requires users of the object to constantly use elaborate and time-consuming math, when in fact that math could have been done in advance and its results carved directly on the object. In other words, if someone were using this object for that purpose, the instruction manual would be written on it (as it was on the Antikythera computer). You wouldn’t have to do the math. The results of it would be part of the engraving on each object, so you could immediately make use of it. This is especially the case because the objects vary so much in size, necessitating engraving this data directly on each. That we don’t observe any such engravings is thus not 100% expected but in fact very improbable on Sparavigna’s theory. So the angular coincidences she documents, which she mistakes as evidence for her theory, actually weighs considerably against her theory.
That plus the fact that these objects appear to have been strangely limited geographically to colder climates (a fact her theory doesn’t explain), and we actually have several textbooks from antiquity on surveying techniques and instruments, and none mention any device like this. Two more facts that are improbable on her theory. Her theory would instead predict they would be everywhere, and get mention in books about surveying and instrumentation (or at least similar techniques would). In Bayesian terms, that means her theory entails P(~e|h), the probability of the contrary evidence, is getting again close to 100%, which entails P(e|h), the actual probability of the evidence, which equals 1 – P(~e|h), is getting again close to zero (Proving History, pp. 230, 255-56, 302 n.13). In fact we already know what ancient Roman dioptras looked like. They didn’t look like this. They were in fact far more versatile, sophisticated, and easier to use. Whereas any object with differently sized holes along a symmetrical axis can be used by a mathematician as a simple makeshift diopter–regardless of what it was built for.
Similarly, the calendrical theory requires a procedure that is so absurdly elaborate as to be well beyond comical. This you can see by looking at the procedure proposed by G.M.C. Wagemans (search his article for the words “It works as followed” to see the nine steps one must take, each already kind of ridiculously elaborate, but taken together, his theory parodies itself). His article has many uses, though, e.g. he catalogs many examples of these objects and plots their geographical location and discusses the archaeological circumstances each was found in and so on. He himself notes that the objects, if used for the purpose he imagines (determining when to begin sowing crops), don’t work very well (their accuracy is kind of shit). Which is improbable on his theory, considering the Romans had far simpler, more advanced, and more accurate ways of doing the same thing. You don’t buy an absurdly expensive screwdriver to hammer nails, when you already have really excellent and affordable hammers (see Roman sundials).
Indeed, the Romans had portable cylinder dials that were not only highly accurate, and did far more things than just tell you when to plant crops, but that could fit in your pocket and weighed next to nothing (being carved from bone, also a far cheaper material than bronze). Moreover, everyone needed to know when to plant crops–everyone, everywhere. Why buy your own little crappy way of doing that, when whole towns routinely invested in highly effective public calendars for the purpose? Anyone who wanted to know planting season just had to take a donkey downtown. And if for some reason you needed your own sundial, you would just get one of those–or even one of the pocket ones. It’s extremely improbable that anyone was using the dodecahedrons do do a worse job, at greater expense, and with vastly more difficulty, than they could already do with the tech at hand. An extraordinarily low P(e|h) thus results, spelling doom for this hypothesis. Making this an extraordinary claim…without extraordinary evidence.
John Ladd has attempted to argue the objects were what we would call a volumeter: any symmetrical object inserted into one axis would have a known displacement of water (and thus a known volume and density). Romans had volumeters (mensa ponderaria, to the right is an example recovered from Pompeii), but this object can’t have had this function for the same reason it can’t have been a surveying instrument: the objects vary enormously in size, so one would have had to engrave each axis of one with its volumetric values. Moreover, the knobs ruin the geometric value of the object for this purpose (by elevating it needlessly when immersed in water). This also doesn’t explain the geographic distribution, or peculiar variation in the shape of the circles (always the same pattern, yet the objects, and thus their holes, are never the same size). And so on. Again, all this is improbable on his theory, or not made probable by it. That makes his theory a bad explanation.
Same problem. The objects vary so much in size, the calibrations would have to be inscribed or etched on each example. Otherwise, what are you calibrating to? Moreover, one would have had no use for this kind of a calibrator. Pipes were cast, and this object would have no use in molding a cast. This object could only tell you the size of a pipe that was already manufactured. Which you could tell just by looking at the pipe. You wouldn’t need an instrument like this. This also doesn’t explain the geographic distribution. And most importantly, this theory simply doesn’t explain the twenty knobs. It thus again doesn’t get a high P(e|h) but in fact a low one. So it’s not a good explanation.
