7/30: Trials update, over 10,000 trials so far!

Thanks to Justin Smith for generating the majority of this data. And here is a ptr to his 3-sided-coin page.

A fair 3-sided-coin is a coin that is thick enough that it lands on heads, tails, and sides with equal probability. This topic has cropped up a few times in the newsgroups sci.math and sci.physics. On this page, I hope to extend the discussion a bit and give some empirical data.

At first, the key question seems to be: What is the critical ratio of thickness to diameter that makes heads, tails, and sides equiprobable? Experience helps bound the problem a bit: I have never seen (or even heard about) a real coin landing on its edge on a hard surface - so the edge must be too thin. A spool of thread will land on "sides" almost all the time. For a more extreme example, I recall being quite surprised once when a dropped can of Pringles chips landed on one end!

The Possibilities:

Before we go empirical, let's run through a couple possible theoretical arguments for the critical ratio T:D (Thickness to Diameter)

T=D. The key to this argument is observing that one projection of this cylinder is a square - and the square would obviously land on all sides equally.

T=D/4: This is the equal area argument, the area of one of the circular caps is D*D*pi/4, the area of the round side is piDT. Set these equal, and the above relation drops out.

T=D/(2 + sqrt3): This is the equal-angle argument, in one dimension. Project angles from the center of the cylinder - the circular sides must subtend a 60 degree arc, so the other side subtends a 30 degree arc (twice). Crank the trig to get the relation shown.

T=D/(sqrt 8) Equal angle argument in two dimensions: Set the surface area of a spherical cap equal to a third of the surface area of a sphere and you get this relationship

All of the above can be scaled by the ratio of potential energies T/D, or even by T^2/D^2. This gives us a lot of flexibility, we can see what the empirical data is, then find a hypothesis that fits the data best, then argue that that hypothesis is cleanly derived from first principles and present the supporting data later. This is called "science" :-)

But first, some nagging details:

There is a sub-problem to be solved first: how does one flip a 3-sided coin fairly? For a 2-sided coin, it seems pretty easy - but for the 3-sided coin, we have 2 unique axes of rotation: the axis of the cylinder, and an axis orthogonal to that. The standard coin flip imparts angular momentum around only one of these axes, and it is not at all clear that this is a fair flip. Perhaps we need "random" angular velocities about both axes. But now, what is random? Uniformly distributed across some interval? How could we choose the size of the interval? Should it be the same interval for both axes, or should it be scaled by the rotational inertia, or something else entirely? Of course, there are practical limits to the rotational velocities as well.

The surface we flip the coin onto would seem to affect the distribution as well. Is it flat or slightly irregular? Is it hard or soft? Irregular surfaces are capable of tipping a rolling coin from the edge to the face (in fact, if the irregularities are of comparable size to the thickness of the coin, there may be no stable edge positions), and soft surfaces would seem to favor face-landings as well.

The construction of the coin should be considered as well. One possibility was to machine disks, and another was to glue together some washers. But a washer with a central hole has a different rotational inertia than a disk, and it is possible that this could affect the landing distributions.

Finally, it is nice if the landing surface is an infinite plane, since contact with boundaries could also affect the distribution. But, infinite planes are hard to come by (not to mention impractical to use), and my dining room table is only a poor approximation. I wondered how re-rolling the ones that went off the edge affected my data.

What would we do with data if we had some?

If we are sucessful in getting some experimental data we should be able to use it to model the edge-landing probability as a function of the ratio T/D. Obviously if T/D is very small, the edge-landing probability goes to zero. If T/D is large, the probability goes to one. Let's first assume that the probability is a sigmoidal function of T/D. If we get proven wrong with this assumption we can change it later, but it seems like a reasonable first guess.

It was time to go play in the machine shop. We made three coins initially: T/D = .25, T/D = .375, and T/D = .5. We machined the disks out of acrylic plastic, and did thousands of drops. The two larger coins are pictured above. It turns out that the smallest one (T/D = .25) cannot work, we got one side out of 150 trials and gave up on it! Here are the results for the other two coins:

The data on the left is for drops on a hard surface, and the data on the right is for drops onto a rug. The T/D=.5 coin is pretty close to a perfect 3-sided coin already!

We then tried changing the landing surface from hard (hard wood/laminate table top) to soft (rug) and the .5 disk landed on sides 214 out of 800 times (26.7%). This is a bit lower than our hard surface data, weakly confirming the guess that softer landing surfaces favor the lower potential-energy states. Subsequent data from Justin may have indicated that this was just a stistical fluke.

Sigmoid?

Let's see if the fraction of sides as a function of T/D really is sigmoidal. We will use our data for the .5 and .333 disks to fit a sigmoid, then produce a prediction for a .667 disk. We'll test the prediction with a real .667 disk. Of course, if we get a correct prediction, it doesn't prove the sigmoidal relationship - but if the prediction is significantly off, we may be looking for other kinds of relationships.

[Stay tuned]

Variations in the Throw

Stay tuned for more results here. Nothing so far!

Flip This!

If you want to generate some data yourself, I'll send you the disks if you send them back to me, along with your results for at least 2000 flips.