Three-Sided Die

Having nothing better to do with my time, I was wondering recently about how one would fashion a three-sided die (an unnecessary problem, as one can trivially assign two sides of a six-sided die to each of the three outcomes). The most obvious shape that occurred to me was a cylinder (essentially an elongated coin), though there are more exotic possibilities. The question is: What dimensions should that cylinder assume? Specifically, what should the ratio of the diameter of each (equally sized) face be to the length of curved side so that it is equally probable that when the die is cast the die lands on one face, the other, or the curved side? (Uniform material composition of the die is, of course, assumed.)

My preliminary thoughts were that the length of the diameter should be twice the length of the curved side. If you put such a cylindrical die in a sphere (or, in 2-D, a rectangular die in a circle) there would be, I believe, an equal chance of the die tending to fall *towards* one face, the other face, or the curved side, if the die were randomly placed in any of an infinity of possible positions. However, as intuition suggests, this does not solve the problem: the die is in a state of less stable equilibrium on its curved side, and may even always fall over with minimal lateral momentum. Oddly, a ratio of 1:1 seems about right. Can any more mathematically gifted site member demonstrate what the ratio must be? (There could be a big market among Dungeons and Dragons enthusiasts!)

Aiden

I just thought of a simple alternative possibility suggested by the icon for posers and puzzles itself, even if it is a bit of a cheat. You take a triangle and elongate it so that it has three rectangular sides. You then place a hemisphere within the two remaining triangular sides, jutting out so that it makes the die fall on to one of the rectangular sides.

Aiden

Originally posted by Pawnokeyhole
I just thought of a simple alternative possibility suggested by the icon for posers and puzzles itself, even if it is a bit of a cheat. You take a triangle and elongate it so that it has three rectangular sides. You then place a hemisphere within the two remaining triangular sides, jutting out so that it makes the die fall on to one of the rectangular sides.

Aiden

Originally posted by iamatiger
That's a better plan. The tubular dice would have a tendency to roll off the table! It does make me think of an interesting problem though - is there any three faced object whose three faces all have exactly the same shape? The faces would need to be curved of course... (I'm considering a sphere to be a one-faced object)

Take a solid sphere. Draw three lines from top to bottom such that you get three equalsized/equalshaped parts. Then remove as much of the spere as posible without removing the lines and by keeping it convex (not realy nesesary). et voila. there's your dice.

Originally posted by Fiathahel
Take a solid sphere. Draw three lines from top to bottom such that you get three equalsized/equalshaped parts. Then remove as much of the spere as posible without removing the lines and by keeping it convex (not realy nesesary). et voila. there's your dice.

You should of course numbe the die as one numbers the 4-sided die

Originally posted by Fiathahel
You should of course numbe the die as one numbers the 4-sided die

Some further comments on the elusive 3-sided die:

First, a disadvantage of the prism-shaped 3-sided die is that, when thrown, two faces are visible, and one invisible. Thus, there is no unique face indicating a single value. Obviously, one can code the two visible for different numbers [(1,2)=1, (2,3)=2, (1,3)=3)], but this is unwieldy.

Second, the cylindrical die will only roll of the table if it ends up on rounded portion. However, once it is on the rounded portion, it has already assumed the value associated with it, and can be potentially intercepted becore it rolls off the table. In any case, the cylindrical die could be safely thrown inside a boxed area.

Third, I am not sure that rounding off the edges of a six-sided die works. This would reduce the area of the adjacent faces, with the result that all edges get faces get reduced. What sort of object would tend to result? I can't mentally imagine it. However, I can sort of imagine linking three strips of paper together, with some curvature at the edges, to make a 3-sided die.

Still, the challenge of determining the dimensions of the cylindrical die stands. I am surprised that the answer is not obvious to one of the very mathematically able members of redhotpawn. Maybe it isn't trivial to work it out after all. (Or maybe the problem is deemed trivial in another way!)

Aiden

Originally posted by Pawnokeyhole
Some further comments on the elusive 3-sided die:

First, a disadvantage of the prism-shaped 3-sided die is that, when thrown, two faces are visible, and one invisible. Thus, there is no unique face indicating a single value. Obvious ...[text shortened]... (Or maybe the problem is deemed trivial in another way!)

Aiden

Originally posted by Phlabibit
You should make one and roll it... do some studies... sounds also like the number will always be on the bottom????

Phla-

Originally posted by Cheshire Cat
LOL!! I think that that would take a lot of free time. Not that I can talk :-) Anyway, this is just an idea: using a triangular prism where the ends are equilateral triangles, connect the edges at the center of the triangles thus making three even sides, albeit curved and the when you roll it, only count the number that lands on the bottom. :-)

I actually have a 4 sided die, 6 sided die (duh), 8 sided die, 10 sided die, 12 sided and 20 sided...

For a three sided one, i'd make a prism.

The cilinder isn't a fair die. If it lands on the biggest surface (bigger surfave=bigger chance or getting it) it'll turn and roll along that side. though when it lands on one of the circular sides it'll flip over and make for the bigger surface again....

Originally posted by TheMaster37
I actually have a 4 sided die, 6 sided die (duh), 8 sided die, 10 sided die, 12 sided and 20 sided...

For a three sided one, i'd make a prism.

The cilinder isn't a fair die. If it lands on the biggest surface (bigger surfave=bigger chance or getting it) it'll turn and roll along that side. though when it lands on one of the circular sides it'll flip over and make for the bigger surface again....

Originally posted by Phlabibit
Prism with even sides.... rounded on both sides, rounded corners to get a good random toss.

What band, what song lyrics are these?

I've got a Dungeon Master's Guide
I've got a 12-sided die
I've got Kitty Pryde
And Nightcrawler too
Waiting there for me
Yes I do, I do