The mysterious dodecahedrons of the Roman Empire
I think the “proof of metalsmith skill” theory is most likely. Perhaps slightly higher than simply journeyman&#x2F;apprentice finishing project; since the particular difficulties lie more in geometry than metalworking, this might be a kind of “Geometer’s Guild” entrance exam piece. It reminds me a little of the more modern “Turner’s Cube”, a challenge of machining skill given to apprentice turners not too long ago (<a href="https:&#x2F;&#x2F;makezine.com&#x2F;article&#x2F;digital-fabrication&#x2F;3d-printing-workshop&#x2F;the-turners-cube-classic-machinists&#x2F;" rel="nofollow">https:&#x2F;&#x2F;makezine.com&#x2F;article&#x2F;digital-fabrication&#x2F;3d-printing...</a>).<p>The process to make one of these dodecahedrons involves geometrically constructing a pentagon, forming it into a mold, casting from that mold 12 times, centre-finding either the mold or each pentagon, forming a centered circular hole on each given center, joining the 12 pentagons at precise angles, casting 20 spheres, and joining one sphere to each vertex.<p>The ancient Greek mathematicians were fascinated with compass-and-straightedge constructions (<a href="https:&#x2F;&#x2F;en.m.wikipedia.org&#x2F;wiki&#x2F;Straightedge_and_compass_construction" rel="nofollow">https:&#x2F;&#x2F;en.m.wikipedia.org&#x2F;wiki&#x2F;Straightedge_and_compass_con...</a>), and posed (and sometimes solved) many such constructions, particularly constructions of regular polygons (<a href="https:&#x2F;&#x2F;en.m.wikipedia.org&#x2F;wiki&#x2F;Constructible_polygon" rel="nofollow">https:&#x2F;&#x2F;en.m.wikipedia.org&#x2F;wiki&#x2F;Constructible_polygon</a>). Constructing a right angle from a line is almost the most basic operation, and repeating that two more times constructs a square, so squares are trivial to construct (<i>4 sides, trivial</i>). The process for an equilateral triangle is not quite trivial, but it is simple (<a href="https:&#x2F;&#x2F;en.m.wikipedia.org&#x2F;wiki&#x2F;Equilateral_triangle" rel="nofollow">https:&#x2F;&#x2F;en.m.wikipedia.org&#x2F;wiki&#x2F;Equilateral_triangle</a>) - simple enough that its proof is given at the beginning of the first book of Euclid’s Elements (<i>3 sides, simple</i>). Doubling the sides of a given polygon is also simple, so from a triangle you can get a hexagon (<i>6 sides, trivial</i>), and from a square you can get an octagon (<i>8 sides, trivial</i>).<p>That’s 3, 4, 6, 8 - what about 5 and 7? For constructing the heptagon of 7 sides, the Greeks considered this problem to be “obstinate”; I believe Gauss was the first to prove it is actually impossible.<p>For 5, the regular pentagon, constructions using just a straightedge and compass were given by Euclid in Elements and by Ptolemy half a millennia layer, but they are certainly not simple (here is an animation of one such method <a href="https:&#x2F;&#x2F;www.mathsisfun.com&#x2F;geometry&#x2F;&#x2F;construct-pentagon.html" rel="nofollow">https:&#x2F;&#x2F;www.mathsisfun.com&#x2F;geometry&#x2F;&#x2F;construct-pentagon.html</a>).<p>The full list is something like<p>3: simple<p>4: trivial<p>5: hard<p>6: trivial, just double a triangle<p>7: obstinate, later shown impossible<p>8: trivial, just double a square<p>9: obstinate, later shown possible but extremely hard<p>10: trivial, just double a pentagon<p>11: obstinate, later proven impossible<p>12: trivial, just double a hexagon<p>So among the first several regular polygons, the pentagon is by far the hardest to construct that wasn’t obstinate&#x2F;impossible at the time. This makes it the <i>perfect</i> shape to test a geometer’s skill; they would have had to understand the hardest shape in the first book of Euclid’s Elements.<p>(Centre-finding on a pentagon is also somewhat complicated relative to the other shapes, though not by as much as construction: <a href="https:&#x2F;&#x2F;youtu.be&#x2F;RxYfdNrt4Sw" rel="nofollow">https:&#x2F;&#x2F;youtu.be&#x2F;RxYfdNrt4Sw</a>)<p>Next we move to the dodecahedron. There are only five Platonic solids, believed at the time to have special properties such as being the building blocks of all matter. Three of these are built from triangles. Of the other two, one is the cube built from squares, and the other is the dodecahedron built from pentagons. The cube is fairly trivial and triangles can build multiple Platonic solids, but the dodecahedron feels more unique: it can only be built from pentagons, pentagons only build it, and pentagons are the hardest shape that build a Platonic solid. I suspect this synchronicity would have inspired exactly the kind of awe that a master craftsman&#x2F;geometer would have liked to command.<p>Why not stop at the dodecahedron? Making and attaching the spheres to each vertex demands another kind of metalsmithing skill, that of forming spheres, which is largely unrelated to all the constructing of polygons done before. It also makes the object testable by non-geometers: place it on any flat-ish surface (it doesn’t need to be perfectly flat as long as you always have the base spheres in contact with the same five points) and the sun passing through the two differently sized holes will form a specific shape, a lens (sort of the like the shape of an eye, the shape made when two circles intersect a bit). Each lens would be a different size but the same shape, you could visually confirm they were all the same, and more rigorously test the accuracy with just a compass.<p>Essentially, this object would let you prove to a smart non-geometer (such as a property owner looking to hire an architect to oversee building a house, or a factory owner looking to prove you could manufacture precise and repeatable objects…) that you had excellent geometry skills, and likely that you had read and mastered at least several parts of Euclid’s Elements (the first major “textbook”, widely considered one of the most important and influential books ever, by far the most logically rigorous [important mindset for engineers…] text around and not surpassed in that regard until the 15th century or so, etc…).<p>If it was made of metal, it also demonstrated important metalworking skills like accurate mold forming, repeatable casting from molds, and joining at precise angles - since errors in metalsmithing at almost any stage would have deformed the resulting verifiable lens shape sunlight makes when passing through it. If it was made of stone, it would demonstrate the same geometry skill and similar skill in stone-working; if it was made of wood, it probably wouldn’t survive to be found as a historical artifact. On top of all of that, it lies at the confluence of solutions to several hard problems in contemporary geometry at the time <i>and</i> it happened to have the same number of sides as there were Gods in the pantheon <i>and</i> it invoked the (religiously-imbued) Sun in verifying its accuracy, all things which would have carried cultural importance and increased the standing and respect of geometers in society.<p>The calendar theories are certainly interesting as well, but if it was used for such utilitarian and important purposes, there should be correspondingly more utilitarian examples of this object, since there were so many farmers.<p>The biggest point against my theory is that if it were true, it absolutely demands that these objects would bear a maker’s mark, and they don’t seem to have those. I don’t know how to resolve this fatal objection; I could say it was considered heretical to mark (a representation of) a Platonic solid with a mere human’s name, but this seems a meager excuse.