Roman Dodecahedrons Part II

This is a continuation of part I on Roman Dodecahedrons.

Games on a Dodecahedron:

So I’ll go on to explain why I think it’s a string puzzle, but first I want to mention what my criteria for an acceptable answer is. To my mind, these are the observed things that need explanation,

  • The dodecahedra are (fairly) widely produced, over a wide geographical area
  • They are fairly expensive to make, both the bronze cost and fabrication time would make them more than just a disposable item.

So if it’s a toy, or puzzle, it must be one that is sufficiently engaging. Any puzzle which is too easy or too hard is not a good candidate, since it wouldn’t explain the observed popularity of the devices. And considering that the owner likely spent a reasonable sum of money on purchasing their own, then it’s probable that it’s a device they’d wish to use again and again. (Otherwise, you’d just wait till your friend got bored and borrow theirs)

So, my test for success with evaluating possible game ideas, is basically whether it’s the sort of toy that gets played with only on Christmas morning, or afterwards as well.

Back to the dodecahedron itself. The first thing that occurred to me is that the dodecahedron’s vertices form a graph, and that you can wind the string to traverse the graph.

https://en.wikipedia.org/wiki/Hamiltonian_path#/media/File:Hamiltonian_path_3d.svg

Variations on the game include:

  • Visit all cities exactly once, no crossing over allowed. (Equivalent to a Hamiltonian Path)
  • Visit all cities once, but return to where you started (Equivalent to a Hamiltonian Cycle)

Example of a Hamiltonian Cycle

And I play-tested them and found it nicely challenging, although once you figure out the algorithm it becomes a little more straightforward to solve, and you can just reuse the same solution after a while.

What suddenly makes it more interesting is a two-player variant. Player A makes the first five moves, then hands it to player B, who has to complete it. The randomness of the start shakes it up nicely, since you can’t just use your remembered solution, but instead have to think it through each time.

A multiplayer option is also made possible by the string, the first player can make the first five moves, finishing by making a knot around the final nodule. Player B then can attempt the puzzle, (perhaps while being timed), and when finished they restore it to the original state just by suspending it by the string and letting it unwrap until it hits the knot. Then player C can proceed, etc. In this way you could have a lineup of people all given the same puzzle, and see who is the fastest.

(I could imagine this being used as an impromptu skill test, or as a way to see how people deal with unusual situations. It’s not exactly the world’s best management metric, but compared to contemporary tests of character, such as apparently killing people that didn’t understand obscure riddles, it’s practically objective!).

And having played around with it, I can say it’s quite a satisfying object. Both the game itself is reasonably challenging, and the act of winding it around the posts is satisfying and somewhat meditative.

Hard Mode – Double Eulerian Walk:

I’m sure there’s lots of games you can play with the dodecahedron, but one I had fun working out was this. Use every road exactly twice, and end up back at home. (I don’t know if there’s an official math term for it, but for the sake of argument I’m going to call it the Double Eulerian Walk).

This gives a very pleasing pattern with the string, and is a much harder game than the 2 player Hamiltonian Cycle.

‘Double Eulerian Walk’ on a Roman Dodecahedron

At this point I was now convinced that the Roman dodecahedron is a string toy, and that gameplay involved one or more variations on the graph traversals. I started looking around on the web to see if anyone had previously had the same idea. And I was delighted to find this article here by David Singmaster:

https://www.comap.com/product/samples/UMAP_37_4.pdf

Which makes the connection between the Roman dodecahedra and a game actually invented by Sir William Rowan Hamilton himself in 1857:

http://puzzlemuseum.com/month/picm02/200207icosian.htm

(Adorably, there’s not just a tabletop version, but he also invented the travel version as well).

More info in a paper here.

 

Making the dodecahedron:

I went through several designs. First I started with an acrylic sphere I had laying around, and tried marking out the vertices on it evenly. That wasn’t too easy, so I worked out the diameter as a circle that would just enclose the pentagonal faces, and then lasercut a wooden template of the same size. In my mind this was a simple matter of doing a geometrical construction in the style of Euclid, but on a sphere. That didn’t turn out as easy as I expected either, and a lot of the vertices ended up kind of squashed. Sigh.

I’m amazed how much the ancient Greeks figured out, when they didn’t have any lasercutters at all.

I then decided to try printing another set of vertices from my lasercut icosahedron/Dymaxion maps, but that would have taken several hours, so while the printer was running I kept on working on designs for the lasercutter.

I tried a cable tied verted model, A snap-together model, a flat model, and one other one I didn’t bother to get a photo of.

Roman dodecahedron prototypes v01

The bag of rejected prototypes

And finally settled on this version, which most closely resembles its Roman ancestors.

 

Files up here for anyone that wants to make their own:

https://www.thingiverse.com/thing:4463142

 

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1 Response to Roman Dodecahedrons Part II

  1. Pingback: Roman Dodecahedrons Part I | Tinkerings

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