Wikipedia:Reference desk/Archives/Mathematics/2013 August 13

Welcome to the Wikipedia Mathematics Reference Desk Archives
The page you are currently viewing is an archive page. While you can leave answers for any questions shown below, please ask new questions on one of the current reference desk pages.

August 13

Odd sided fair die?

I was looking at a equilateral triangular prism today and was wondering if there is any way to *calculate* the ratio between the length of the edges of the equilateral triangles vs the length of the edges *connecting* the triangles so that the resultant prism would have an equal chance of ending up on any of the 5 sides. (If that ratio is too small, the three rectangular faces are more likely to come up, if too large the two triangular faces are.) Similarly any chance of determining the relative side lengths in a square pyramid so it would be fair die. I know that a functional 5 sided die may be made by numbering the sides of a Pentagonal trapezohedron twice 1-5, but that's not what I'm looking for. If it isn't possible for 5, is there *any* solid shape with an odd number of faces where each side is equally likely to show up if thrown?Naraht (talk) 02:46, 13 August 2013 (UTC)

Well, it occurs to me that you can do it easily, if you're willing to allow curved sides. For a fair three-sided die, take an equilateral triangle, and position a point directly above the center, and another the same distance directly below the center. Now three points define a circular arc, so draw one each defined by each vertex of the triangle combined with the top and bottom points. Fill in the (curved) faces in the obvious way. --Trovatore (talk) 02:54, 13 August 2013 (UTC)

Re the three sided prism idea, I actually did an experiment similar to this a long time ago and mathematical computations are not enough. You have to take into account the amount of kinetic energy the die has when in lands, speed of rotation, elasticity of material it's made of, friction, etc. All these factors balance out with a regular solid but make the computation unreasonably complex with an irregular one.--RDBury (talk) 03:45, 13 August 2013 (UTC)

I have an alternative: Make it a long, pentagonal prism, and roll it such that there's no chance of it landing on either end. (A long, pentagonal pyramid would also work, although it would tend to roll in a circle.) StuRat (talk) 10:15, 13 August 2013 (UTC)

A pentagonal teetotum? Sławomir Biały (talk) 11:41, 13 August 2013 (UTC)

I've seen a game with a small number of die that can be used for 1:2, 1:3, 1:4, 1:5, 1:6, missing a few values up to 1:20. In some cases, count pairs of faces as the same (e.g. the dice with 20 faces had two 10-faced pyramids base-to-base, and pairs of faces could be used to get 1:10). The key principle is symmetry (which is missing in the OP's suggestion), but as per the suggestions above, this is easy to achieve. A prismatic rod with rounded ends will work for 3 up, but as in the game, combining faces allows shapes to do double duty. A normal six-sided dice can be used for 1:2 and 1:3. Products of the probabilities on a par of faces can also be used. (e.g. 1:8 can be obtained from three rolls of a six-sided dice). — Quondum 11:45, 13 August 2013 (UTC)

OK, none of these quite answer what I was looking for. The faces for the solid I'm looking for would all be flat, so I'm excluding the curved sides and the pentagonal prism-like shapes and pentagonal pyramid-like shapes with the odd side curved so that it can't actually land on there. Any odd number N can be done by double numbering a 2N face Trapezohedron and anything of the form 2^N*3^P*5^Q by double (or more) counting the the regular solids (and in fact cubes and icosahedra are sufficient.) (so 7,11,13,17 & 19 would be the gaps)Naraht (talk) 14:36, 13 August 2013 (UTC)

RDBury, in regards to your comment, does that mean that although a rubberized cube and a solid plastic cube would both give 1/6th chance on each side but that the fair solid plastic square pyramid might be of different height (relative to the square sides) than one made of the the rubberized material?Naraht (talk) 14:36, 13 August 2013 (UTC)

I see it as intuitively obvious that no non-symmetrical polyhedron will give equal probability of face occurrence under all conditions of throwing, so given that no odd-sided regular solid exists and that you reject all suggestions such as an odd-sided cylindrical prism (whose ends would be ignored), then what you ask can't be done.→31.53.1.113 (talk) 18:59, 13 August 2013 (UTC)

Your constraints do not seem to exclude two identical pyramids with base that is a regular polygon, stuck base-to-base. With this, you can get any 1/n ratio that you choose, within practical constraints. — Quondum 15:58, 13 August 2013 (UTC)

The original posting specifically mentioned that I didn't want the double numbering of a Pentagonal trapezohedron , the double numbering of the Pentagonal bipyramid is more or less the same. What I'm looking for specifically is a solid with an *odd* number of faces, not a solid with 2*(2n+1) faces which can be double numbered or a solid with k*(2n+1) sides that can be k-tuple numbered.Naraht (talk) 16:59, 13 August 2013 (UTC)

