In my OP I did expect the fat coin to be a fair d3. Master Blaster did a good job of explaining why a fat coin could not be a fair d3. After that I insisted in the best approximation I could get with the fat coin anyways. The other better d3s are well known to me and I am not particularly interested in them at this time.
If it matters, the scenario I was trying to solve was having to roll a d3 in a lost-in-the-woods situation, where you have some cans among your rations. This is why I can take the floor to be soft and not given to the can bouncing and rolling. Anyways, the question is more academic than practical.
So, are we agreeing that d/h=1.5 is close enough? This sounds like a fairly easy to find can model. Time to go to the supermarket with a measuring tape.
Sorry for resurrecting a zombie thread but I was digging through my old video collection and I realized I meant to upload a video of my simulation but never did.
I don’t see why a tuna can shape couldn’t be fair. I see that it’s not a straightforward calculation. But couldn’t you experimentally flip a bunch of different tuna cans 10,000 times and see what shapes empirically gives you the right results?
Yes, but the empirically correct result would depend on things like what surface you’re flipping them onto. The optimum shape would be different for a hard surface or a bouncy one or a squishy one.
This made me wonder about the fairness of ten sided dice. Are they really as likely to roll onto any adjacent face, or are they less likely to switch from a “north” face onto a “south” face?
You might be able to come up with a way to roll it so that it would tend to land either north or south. But with a regular die you can do this too. If you drop it without rolling, with the six on top, you can get it to roll sixes more often, just from the cases when it lands flat and doesn’t roll.
If you shake it loosely in your hand before rolling, the tendency to switch from north to south, and from south to north, will cancel out.
It is trivial to make fair even sided die. Construct 2 n-sided pyramids and glue the bases together. The picture on the left here seems to show a related 10-sided die. The bases are not aligned, so some rounding has been done to the edges. An less rounded version is shown on the left but note the sides are not flat.
To get an odd sided fair die, construct one with twice as many sides and pair up sides. Whole lot easier, and more accurate, than futzing with tuna cans.
It all comes down to symmetry. Can you rotate/flip/etc. the die from one face up to another face up so that it looks exactly the same except for labelling?
The skew of the two pyramids is so that there’s a flat face on top when it lands, instead of an edge. This does not require rounded edges or curved faces; what it does require is non-triangular faces (they come out kite-shaped).
