I love this guy's name. It's like the name someone would have in a medieval fantasy story. "Quick, m'lord! We must reach the king's statistician Persi Diaconis before sundown or all hope is lost! He's the only one who knows how to make a fair die out of non-regular polygons!"
It is rumoured that the legendary artificer, Persi Diaconis of the Order of the Logician, once created an object that appeared to be a normal dice. In truth, this dice was filled with magic, and power, and quite a lot of hatred, and its roll would influence the very fate of the world...
For anyone interested, here are the names of the shapes shown at 7:20 Left to right, then top to bottom; cube/hexahedron, octahedron, pentagonal hexecontahedron, pentagonal icosahedron.
Yes. I did wonder if the d20 was in the standard platonic solids set... I suppose the most questionable die used in D&D (aside from one for which no dice formally exist, such as d100, or d2) would be the d10. All the others are platonic solids. d4, d6, d8, d12, and d20, and then d10...
Josselin Luneau If I undertand well, larger faces would mean more probability. Then, 7 should be in the larger one because it's the most probable result, while 2 and 12 (not 1 and 12 like I said incorrectly before) should be on the smaller ones. It's easier to get a central number on a two dice throw because there's more possible combination outcomes (7: 6+1, 2+5, 3+4, 4+3, 5+2, 1+6; 2 and 12: just 1+1 and 6+6)
I once saw 7-sided dice, that were basically extruded pentagons. And my initial reaction was that there's no way such a die could be fair, but then I thought about it for a few minutes. If you have a pentagon that's extruded very thinly, like a wafer, then it'll be biased in favor of the two pentagonal faces and the other 5 faces will hardly ever show up. If it's extruded several feet, then the two pentagonal faces will hardly ever show up and you'll usually get one of the 5 others. So there MUST be a sweet spot in the middle where the biases cancel out, and you'll get one of the pentagonal faces 2 out of 7 times!
I think standupmaths (a side channel for Matt Parker) did a video on a similar problem, finding the dimensions that would make a cylinder work as a 3-sided die. I remember that when they were trying to calculate it, they got two different results based on what area was selected for the random distribution, and they tried rolling a bunch of such dice themselves, but I don’t remember if they got any results from that. There were a bunch of people in the comments (including me) who were saying that it would never work due to the fundamental lack of symmetry between the sides of the cylinder. In particular, the way you throw it majorly affects the results; ie, throwing it so it rolls along its axis means that you would get far fewer of the two ends and too many of the band. Essentially, isohedral dice are fair because all the sides are interchangeable. However you throw the die, it can be oriented beforehand so that it has the same overall shape and thus rolls the same way, but any other side you want ends up on top. Because of this, if the original position of the die is random and unknown, so is the resulting roll. This is not true for the cylindrical die; you can’t put the band in the place of one of the ends, or vice versa. So it can never truly be fair. Just use a top for stuff like that, or try cubical dice and count opposite sides the same. PS: speaking of the starting position being unknown, some people there tried to argue that this argument was wrong because ordinary dice can also be affected by the way you throw them. For example, some people have become well-known for cheating at craps via something called the blanket roll; basically, they throw the dice so that they roll around only one axis (similar to what I said about the cylindrical die), and like that, the two sides at the end of the axis (in this case, usually 1 and 6) were less likely to end up on top. My answer to that is that this effect depends on the original position of the die being known; the blanket roller has to look at the dice and put the 1 and 6 where they want them for this to work. If they just grabbed the die and threw it without looking, their chances of getting, say, a 1 wouldn’t be any different from before. Again, this is because the dice’s sides are interchangeable; the cylindrical die’s aren’t, so they can’t be thrown fairly.
