Sunday, September 3, 2017

Tetrahedron Success #2! (Part IV)

Right off the saw!
Late last night I got an email from a reader:

 Make a cube. Label all the corners as A or B so that each edge connects an A with a B. Now draw straight lines connecting the A corners. Saw off the B corners using those lines. You have an accurate tetrahedron without needing to make any length or angle measurements. The edge length of your tetrahedron is 1.414 times the edge length of your cube. You recognize the square root of two there.  


This got me thinking that perhaps there is a good way to lay out the tetrahedron with no measuring at all. Just my cup of tea.

I asked Tim for a sketch, and while I waited I found this on YouTube:
It turns out this is exactly what he was talking about. This second way is a great way because there is no measuring, no math calculations, no angles that aren't on my combination square to worry about.

Right up my alley. I decided to try it out.
Start with a block of wood. Duh!
This way is very simple. Start with a four-square block of wood, and ensure one end is also square. (Five square?) Mark the remaining face with a marking gauge, but don't cut it off yet so your clamp has something to hold.

Then, mark the diagonals on each face as per Tim's instructions.
Then it is just a matter of sawing them out like on the video!
 I left each of the diagonal cuts a bit short so I could continue to hold the block with a regular clamp. Once I crosscut the cube free, I could easily finish the saw cuts I started.
Cross cut

Perfect cube off the saw.

Finishing the saw cuts.
A pretty darned good tetrahedron!
With just a little care, I was able to make this tetrahedron without any trimming with a plane. It is plenty good enough for my purposes. I am excited now to try it on the real thing: ebony.

The only reason I want to use ebony is I happen to have some laying about. I think pretty much any hardwood would do for these dice. I think the heavier and tighter grained, the better. We'll soon see.

What I will do after cutting the ebony out, is polish the tetrahedrons (tetrahedri?) with fine sandpaper. That is, if I can get them as accurate off the saw as I did this pine one. I also expect it to be a bit more challenging because the ebony ones will be smaller.

In addition, I have an idea that I got from another YouTube video for making some appropriate dice from some regular rectangular stock. There's no reason this has to be so complicated, other than I think the tetrahedrons were a more high-end version for the wealthy, just like the original inlaid game board. There is evidence this game was played with knuckle bones from goats.

This can be a simple game that can be made in a hurry. Stay tuned...

Lastly, I would like to give my most humble thanks to both Sylvain and Tim for sharing their knowledge with a simpleton like me. My first instinct was to start whittling on a stick until I came up with something that more or less suited. Both of these methods are highly superior and are capable of repeatable, accurate results. Guys, have a beer on me.


  1. Tim's tetrahedron is not a "regular" tetrahedron,
    it has three ridges of length "a" and three ridges with length "a X 1.418...".
    As a result all vertices do not have the same probability to come on top.
    It is very nice as decorative tetrahedron but not suitable as a dice.

    1. Are you sure? It looks to me like every single edge is the same, as they are all the diagonal of the square in the cube. However, you know much more about these things than I.

  2. Oops
    My apologies to Tim.
    Tim's method is correct but what is shown on the start image of the video is not what Tim says as it uses 3 A vertices and one B vertex instead of the 4 A vertices.

  3. I should have looked at the complete video instead of just the first picture.

    1. Not to worry. I wasn't sure at first how it would work until I tried it.

  4. Ok. All I could find at the local woodcraft(90 miles away from me) were African blackwood pen blanks that are 3/4 in cross section. I got up early this morning and drove there. I'm now on my way back, lol. I think I will need to get home and practice and then get out the very small Zona saw. How big are you making yours? Any thoughts on the white tips? PS, I've now lost my phone to my family who are using it as the shared UR gameboard while we travel alon today. I'm sneaking this comment in while they are inside a store ;) Thanks again for sharing your progress .

    1. Haha! You've totally made my day!

      It looks like my ebony blank is just a shade shy of 3/4". It might be just a bit small, but I think it will work. On a related note, I think Tim's method yields a slightly larger tetrahedron than Sylvain's. you might try your own test, though.

      I think pretty much any hardwood would do. But African Blackwood should be great.

      For the pip, I've been thinking a couple of things. Brass would be good, if you could keep it shiny. Otherwise it will turn black. Aluminum might be good, although non-traditional. Gold really would be the perfect thing. Or ivory, but you won't see me do it. Unless I had some elk teeth or something, but I don't.

      I've also thought of using bamboo, but again, a bit modern. I'll probably use some sycamore, which I have here. I have some ash, but the pores might not look as good.

      Finally, I'll be headed to Dictum in a week or so, and if I haven't done the sycamore thing by then I'll buy a synthetic ivory knife scale. I think that would look best out of the realistic options I have.

      Cheers, and drive safe!

  5. The tetrahedron used in the British Museum video are quite small.
    I wander (once the layout is done) if a juweller/stonecutter method would not be appropriate?
    I would try to do it in bone (which was a dice material before synthetic materials were available).
    Synthetic material [having the same properties in any direction] (and perfect geometry) would ensure correct probability.
    What about a shellac tetrahedron? (don't spill alcohol on it)(not to use plastic in a 3D printer which would be far away from woodworking).

    1. I forgot to mention bone. Or, even horn would be nice.

      Once my dice are done, I plan to test them to ensure they are close to 50/50 white/black.

  6. The game set on display in the British museum is also displayed with flat sticks used as die.

    (Irving L. Finkle : on the Rules for the Royal Game of Ur ~ 2007)
    A paper by Dr Irving Finkle claims they were more prevalent, and in fact only the one game set had the tetrahedral kind of all the sets found.

    The flat sticks were often used for Senet and Hounds and Jackals, though knuckle bones were more often used in the latter.

    1. Nice catch, Larry! I guess I assumed those sticks were just broken off pieces of inlay.

      I did see one photo of a game board hastily scratched on stone, so I would imagine the Game of Ur was simple to set up and you would be expected to come up with your own board. The special one was found in the burial tomb of a rich princess, so I imagine it wasn't the norm.