I have to say, when i watched the stand up maths video, i felt like way in which the coins got thrown would make the results inconsistent. I never imagined that almost every single variable would affect the necessary thickness. I imagine this took quite a while to figure. Thanks for satisfying my curiosity.
I looked at the title and thought "a coin pretty much allways lands on the faces, a piece of wire pretty much allways lands on its side. Making the coin thicker influences the likelyhood of it landing on its side, so we can assume we can "select" a value, and there is no reason that we couldn't select equal chances, so the answer must be 'yes' " Turns out, you wanted to find the ratio and not just figure out if it exists...
Using the cube to represent a 3 sided die is probably the best solution, although you probably have to figure out which values should be opposite. Does having the 1s opposite each other while the 2s and 3s are adjacent have any real impact on probabilities? In some ways I would think the answer is no, but at the same time that does mean if you had consistent roles, you might be able to take advantage of that geometry to slightly bias the results. I think the biggest discovery made here unstated is that you could also repurpose the cube to create a 2 sided die. Now you'd just need to use 1s and 2s on different faces. I'm not sure you'd even have the same sort of biases you might have with the 3 sided cube because there's not an alternative layout. Completely fair!
For the 2-sided die there's only two layouts: either all faces with the same value share one corner (i.e. one face is perpendicular to each axis & forms an "L" shape if unrolled), or all faces with the same value are parallel to the same axis (i.e. two opposite sides plus one side to connect them by, forms an "I" shape if unrolled).
@@Stratelier oh... I guess so. Something like a tennis ball or on each corner. Still, with both layouts there isn't a difference between them, but it might not be fair.
0:37 when I was in grade school, textbooks always called dice “number cubes”, I assume due to some moral panic about gambling that was going around (do they still do this?)
I feel like there’s almost certainly a way to do this rigorously. Assuming any angle is equally likely, the chance of it landing on a certain face is the chance of the centre of gravity being above the opposite face.
Great video! Have you seen the d3s that have a cross section almost like the inside if a mercedes emblem? I got some like this for playing DCC and think they're a great solution and kind of keep the spirit of a coin
By intermediate value theorum, for every "throw" (initial velocity and angular momentum or whatever the physics word is), there must be some thickness of coin that gives an equal probability for all three It may be true that it makes it too easy to rig your result by rolling in a certain way though
You could take a standard die and use a sharpie to write 1, 2, 3, to replace 4, 5, 6 and that's the easiest and most fair three side "coin" you could possibly make. Giving you a mathematical 1/3rd rounded down chance on each roll to get 1-3.
For what proportions are all three sides equal? Such that πr²=2πrh? For r=1 to find h, π=2πh, 1=2h, ½=h: therefore, the surface area is equally divided between the three sides when the height of the cylinder is equal to the diameter.
When accounting for the angular momentum, does it make a difference which axis you spin the coin at? I assume the answer is yes but you don't specify. Also, I would set as convention that the coefficient of friction and restitution to be set at 0.5. If we follow those conventions, how does the interplay of initial angle, initial linear velocity, as well as angular velocity along the two relevant axis determine the likelihood of the coin landing on its side? You could write an academic paper on this! Another thing is that an extremely heavy mass, would increase the likelihood of landing on the faces, why is that not the case?
I would have to say that I believe a 3 sided coin has to exist. It might not be physically realistic but as long as the height of the cylinder is right it should be able to be a 1/3 probability of any side.
Hold up, there's a bias here. Of course the side the coin lands on is going to greatly vary depending on spin angle coefficient of friction, and others. Since you're not throwing it randomly every time I bet you're going to see the same type of data when just throwing an actual 2 sided coin with set values as the set value is more likely to land on either one side or the other, or somewhere in between.
@@pressurisedcarbon6238 what do you mean? all three sides are rectangles, and all three are equal in length. like I said, just make it really long, and maybe have the ends rounded or pointed so it cant balance there. you're basically just confining that "die" to a 2d set of movements instead of 3d.
Here's my take. Number the faces 1 and 3, and the edge 2. Now if I'm rolling a d6 usually I don't want to roll specifically a 4. I usually want to roll above or below a 4. Yes there are many cases where you need to roll a specific value, but the preponderant use of rolling dice is to yield a high or low result. If that is accepted, then making the middle value volatile might actually mitigate the variable probability of a side.
The thing is, this isn't how coin flips are done. When you do a coin flip, you don't roll it like a dice. you toss it, it flips around in the air, you catch it. So the perfect 3 sided die would just be one with a side thickness equal to the face diameter
It wouldn't work the way you've described. Tosses can happen anywhere from a perfectly perpendicular to a perfectly parallel. Which means if it flips along the edge, it never lands on a side.
Just because you catch a coin during a flip, doesn't make it standard practice. Plenty of people let it hit the ground... Also, that isn't the point of OP's video...
I thought you were gonna design and make a coin with 3 sides like some kind of 4d klein bottle, turns out it was just probability counting whether or not it can land on the edge 😅
A 3-sided "coin" might not exist, but a 3-sided dice does exist. It takes the form of an ovoid. You can modify the ovoid so that it has 3 ridged faces separated by 3 ridged edges. Those faces will still be rounded, somewhat. (I'm sure there is a name for this object, but I am not aware of it.) The ridge ovoid can't give you unfair rolls. The ovoid shape can be used for n-sides. There is an upper limit as you approach pi. The lower limit is 2, and looks like a coin.
iirc there are actually at least 2 fair 3-sided die. One is a d6 with the faces joined into groups of two, and the adjoining edge being rounded off. The other looks kind of like a purse, and I think it is the one you are talking about... but I had a hard time visualizing it based on your description.
a triangular prism with pointy ends (so that it wont roll onto them) and you throw really high, i'd say thats a fair 3 sided dice but i dont think there is a 3 sided fair coin
I can’t believe I just found this video now, but those simulations were a surprisingly good way of understanding how the factors actually influence the result. I had emailed Matt Parker a few years back and after the video came out, my only real understandings was that it was messy and probably a single value didn’t exist, but not much understanding of why. This I think really helps give a better idea why those things influence the final value. Fantastic video!