So I tried making an actual list of the fair dice as shown at 7:23, using these visuals and the original paper. Here's what I've got: D6 - regular cube D8 - regular octahedron D60 - pentagonal hexecontahedron D24 - pentagonal icositetahedron D60 - pentakis dodecahedron D12 - rhombic dodecahedron D30 - rhombic triacontahedron D24 - triakis octahedron D4 - regular tetrahedron D24 - tetrakis hexahedron D60 - triakis icosahedron D60 - deltoidal hexecontahedron D12 - triakis tetrahedron D24 - deltoidal icositetrahedron The infinite family of bipyramids (pictured is the triagonal bipyramid I believe) D48 - disdakys dodecahedron D120 - disdakys triacontahedron D12 - regular dodecahedron D20 - regular icosahedron After that I am kinda lost. The visuals are confusing me a bit, cause the deltoidal icositetrahedron and disdakys dodecahedron seem to be there twice (at positions 14 and 22 and positions 16 and 21 respectovely). The second to last shape also looks like just a regular octahedron, which is already listed before. The last one also looks like a rhombic dodecahedron, also already listed. Furthermore, after reading the original paper, I've come to understand that the fair dice are: 5 Platonic Solids, 13 duals of Archimedean Solids (known as Catalan Solids) and 2 infinite families. Based on the paper I think the infinite families are supposed to be bipyramids and trapezohedra. But that's all I got and that's just 20 families. The video says there should be 30, but I can't figure out what the remaining solids in the video are supposed to represent and the paper seems to be talking about only 20 families as well, unless I missed something. If anyone has any ideas, please let me know.
4:20 reminds me how a standard 3x3x3 Rubik's cube works. When you make one full turn on one side it stays in the cube shape. But when you change the shape into lets say a Rhombohedron while still keeping the same turning cuts as a standard 3x3x3 Rubik's cube it starts to change shape when mixing it up.
What he described is not a traditional d10, a d10 is a dodecahedron with two faces extruded to points. On a d10, the faces interlock with two opposite faces. His d10 only has one opposite face connected to each.
Only if it were possible to space every dimple evenly apart, and make sure all dimples have the same number of adjacent dimples in a way that is identical and symmetrical to every other dimple. Someone who know more about the design of golf balls could probably say for sure. EDIT: After watching a few videos the best I can really say is; maybe? I'm pretty sure golf balls are carefully designed to meet the right criterion for fairness, because a fair dimple placement is required for aerodynamics, which is the whole reason golf balls have dimples in the first place.
Probably not. If you look closely, you see that some dimples are next to five other dimples, and others are next to 6. This means that the symmetry of a golf ball isn't face-transitive. It could be possible to create a golf ball to be experimentally fair despite this, but why would you want to? Just use a random number generator or something.
Since it's round, it wouldn't stop turning until something stopped it or it finished all the momentum, so it wouldn't land and you wouldn't be able to recognize the face it shows
So, the d30 is somewhat similar to a soccer ball, in that it's a shape made of a pattern of two different subshapes, namely 3x5-rhomboids and 5x3-rhomboids, and because of that, some values are more likely to show than others. Now I'm really curious as to the frequency distribution of a d30. If anyone knows where to find that, please drop me a line!
No, the d30 is a rhombic triacontahedron, it's made of one shape - the golden rhombus - and all values are equally likely to appear, assuming the weight is evenly distributed.
+31nar288 Correction, all 8 _cubes_. (Nobody says that a 3D die lands on one of its _areas_.) Liked anyway. Cheers ;) [Edited to add] On second thoughts. maybe even _cubes_ isn't right. What (3D) corresponds to 2D _sides_?
You'd have to make sure you were throwing it in 4 dimensional space too, though, since in our 3D space it would only flip in 3 different directions which would make it unfair.
Actually, all the edges on a rhombic triacontahedron (the 30-sider) are in fact the same. You can map any edge onto any other edge the same way you can for the faces. The fair dice shapes are sometimes called "Catalan Solids" or "Archimedean Duals".
@@DanielFerreira-ez8qd I'm not arguing. I'm just stating a fact. And why in the world should anyone need to be "warned" not to argue with a mathematician?
@@PhilBagels I didn't mean "argue" in the aggressive manner, just that you corrected the math man, which is a humorous thing to do in a scenario where you could be corrected immediately. This ain't one of those obviously, I'm just messing around
Yeah, right, because Casinos make so much less money before him than after him. What the heck are you talking about? I'm not sure you know how Casinos work. Either that, or you haven't thought the statement through. The point where the rubber meets the road is the point where the rubber meets the road, and I can prove that mathematically.
Well you could determine a fair number in 1-120 with 2d10, 1d20 and 1d4. Roll the d4: at [1,2,3] it's up to 1-100 at [4] its 101-120 In case of [1,2,3] simply roll 2d10 to get the digits for a d100 _or_ In case of [4] just roll the d20 for your 101-120
I took discrete probability at SUNY Albany with Professor Martin Hildebrand, whom I think had this Professor Diaconis as his PhD advisor. I imagine this professor has advised many PhD candidates, but Hildebrand seemed to stand out as pretty brilliant (Harvard PhD after all). Any of you guys take classes with Professor Diaconis or any of his "descendants"? I do believe my prof at UAlbany has (obviously) published with Prof Diaconis as well....
