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JamesCarroll
11-28-2003, 04:05 PM
Would it be possible to create dice with n sides or n equally probable outcomes?

Is there any interesting math that goes along with it?

Thanks!

Matchka
11-28-2003, 04:08 PM
Sheeesh...and I was impressed with the D&D 30-sider.

Mac Guffin
11-28-2003, 04:15 PM
I have a friend that had a 100 sider.

It's about the size of a raquette ball.

I've never seen it used to roll percentage, but I have seen it used as a threating weapon.

wolfstu
11-28-2003, 04:31 PM
Using natural numbers, (The only numbers for which I can see this making much useful sense)

n = 1 : A sphere
n = 2 : Can be very closely approximated by a disc of finite thickness with both faces of equal diameter and the edge so contrived as to minimize the probability of the disc falling on its edge. You could round it off, for example.

n = 3 : Uh... yeah, you could make something shaped like a... three-sided lemon. Like, take three wires of equal length, tie them together at their ends, and then bow each of them outward an equal amount. Now you've got the frame of the shape I'm imagining.

For larger numbers, just deform polyhedra like this. Take the polyhedron with the fewest regular sides greater than n, and deform some of the sides to make them impossible to land on. My lemonoid can be thought of as a deformation of a... well, not of a regular polyhedron, but of a triangular prism, I guess, with the areas of the ends reduced to zero (and the middle cross-section maintained at a finite value greater than zero).

This should be possible, but they would have surprising shapes, and care would have to be taken to insure that you don't interefere with the 'landability' of one side when you deform another. Or that you create supplemetal landing modes.

As an example of this last, if I deformed three sides of a dodecahedron by extending them outward with antennae, the die could come to rest on the three antennae instead of on a face.

Some numbers might prove challenging, as to keep the probabilites of the faces equal, you have to keep the mass and volume distribution rotationally symmetrical, I'd think.

On the other hand, you could generalise the lemonoid example for any number of sides, though eventuaally you have to make it bigger or the sides are so small that it practically becomes round.

Really Not All That Bright
11-28-2003, 04:37 PM
your n=2: the disc with a fat edge (at least that's the image i got) would really be n=3, wouldn't it?

If the edge is one side, one face is another side, and the other face is a third side.

Any planar figure which doesn't balance on its edge, though, would work for n=2.

Bryan Ekers
11-28-2003, 04:46 PM
Originally posted by wolfstu
n = 2 : Can be very closely approximated by a disc of finite thickness with both faces of equal diameter and the edge so contrived as to minimize the probability of the disc falling on its edge. You could round it off, for example.

Or just toss a coin.

KP
11-28-2003, 05:20 PM
Actually, the thickess of the "two-sided die" isn't very important. It can be quite thick, as long as the *edge* is exceedingly thin. Taper the to edge to a knife-edge or round it off, to eliminate on-edge balancing, then confirm symmetrical 50-50 outcomes afterwards (asymmetric rounding could slightly skew the die)

Of course, when it comes to "unintended landing modes", a hard sharp disc has the potential to make its own. It could embed itself in a wood table (or, given the reported use of 100-sided dice as weapons of intimidation, embed in an opponent's forehead, like a round shuriken)

KP
11-28-2003, 05:32 PM
I also meant to note that simply having equal area faces (or other simplifications) might not assure equal outcomes. In any shape that is not uniformly convex (i.e. not rotationally symmetric around all vertices) or does not have all sides the same shape, the landing modes may be skewed in unexpected ways. Such shapes are surprisingly common. A soccer ball, for example, has hexagonal (white) and pentagonal (black) faces. A buckyball (C-60 buckministerfullerene) molecule shares this configuration, though its faces are not obviously conveniently colored.

Most values of N won't correspond to a physically possible convex regular polygon. I suspect that empirical or computer-simulated adjustment of the exact area of the sides could produce the desired uniform outcome, but it would be nontrivial, and manufacture of such shapes to the desired accuracy would also not be trivial, especially at high N. You couldn't use cheap plastic casting, for example; normally negligible variations would noticeably skew die outcomes.

LSLGuy
11-28-2003, 05:45 PM
JC,

Only certain N-sided regular convex poyhedra are possible. 3 isn't one of them and as the numbers get larger they also get farther apart. I don't have a ready reference to which are and which aren't possible.

If you're trying to make a die, normally you need polyhedra to be both regular and convex in order for the probability of each side to be equal (up to the limits of your manufacturing precision).