I had this idea in the car once. I assumed there exists a value since you can write the probability as a function of length and use intermediate value theorem. Neat to see it in action
I think the problem here is that different way of throwing the coin would result in different result. On a symmetric dice or coin this won't be a problem, but symmetry can't help us here so I don't think it's gonna work
Why not use a triangular prism, but instead of the projection being straight, have the edges pinch into each other on each end of the projection, creating a shape similar to a rugby ball?
Fair dice seems to be a modern invention. E.g. most roman dice were wildly "unfair", having a long side up to 1.5 times longer than the short side. I assume they could have made dice much fairer than that, but didn't want to.
If I'm not mistaken, this was because dice were pieces of natural bone, rather than crafted objects. Also, in Europe, probability wasn't really understood as a mathematical discipline until around the Rennaissance. The book " The Drunkards Walk" goes into the history of western thought on probablity
@@dr.limeade3035 Or it's because they wanted to add more strategy into their games. If every possibility is equally likely, then it is purely random, which means there is no strategy: you just hope you get lucky. But if some possibilities are more likely than others, then you can strategize around that. You can take a risky gamble and bet on a lower probability, or play it safe and bet on the higher probability. This creates a risk vs reward mechanic.
@@SchemingGoldberg There are a couple problems with what you are suggesting is that this dynamic exists in dice games with equal probabilities. Craps gives bigger payouts for more unlikely rolls. Second, a lot of dice games ARE about just luck, especially in a gambling context. Third, probability was not an understood concept in antiquity, and dice games were often viewed in a metaphysical and supernatural context
@@dr.limeade3035 They must have had some understanding of probability. Humans and many other animals even have an intuitive understanding of probability, look up statistical learning. Their understanding of probability was probably as good as that of any average modern human, no pun intended. They might not have any academic understanding of probability, but neither do 99.99% of modern day humans.
Would have been nice if you've had written the value meanings on the respective axes of your graphs. That would have made it a lot easier to understand and less ambiguous
Yup that would work, there's a whole class of dice which do exactly that (they look like 2 cones stuck together), but I'm not sure that would actually be as "fair" as just dividing a 6 sided dice into 3 groups of sides.
One more interesting test would be to repeat all experiments with different numerical precision. (64bit/32bit/16bit/8bit floating point numbers) The size of the time steps in the simulation could also affect the outcome. Testing these could prove the simulation trustworthy or not.
First off, thank you very much for your work on this. Given that the coin flip depends on the precise starting conditions, I think it could be worthwhile exploring the necessary coin thickness when integrating over the range of 'plausible' (human thrown coin conditions? ground is mud, gravel, ...), but different starting parameters. Seems to me, that once you either define a range of starting conditions and assume a prob. distribution (e.g. just constant in the range) for your starting parameters you will get an answer for the coin thickness. Stand up maths assumptions were to calculate the initial point of ground impact (if the coin would be covered by a sphere or circle) and then state that the face the impact point is projected on is where the coin would land finally (always given in understood it correctly, which is not always the case!). These assumptions are quite strong and lack to deal with the involved dissipation, velocities bounciness as you described. I would really appreciate a small follow up or alternatively if you could make the simulation results available for others to crunch the numbers. Thanks again
wow, as someone who loves thinking about these kind of problems, you covered everything I would have thought of to discuss, and some things i never considered, in less than 4 minutes. Phenomenal job!
initial angle, speed, and rotation of the dice should have been simulation parameters because that is where the pseudo-randomness of a dice resides. You would have gotten rid of multiple parameters, and it may as well have influenced your result curves concerning other parameters. The downside is, you should get more uncertain result, or just as uncertain but with more throws. You could try setting all parameters to random and see if the dices converges to a set of parameters. Also, you could implement machine learning and try n times each dice from a set of q dices with randomized parameters. Those with best scores (closer to 1/3 chances) survive to the next generation, made from small randomizations for all parameters of the best dices. Repeat the generation until all dice converges to a set of parameters assuring approximately 1/3rd of a chance for each face (that is just machine learning applied to dice throwing).
Isn’t Unity’s physics system focused towards speed instead of realism? I think blender simulation is slower because it’s much more accurate. Unity’s physics is buggy as hell.
Curiosity box once had a set of dice that contained a different design for a 3 sided dice than the one you used in the video. But then it wouldn't be a coin. Maybe it would be interesting to test that one.
Try doing the calculations again but shoot it way up in the air and at like >5 rotations per second like a real good coin toss and i reckon your graphs would look quite a bit different. These are more like coin drops (which is still interesting, but I think we both know u wanna see a REAL GOOD coin toss)
Thanks for making this video 3 minutes. I saw this video title and was expecting a 30-minute video (and I don't like watching those). Thanks for not being like other channels. Subbed.
video from start to near the end: Maths!!!!!!! SIMULATION! 3 D G R A P H! DDDAAAAAAAAAAATTTTTTTTTTAAAAAAAAAA!!!!!!!!!!!!!! end of video; but a simple, practical solution is have 3 values on a 6 sided die.
With this kind of a result to the simulations, it would be really interesting to repeat them for a known fair die shape. Just for a convenient plausibility check.
2:56 Why this music... Am I insane? Is this just copyright free music that you selected out of nowhere and I'm making this up? Please don't say I'm the only person that this soundtrack triggered something in me.