Fun bit! The dice they are tossing in the examples, the ones with no paint in the numbers and very sharp edges? Those are from Gamescience, a company that prides itself on making them that way because the rounding of the edges can cause flaws with their balance and make them less random.
4:45 as you can see, there is a small triangle at each corner of the tetrahedron, which means there is a chance, an astronomically small chance, that the die can land exactly on the small triangle.
Make it transparent and have a multi-directional laser in the centre that fires out vertically so you can better see what number it's landed on. Cheap and perfectly safe. Problem solved.
In my opinion its not because the way it rolls. The sphere if you look in terms of phisycs it only rolls at 1 direction, by the other way cubes turns on all directions. I don't know, i'm just giving my thoughts
Of course, it depends on the rolling substrate. If it's lumpy, the sphere is super fair (if you've just labeled sections as sides), but as Jorge said, you completely control the one axis of rotation if you roll it on something flat.
A fair throw is where you didn't intentionally manipulate the chances. The matter of the dice throwing to get a result from a fix pool that you didn't know before the throw. But, it's needed to make it "unfair" to make it possible to determine a value. In a theoretically perfect world a theoretically "perfect" throw would cause the dice infinitely rolling without stopping, because that perfect you throwed. If not, and we assume that the dice is not mathematically perfect object, it would stand on an edge, like a coin. Throw a coin. Heads or tails, but you throwed so perfectly it didn't lean to either side.
I saw a 7-sided die. It was a pentagonal prism, so two sides were pentagons and there were 5 sides that were connecting them. The guy who made it was talking about how if you look at it, the pentagons look so much larger than the smaller connecting sides, and you'd think there was a higher chance of the pentagons landing up, but there actually wasn't. He made bets with people where if it landed on a pentagon, he'd give them one dollar, and if it landed on the others, they'd give him two. Of course, he ended up winning because it was a fair die.
What about a case for the three sided die... you can see three sided dice shapes within the old pieces for risk which symbolized 10 men. The 1 man pieces were squares, and the 10 man pieces aka cannons were the shape of what seemed a fair three sided dice.
The long story short is that as long as the individual panels of the polyhedrons are made of "equilateral" shapes (All sides AND angles are the same) then the die should be fair in terms of symmetry.
Maybe, there's an other parameter. Which side do you recognize as the "result"? We usually use the upper face of the dice, but for example the dodekahedron doesn't have two sides parallel. Isn't it necessary to have a "result face" looking upwards?
You build a machine to do it, with maybe 20 or 50 dice being tossed at a time, incorporating a camera to record them after each toss, and recognition software to count each outcome from each die, at each toss. At the beginning of Part 2 he shows how a student did this as a project; he used 12 dice at a time. Or you vedge out with dice while watching TV for 15 years.
You roll it once, then you wait until it comes to rest, then you pick it up and do it again. Do that over and over until you reach the number you are after. I hope this helps.
Here's a fun question... I often think about "true/pseudo" random numbers and how random certain things may be. So, i know that pseudo random generators can be pretty bad and there are some that are fairly good at even histogram distributions. So, what if we had a "perfect" die rendered in 3D and thrown in a physics engine. If the physics engine dropped a die exactly the same every time (using some sort of pseudo random starting face), would this create a better distrobution than just a psRNG on it's own (assuming the physics engine produced different results when dropped)?
Whatever you decide to make random (the starting position, the way to throw, etc.) will depend on the RNG, so the result couldn't go any further than that
There's a clever trick to ensure you are flipping a fair coin (the logic can be generalised to dice). The idea is that you start with a H/T coin with roughly 50-50 ratio for landing: and by 'roughly', I mean that 0.01
You can start by proofreading your comments before posting, so that you don't come off sounding like an ignorant moron. Morons don't fare too well in casinos. Next, go to videos by 'Dangerous Arm Craps' and watch and listen. Then put it all, everything you've got, across the numbers as soon as you get the dice and don't work them on the Come Out roll. Hit 4 numbers and pull the bets down. Send what you came with home and play only with the profit. You're welcome.