Most non-regular objects, such as those which folks proposed above which have certain faces disallowed by bumps or rods, will in fact have non-uniform probabilities. Faces adjacent to disallowed faces will be more likely to end up as the base when it stops moving versus faces not adjacent to disallowed faces.

Although as wolfstu pointed out with his 3-sided gizmo, for some shapes you can come up with a shape that's 3D non-symetrical but still has the desired properties.

In fact, you can create an N-sided die for arbitrary N by simply making an bar-like extrusion whose cross-section is a 2D N-gon.

Some items of Toblerone brand chocolate come packaged in a 8"-long, 1"-sided triangular tube. That's a model of the 3-die. ordinary metal bar stock or a 2x2 board is a 4-die, a coupling nut like this (http://www.imperialinc.com/items.asp?item=0713310) can be a 6-die, etc.

The difficult with the extrusion-die is you're only using 2 of the 3 available dimensions to encode the values and so quickly it becomes difficult to manufacture. For example, a 36-gon with 1/4" wide faces will be about 3" in diameter. Pretty unwieldy. But if you reduce it to 1" in diameter, each face will be very narrow and results won' tbe reliable unless your rolling surface is very very smooth.

And with enough sides (100-gon anyone), it becomes difficult to even determine which face is really the up face. Perhaps you could roll it on a glass coffee table and read the bottom face as the winner by looking up through the glass from below.

Just some silly musings here, not meant as a deep treatment of regular n-hedra theory.

wolfstu
11-28-2003, 05:55 PM
Originally posted by Bryan Ekers
Or just toss a coin. Well, right, that's what I was saying. :) But the point is that some coins have a small chance of falling on their edge. A carefully-dropped Canadian nickel, for example, can fall onto its relatively thick, smooth edge, and not always fall over. We want to ensure this doesn't happen. KP has suggested some ways.

Polycarp
11-28-2003, 06:11 PM
There are, I believe, only five regular polyhedra: the tetrahedron, the cube, the octohedron, the dodecahedron, and the icosahedron (4, 6, 8, 12, and 20 identical sides, respectively, IIRC).

If I'm understanding the statements above correctly, people are suggesting that a properly engineered, say, 15-sided solid can be so designed that the odds of it coming to rest with each side up are 1:15 for each side, even though the sides will not be identical. Is that correct?

11-28-2003, 06:17 PM
Originally posted by LSLGuy
Only certain N-sided regular convex poyhedra are possible. 3 isn't one of them and as the numbers get larger they also get farther apart. I don't have a ready reference to which are and which aren't possible.How do you define regular here? The answer I was expecting was the platonic solids, but you don't seem to be thinking of these, am I right?In fact, you can create an N-sided die for arbitrary N by simply making an bar-like extrusion whose cross-section is a 2D N-gon. Actually, that's a really good idea. Anyone see any flaws? It might get unweildy for 100 sides, but so would anything else.

(BTW the rest of your post was I thought a good explanation of what can and can't be done; I just cut it because I had no specific reply.)

11-28-2003, 06:29 PM
Here's an idea: divide a sphere surface into n (convex?) 'polygons' with great arcs. Join all the corners with straight lines, and the lines to form faces. If you throw it straight up in the air spinning, would the chance of the bottommost point when it reaches the floor being on a face be the same as of being in the corresponding polygon the original sphere? And if so would it always settle on a face if a point on that face was downmost (on a line btw the centre and the floor)?

Bryan Ekers
11-28-2003, 06:31 PM
Heck, have a series of coins with a binary arrangement. The first coin represents the first bit, etc. With four coins, you effectively have a D16. If all you really want is a D13, just "reroll" any results that get you 14 or better.

Of course, it might be easier to just use two D10s to represent any number from 00 to 99, and ignore any value outside your accepted range.

TJdude825
11-28-2003, 07:37 PM
Originally posted by wolfstu
n = 2 : Can be very closely approximated by a disc of finite thickness with both faces of equal diameter and the edge so contrived as to minimize the probability of the disc falling on its edge. You could round it off, for example. I love this board. Only here would someone take this many words to say "flip a coin." :D

I realize this is more of a mathematical diversion than a practical necessity, but the easiest way is to use a computer simulation, or better yet, just don't count one side. (For example, roll a regular 6-sider, and call a 6 "roll again.")

Also, I've seen things where there's one die inside of another, and the outer one is transparent so you can read both of them. If the inner die has m sides and the outer has n sides, you can throw this combination-die to get mn different possibilities. This isn't inherently different from rolling two separate dice, but it's contained in one object.

ultrafilter
11-28-2003, 07:44 PM
Take a computer that can generate random numbers in the interval [0, 1]. Let the output of that be r. The integer part of nr + 1 is equally likely to be any of 1, ..., n.