Add a cylindrical prism connecting the two pyramids. A continuity argument shows that you can guarantee equal probability that any of the sides come up. But this construction only works for multiples of three. I think you can get 1and 2 mod 3 as well, by truncating either or both prisms at the extremal vertex. Sławomir Biały (talk) 17:47, 13 August 2013 (UTC)

I think you can extend this idea to multiples of five. Place a pyramidal frustrum between each of the pyramids and the prism. By continuity arguments, you can match the probability of the prism sides to the probability of the sides of a given size frustrum (regardless of the size of the pyramid). You can then use continuity to adjust the frustrum size (while simultaneously changing the prism size to match the frustrum probabilities) to match the pyramid probabilities. I get the sense that you can continue adding frustra for any odd multiple, but I don't know if there's some limit I'm not seeing. -- 205.175.124.72 (talk) 22:03, 13 August 2013 (UTC)

Actually I thought of a modification of this that should work for all odd cases > 5 (Quondum's example covers all even cases). Glue together two pyramids with the same base but different heights. By truncating the larger pyramid and invoking continuity (in three parameters: the heights of the pyramids and the proportion of the height being truncated) you can ensure an equal probability for all sides to come up, I think. (It's anybody's guess how to do it though.) Sławomir Biały (talk) 23:12, 13 August 2013 (UTC)

Sigh, by a continuity argument, you could do a pyramid or prism with the correct number of sides for any n, odd or even, no need for fustrums, but that doesn't get us any closer to something that it actually is true for. Existance proofs and giving an example are *very* different things.. And to respond to an earlier comment, there are many shapes whose faces are all equivalent that aren't regular solids starting with the trapezohedrons or bi-pyramids. Has anyone determined if there are any odd faced Isohedral figures? — Preceding unsigned comment added by Naraht (talk • contribs) 01:53, 14 August 2013‎

Well, an existence proof is a start. Often in mathematics existence proofs precede actual construction. So you shouldn't dismiss them as you have done. And in any event, these existence proofs give several ideas about how a construction might proceed. So they do in fact get us closer to something that is actually true. Sławomir Biały (talk) 11:23, 14 August 2013 (UTC)

Small OT spelling quibble here — frustum has only one r. I know it looks like it should have a second one, but it doesn't. --Trovatore (talk) 04:46, 14 August 2013 (UTC)

Isn't it fair to assume that the probability that a die will land on some face is proportional to the solid angle of that face as seen from the center of gravity? Bo Jacoby (talk) 05:31, 14 August 2013 (UTC).

As I mentioned, I've tried that and the experiment isn't close to the theory. It wouldn't be too hard to find a prism where the solid angles are equal, but I doubt it would work in practice. Here's an idea, use a regular die and if you roll a six then roll again :) --RDBury (talk) 08:39, 14 August 2013 (UTC)

How many standard deviations away from your observation is the theoretical probability? Bo Jacoby (talk) 09:48, 14 August 2013 (UTC).

Slightly more seriously, how about a 5-sided dreidel? Slightly less seriously, the pentachoron, assuming you don't mind moving to a 4-dimensional universe. To answer what the OP might have been trying to get at, all the face-regular convex polyhedra have been enumerated (Platonic solids, Archimedean duals) and none of them has an odd number of faces. There are several with the number of faces a multiple of 5 though, so, as mentioned above, it's possible to get 5-sided die in effect. This is the way D & D does 10-sided die anyway (icosahedrons) even though regular 10 sided solids exist.--RDBury (talk) 09:43, 14 August 2013 (UTC)

A dreidel won't work, since the faces that meet at the apex of the top have zero probability of coming up. The problem asks for a die so that the probability of all the faces is equal. Sławomir Biały (talk) 11:23, 14 August 2013 (UTC)

Is there a more analytical way of looking at it? The first question is is it it theoretically possible? Could you mark a uniform ball with an odd number of regions of equal size. As there is a uniform probability of resting on any point it would imply an equal chance of landing on any region.

The Gömböc is interesting, as it has only one stable equilibria it could be considered a one-sided die. I'm wondering if results like Four-vertex theorem have bearing on the theoretical possibility.--Salix (talk): 10:13, 14 August 2013 (UTC)

"One equilibria"? --Trovatore (talk) 19:26, 14 August 2013 (UTC)

Actually the Pentachoron wasn't something I had considered. The n-simplex will work for this in any even dimention. (so the 6-simplex as well). Also, I'm not sure that face-regular convex polyhedra is what was meant, that redirects to the Johnson Solids and there are certainly Johnson Solids with an odd number of faces (square pyramid for example). Naraht (talk) 20:07, 14 August 2013 (UTC)

latex question

(maths desk? or is this computing?). Question moved to computing desk. Robinh (talk) 19:58, 13 August 2013 (UTC)

IMHO this would be much better suited to the Computing desk. Rojomoke (talk) 16:40, 13 August 2013 (UTC)

OK I'll remove this and ask there. thanks, Robinh (talk) 19:58, 13 August 2013 (UTC)