In more simple terms, it's easier to "game" a dice that was dihedral like at 8:45 than a simple cubic dice. So in the dihedral dice, it's easier to do things like unfairly rolling a simple cubic dice carefully so that 2 of the faces don't appear.
Absolutely fascinating - I'd love to see a big venn diagram of face transitive, edge transitive and vertex transitive solids (maybe I should get to work on it myself...)
What about a 'cylinder shape' with rounded ends and an 3 or more flattened faces? This is how d5 and d7 are made, it can't land on the ends because they're rounded and each side around the axis is equal chance. Right?
One question for this brilliant video: So, I naturally had the question whether the simple differentiation between our symmetry group objects and the "real" platonic solids is simply that for the platonic solids their moments of inertia are the same for all coordinate axis around the center of mass of the object. Our symmetry group object f.e. is shaped such a way, that there is a specific "long" (or "short") axis, which would suggest different moments of inertia for rotation around that axis and in the plane perpendicular to it. That would mean that the object would prefer to rotate either around the axis or perpendicular to it, both making some faces more likely than others. On the other hand, an object with all moments of inertia the same for any rotation around the center of mass would keep its initial rotations and such. No rotation axis would be more likely than another. I don't know all platonic solid's moments of inertia, but I assume due to their high symmetry in vertex, edge and face, that they should have the same moments of inertia. That way, only some dice would be fair in their theoretical way of flying through the air with a given angular momentum. So just maybe that might be another thing to consider. Anyway, I liked the video. Good job :)
Anyone see where the D10 die fits into the classification scheme at 7:21? It looks a bit like the second one but D10 isn't symmetric about the horizontal plane. I'm thinking maybe it's second row from the bottom, second from the right.
you can create fair dice that have symmetrical edges, vertices and sides IF you allow the faces to have even a tiny amount of curvature. For example a toblerone that doesn't have flat ends, but points.
Hey, random relevant question: I've been getting into Dungeons and Dragons recently and I was faced with a decision. You are rolling a 20-sided die, trying to get the highest number possible. You have three options: you can either roll three times, roll twice with a +2 on each roll, or roll once with a +4 on the roll. Which do you pick?
The answer is either 3 or 2 rolls. Rolling twice has been mathematically analyzed by countless players and found to be worth about +5. I could analyze the other case for you, but I'm too tired for that right now.
If you made a square die that had the numeric values of 0-5 and rolled that with a 1-6 die, could you get every number from 1 to 11? With rolling two 1-6 dies you won’t ever get 1, is there a number that can’t be made between 1-11 that can’t be made with the 0-5 and 1-6 dies?
Friendly suggestion, get to the point like your earlier videos. Your edits have become boring. I used to get excited watching your videos now I'm watching them to support you.
to extrapolate the theory, BLOWING on the dice for luck, could influence the outcome of the dice roll ! one side of the die would be warmer than the other, influencing the dynamics of the roll ... great video
by only applying to the simple fairness definition, you can make a 7 sided dice, if you allow curved planes: start with a 7-gon like they did with a 5-gon at 6:49. now insted of connecting the two points with straight lines (like they did) use curved lines. you end up with a somewhat football like looking 7 sided dice, that should just work fine. of course this doesn't only work with 7 sides, but i picked that one because it's such an undicey number :-)
I love this guy's name. It's like the name someone would have in a medieval fantasy story.
"Quick, m'lord! We must reach the king's statistician Persi Diaconis before sundown or all hope is lost! He's the only one who knows how to make a fair die out of non-regular polygons!"
Finally, a name for my NPC wizard.
It is rumoured that the legendary artificer, Persi Diaconis of the Order of the Logician, once created an object that appeared to be a normal dice. In truth, this dice was filled with magic, and power, and quite a lot of hatred, and its roll would influence the very fate of the world...
Yeah his name is awesome! :D
Be careful not to attack him though, he does 2d10 math damage.
It sounds like something out of the game twisted wonderland and I love it
Your channel turned me from a person who thought they hated maths, to someone who appreciates its beauty, thanks!
I've always appreciated math. Its basically our way of explaining the universe.
I'm just absolutely garbage at it, and that makes me bitter.
@@Guru_1092 I know just how you feel. I share that bitterness as well.
"There are 5 fair dice."