This theory is based on a single example having been found with some melted wax in it. But that’s rather like finding some melted wax in a coke bottle and inferring coke bottles were designed as candle holders. Even if someone used one of these objects as a makeshift candle holder, that doesn’t tell us what it was actually designed for (and wax being in one doesn’t even mean it was used that way; just that some wax object was nearby it that melted). If these objects were designed to hold candles, all or most would have traces of wax all over them. One should also note that, on this theory, the smallest dodecahedrons entail improbably thin candlesticks.
And on top of all that, who makes a candle holder that ensures the wax leaks right out the bottom? And that the candlelight gets blocked by the housing and no longer held up by it when it burns low? We know what candle holders typically look like. It’s not this. See the image to the right (as a typical design): the tray is made to collect melted wax (not let it seep through), and the stand elevates the candle so that when the candle burns low, the flame remains elevated and unobstructed. (Incidentally, glass and shell and mica housings, all transparent to light, were then used when one wanted to shield a flame from wind; such tech is already evident from the 1st century, with crushed examples recovered from Pompeii.) So none of the design aspects of the dodecahedrons are expected on this theory, but quite the opposite. So again we have a high P(~e|h), which entails a low P(e|h), which makes for evidence against the theory.
Army Standard Bases?
A non-starter. They are too small and weak for such a purpose. Roman standards were enormous, with an enormously heavy headpiece atop a very long pole (to maximize visibility at a distance). These objects are tiny little things sitting close to the floor with a distance between holes for supporting an object (if such they did) smaller than a single human hand. Do the math on that. The evidence’s improbability on this theory is therefore off the charts. Those pesky laws of physics, you see. One thing the ancients knew well was the law of leverage. It was indeed the first mathematical law they discovered and explained.
Likewise improbable. The holes make it non-random (each side weighs differently and thus does not have an equal chance of coming up). But more importantly, a game randomizer would not need the twenty knobs. Nor would it need the holes–it would instead be inscribed with numbers, letters, or signs. For in fact we have many examples of dodecahedrons used as game randomizers from ancient Rome. And they look like this (see image to the right). They do not look like those mysterious dodecahedrons. So this theory just does not make the evidence likely, nor does it correspond to the prior probability derived from past cases (other dodecahedral game randomizers).
When in doubt, always punt to “some sort of unfathomable religious significance.” Or so the bad rule of thumb goes. In fact, the objects have not been found in association with religious objects or areas in any significant frequency to justify this hypothesis. And the design is too consistent and specific over a large geographic area. It appears to have a utilitarian function. Not a religious one. Again, we have a low P(e|h), not least because this is, like “God did it,” a non-explanation. For no one can deduce (i.e. make probable) any of the features of the object from this theory.
Yeah. A template for knitting gloves.
This one is probably correct.
It explains every single feature of the object common to all specimens. It explains the knobs (they hold yarn loops during construction), the hollowness (to fit a ball of yarn), the size variance (small for babies, large for men, and all sizes in between–but no sizes smaller, no sizes larger), the symmetry (as knitters will attest, an asymmetrical template would get you a wonky knit glove), the number of holes (two for inserting and manipulating the ball of yarn, two sets of five more holes for knitting two pairs of gloves of five fingers each), and most importantly, the peculiar sequence of hole sizes on each object: they run large, medium, large, medium, small on one half, then mirror that with small, medium, large, medium, large. Which is exactly what fingers do: thumb is large, index finger is medium, middle finger is a little larger than the index finger, the ring finger is smaller again, and the pinky is the smallest; and the sequence is mirrored for the left hand and right hand.
And you know what else it explains? Why these objects are only found geographically in colder climes.
The solution was discovered by amateur Martin Hallett, who printed his own dodecahedron using a 3D printer, and toyed around with it. Then saw how it worked. And knitted some gloves with it. Here is some background, and again the video, showing how it works (it’s incredibly simple, and from a design perspective, a brilliant invention).
No feature common across all specimens is improbable on this theory, but in fact exactly expected. Yet every single one is in fact very improbable on any other theory but this one. The odd geographical distribution is also highly probable. So here the probabilities are reversed: P(e|h) is close to 100% and P(e|~h) is pretty close to zero for any alternative, making a ratio of at least 100 to 1 if not thousands to one, in favor of this theory over all contenders.
We’ve been sitting on super-clever, pagan-invented glove knitting templates for decades, and never knew it.
Update: Or not. There are still some sticking points with this theory. See the comments below for discussion of the issues. The same logic applied to the other solutions could dial back the probability of this solution.