KP
11-28-2003, 07:47 PM

Consider, for example, the minimum radius from the center to any point on a given face. Unless all faces are identical, some will have smaller minima than others, and the entire die will have to lift itself up a larger gravity (potential energy) well to roll onto another face from this one. Every die must dissipate a final block of evergy at the end of the roll, because it doesn't have enough energy to make it over the next edge in its direction of motion (momentum)

Rolling is inevitable in the terminal stage of a real-world die toss. If there were no possibility of rolling, there would be less assurance that a die could not skillfully thrown to land on a certain face.

Also consider the effect of adacent sides: in cases where the faces are not identical, it might be easier to roll between one pair of two adjacent sides than another pair. If several such simple side pairs were adjacent or arranged in a path, they would form a 'sweet spot' that would further skew the die's roll toward preferred sides.

During a roll, each edge crossing that isn't perfectly perpendicular to the edge exerts a lateral force that displaces the center of gravity to the side. This takes energy. Energetically favorable transitions can steer the die towards or away from certain faces. Even with symmetric dies, like cubes or tetrahedra, most of the initial rolling is not perpendicular to an edge, and the final few edge transistions are close to perpendicular, because there isn't enough energy to deviate much from the least-energy transfer path

However, your suggestion did give me an idea for a trivial, infinitely variable die (infinitely variable meaning that you can ceate an indefinite number of regions, each with a known (equal or unequal) probability of coming up:

Simply paint a sphere in (numbered?) regions of different colors, and read the color off the uppermost point. This is not as easy to read as a flat-faced die, and it would take much longer to come to a rest, but it could work in principle with an optical reader device.

Derleth
11-28-2003, 07:55 PM
On the algorithmic generation of random numbers:

It's impossible, folks. If we could do it without an external source of entropy, encryption would be orders of magnitude more practical and tougher to crack. Any algorithm you use can do, at best, a long stream of pseudorandom numbers before it begins to repeat itself. It will repeat itself endlessly as long as you don't re-salt the algorithm (that is, it works on a cycle of a fixed, finite length). `Salting' is the process of adding random perturbation to an otherwise deterministic system. It implies the existence of a source of entropy.

The only way to do it is, as I've said, to find an input stream that is truly random. People have used everything from noisy diodes to radioactive particles to lava lamps as their source: Any source of randomness you can digitize is acceptable here. Good entropy is surprisingly difficult to find: Quantum effects are the best way to go, as expressed in the noisy diode and the radioisotope, or the Brownian motion of the lava lamp. Security concerns dictate that you can't use a source others can tap into, but perhaps that isn't as essential in this environment. (As a crypto-geek, however, it's a concern that is etched into my mindset. ;))

JamesCarroll
11-28-2003, 07:56 PM
What if you started with a sphere and drew n lines from the center to the surface such that they are evenly spaced. Use that intersection as the center and expand each circle on the surface until it touches its neighbor. Flatten the circles leaving the other areas round. There would be "non number" areas, but since they are round the die couldn't settle on them.

But then since when two circles overlap you can draw a straightline between their intersections, couldn't you keep expanding to the point where the "non number" area is the smallest and use the straightline to create an edge?

PS: You see this is why people like me should just stare at the wonders of USA Today's pie charts...

Orbifold
11-28-2003, 11:41 PM
For any even N greater than four, you could use an (N/2)-sided dipyramid. That is, two (N/2)-sided pyramids glued together along their bases.

And you can handle odd N greater than 1 if you don't mind reusing labels on the faces: just construct a 2N-die and have two faces labelled "1", two faces labelled "2", and so forth.

Crook'd Dope
11-29-2003, 12:57 AM
Roulette and Bob Barker solved this problem long ago (analogously to LSL's crosssection approach above). As the manufacturing is stressed you just make the things bigger :).

My personal preference would be to glue a n-sided teepee onto a hemisphere or cone and spin the thing.

Mac Guffin
11-29-2003, 05:51 AM
Originally posted by JamesCarroll
What if you started with a sphere and drew n lines from the center to the surface such that they are evenly spaced. Use that intersection as the center and expand each circle on the surface until it touches its neighbor. Flatten the circles leaving the other areas round. There would be "non number" areas, but since they are round the die couldn't settle on them.

But then since when two circles overlap you can draw a straightline between their intersections, couldn't you keep expanding to the point where the "non number" area is the smallest and use the straightline to create an edge?

this pretty much exactly describes the 100 sider I saw.