*angry d2 noises
*d2 lands on its side
deniz-usta Gedik *angry d20 noises.*
@@The_Murder_Party *Angrily rolls percentile dice*
The Procastinators I mean to be fair percentile are two d10s, but this is fair.
Never had a D2. Is there a die for that? Or is it a coin?
Fair dice: Not the DM's
As a Dungeon master I approve this message.
For anyone interested, here are the names of the shapes shown at 7:20
Left to right, then top to bottom;
cube/hexahedron, octahedron, pentagonal hexecontahedron, pentagonal icosahedron.
*opens gaming shop called "Die, die, die!"*
The missing one is pentakis dodecahedron
@AINIEL YABUT the "not sure" shape is actually a rhombic dodecahedron
You're a legend. This comment should be pinned.
All a D&D player wants to know is whether the D20 is a fair dice. ;D
That's the regular icosahedron, one of platonic shapes which he described as fairest.
Yes. I did wonder if the d20 was in the standard platonic solids set... I suppose the most questionable die used in D&D (aside from one for which no dice formally exist, such as d100, or d2) would be the d10.
All the others are platonic solids. d4, d6, d8, d12, and d20, and then d10...
i guess
It never is...
Wil Wheaton might disagree.
Is it possible to make strategically unfair dice?
I've always wanted to make a 12 sided dice, with the same probabilities as two 6 sided die
Maybe you could make the faces with the central values larger than the ones with the extreme values? Make 7 huge and 1 and 12 tiny.
one problem with that is that two 6-sided dice will never land on a sum total of 1.
There are only eleven possible outcomes to a 2D6 throw. ;)
Still an interesting question, though.
jmiquelmb: I would say the opposite, because big faces are more stable, thus, if 7 is on a small face opposite to 12, probability would be correct.
Josselin Luneau If I undertand well, larger faces would mean more probability. Then, 7 should be in the larger one because it's the most probable result, while 2 and 12 (not 1 and 12 like I said incorrectly before) should be on the smaller ones. It's easier to get a central number on a two dice throw because there's more possible combination outcomes (7: 6+1, 2+5, 3+4, 4+3, 5+2, 1+6; 2 and 12: just 1+1 and 6+6)
I once saw 7-sided dice, that were basically extruded pentagons. And my initial reaction was that there's no way such a die could be fair, but then I thought about it for a few minutes.
If you have a pentagon that's extruded very thinly, like a wafer, then it'll be biased in favor of the two pentagonal faces and the other 5 faces will hardly ever show up. If it's extruded several feet, then the two pentagonal faces will hardly ever show up and you'll usually get one of the 5 others. So there MUST be a sweet spot in the middle where the biases cancel out, and you'll get one of the pentagonal faces 2 out of 7 times!
Ah, I see now he covered that in Part 2! Maybe not so fair after all...
meh, tops are really useful for X sided dice, take the dreidel for example.
INFINET SIDE DIAS!
It'd just make more sense to have a heptagonal prism and round the edges for a 7-sided die. That's what they did for the 3-sided die.
I think standupmaths (a side channel for Matt Parker) did a video on a similar problem, finding the dimensions that would make a cylinder work as a 3-sided die. I remember that when they were trying to calculate it, they got two different results based on what area was selected for the random distribution, and they tried rolling a bunch of such dice themselves, but I don’t remember if they got any results from that. There were a bunch of people in the comments (including me) who were saying that it would never work due to the fundamental lack of symmetry between the sides of the cylinder. In particular, the way you throw it majorly affects the results; ie, throwing it so it rolls along its axis means that you would get far fewer of the two ends and too many of the band.
Essentially, isohedral dice are fair because all the sides are interchangeable. However you throw the die, it can be oriented beforehand so that it has the same overall shape and thus rolls the same way, but any other side you want ends up on top. Because of this, if the original position of the die is random and unknown, so is the resulting roll. This is not true for the cylindrical die; you can’t put the band in the place of one of the ends, or vice versa. So it can never truly be fair. Just use a top for stuff like that, or try cubical dice and count opposite sides the same.
PS: speaking of the starting position being unknown, some people there tried to argue that this argument was wrong because ordinary dice can also be affected by the way you throw them. For example, some people have become well-known for cheating at craps via something called the blanket roll; basically, they throw the dice so that they roll around only one axis (similar to what I said about the cylindrical die), and like that, the two sides at the end of the axis (in this case, usually 1 and 6) were less likely to end up on top. My answer to that is that this effect depends on the original position of the die being known; the blanket roller has to look at the dice and put the 1 and 6 where they want them for this to work. If they just grabbed the die and threw it without looking, their chances of getting, say, a 1 wouldn’t be any different from before. Again, this is because the dice’s sides are interchangeable; the cylindrical die’s aren’t, so they can’t be thrown fairly.
The way this man describes dice reinforces the idea that there's a very fine line between insanity and genius.
In high school our physics teacher used to choose people for oral exams by throwing a 30 sided die lol
that sounds so weird and disgusting...
redbeam_ why?
@Nate his mind is in the gutter, "oral exams"
Emerson Harris I know. Im just saying that it shouldn't sound dirty and that his mind is in the gutter.
is it fair?
So I tried making an actual list of the fair dice as shown at 7:23, using these visuals and the original paper. Here's what I've got:
D6 - regular cube
D8 - regular octahedron
D60 - pentagonal hexecontahedron
D24 - pentagonal icositetahedron
D60 - pentakis dodecahedron
D12 - rhombic dodecahedron
D30 - rhombic triacontahedron
D24 - triakis octahedron
D4 - regular tetrahedron
D24 - tetrakis hexahedron
D60 - triakis icosahedron
D60 - deltoidal hexecontahedron
D12 - triakis tetrahedron
D24 - deltoidal icositetrahedron
The infinite family of bipyramids (pictured is the triagonal bipyramid I believe)
D48 - disdakys dodecahedron
D120 - disdakys triacontahedron
D12 - regular dodecahedron
D20 - regular icosahedron
After that I am kinda lost. The visuals are confusing me a bit, cause the deltoidal icositetrahedron and disdakys dodecahedron seem to be there twice (at positions 14 and 22 and positions 16 and 21 respectovely). The second to last shape also looks like just a regular octahedron, which is already listed before. The last one also looks like a rhombic dodecahedron, also already listed. Furthermore, after reading the original paper, I've come to understand that the fair dice are: 5 Platonic Solids, 13 duals of Archimedean Solids (known as Catalan Solids) and 2 infinite families. Based on the paper I think the infinite families are supposed to be bipyramids and trapezohedra.
But that's all I got and that's just 20 families. The video says there should be 30, but I can't figure out what the remaining solids in the video are supposed to represent and the paper seems to be talking about only 20 families as well, unless I missed something. If anyone has any ideas, please let me know.
My man that was a lot of work
4:20 reminds me how a standard 3x3x3 Rubik's cube works. When you make one full turn on one side it stays in the cube shape. But when you change the shape into lets say a Rhombohedron while still keeping the same turning cuts as a standard 3x3x3 Rubik's cube it starts to change shape when mixing it up.
This was as mathematically and philosophically as beautiful a video ad any other numberphile video as I've watched ever.
"Small changes in the initial conditions change what side faces up"
In othere words, dice are not just random, but chaotic :D
"Talking to me about dice and fairness is like talking to a California wine person about wine - it can go forever."
Please do!!
"There are only five fair dice, d4, d6, d8, d12, and d20" *sweats in white wolf*
D10 bruh
Wild Magic Sorcerer: *Sweats in D100*
D10
@@gillasosaurus d10 isn't fair. Vertices of 4 or 5 depending on position.
@@AlexH274 k, but they're statistically equally likely, which is what matters when it comes to fairness
There is one of the interests of simulating chance: once they're balanced, all virtual die are fair.
what about this dice guy? did he end up with equal results for all 6 sides of a die?
don't think so. they mentioned the weight difference, especially between 1 and 6, shortly after talking baou him.
i guess he had less 6s than 1s.
he would of been infinitely close.
Pretty sure it would be more 6's. 1 is less material removed therefore heavier and more likely to be on the bottom. The inverse is true for 6.
I think you would need an absurd amount of data to notice that difference to be honest.
Well, he rolled it 3.5 million times, so...
6:47 I was waiting for him to mention a d10, then he invented it
What he described is not a traditional d10, a d10 is a dodecahedron with two faces extruded to points. On a d10, the faces interlock with two opposite faces. His d10 only has one opposite face connected to each.
If all the dimples in a golf ball could be numbered, would it be fair??
Only if it were possible to space every dimple evenly apart, and make sure all dimples have the same number of adjacent dimples in a way that is identical and symmetrical to every other dimple. Someone who know more about the design of golf balls could probably say for sure.
EDIT: After watching a few videos the best I can really say is; maybe? I'm pretty sure golf balls are carefully designed to meet the right criterion for fairness, because a fair dimple placement is required for aerodynamics, which is the whole reason golf balls have dimples in the first place.
Probably not. If you look closely, you see that some dimples are next to five other dimples, and others are next to 6. This means that the symmetry of a golf ball isn't face-transitive.
It could be possible to create a golf ball to be experimentally fair despite this, but why would you want to? Just use a random number generator or something.
Since it's round, it wouldn't stop turning until something stopped it or it finished all the momentum, so it wouldn't land and you wouldn't be able to recognize the face it shows
There are 100 sided dice, they look like golf-balls....
Yes, if it were perfectly spherical and dimpled in such a way where one is directly across from another through the center.
That d4 awakened a deep anger within me
"(...)new Tadashi video soon, that's something to get excited"
Oh Brady, you know your audience so well...
6:00 The edges on a rhombic triacontahedron ARE transitive. See Wiki on the triacontahedron
So, the d30 is somewhat similar to a soccer ball, in that it's a shape made of a pattern of two different subshapes, namely 3x5-rhomboids and 5x3-rhomboids, and because of that, some values are more likely to show than others.
Now I'm really curious as to the frequency distribution of a d30. If anyone knows where to find that, please drop me a line!
No, the d30 is a rhombic triacontahedron, it's made of one shape - the golden rhombus - and all values are equally likely to appear, assuming the weight is evenly distributed.
@@RDSk0 Yes, it's all one facet shape. No, they're not all as likely to show up because of how they're grouped.
'Fair dice' feels like it should be a saying ...
I think what you said is fair dice
jmiquelmb
haha! yeah, just like that :)
I cannot talk about probability all day like Prof Persi, but I'll watch any Numberphile video, so- fair dice.
I think what he said is but a parker square of a fair dice
No dice.
Would a tesseract be a fair die though ? =D
A hypercube, yes.
Yeah, it would have the same chance to land on all 8 volumes.
They have made a video called "Perfect Shapes in Higher Dimensions" that is kind of this problem in higher dimensions.
+31nar288
Correction, all 8 _cubes_.
(Nobody says that a 3D die lands on one of its _areas_.)
Liked anyway.
Cheers
;)
[Edited to add] On second thoughts. maybe even _cubes_ isn't right. What (3D) corresponds to 2D _sides_?
You'd have to make sure you were throwing it in 4 dimensional space too, though, since in our 3D space it would only flip in 3 different directions which would make it unfair.
"I have a thirty sided dice" Who wants to bet that he got it to play D&D
What TTRPG needs a d30?
I will.
Just bought one for my son to use as a DM.
6:45 did he just say "fivegon"??? 😞
Great to see Game Science dice in use.
Zocchi dice!
I'm guessing he chose them because of their purported fairness.
I got my game science dice to overcome dice superstition, and have since become superstitious about using any dice that aren't my game science dice
Adderkleet that’s what I thought! I saw the clipped corners on the d4.
I was hoping he would said something about the sprue discoloration.
Dammit! I thought this video had the man with a thousand Klein bottles when I saw the thumbhnail but it was an impostor.
Right?!?
Ah, but this is Perci Diaconis, a magician who studied with Dai Vernon. Just as interesting. Problem is, Brady has no reason to ask him about it.
Cliff Stohlen Identity
Make it stop.
Actually, all the edges on a rhombic triacontahedron (the 30-sider) are in fact the same. You can map any edge onto any other edge the same way you can for the faces.
The fair dice shapes are sometimes called "Catalan Solids" or "Archimedean Duals".
I'm 5 years late to warning you not to argue with the old mathematician.
@@DanielFerreira-ez8qd I'm not arguing. I'm just stating a fact. And why in the world should anyone need to be "warned" not to argue with a mathematician?
@@PhilBagels I didn't mean "argue" in the aggressive manner, just that you corrected the math man, which is a humorous thing to do in a scenario where you could be corrected immediately. This ain't one of those obviously, I'm just messing around
@@DanielFerreira-ez8qd But I'm right,
love this! being a gamer i roll dice all the time, so this is a great video.
*Several D&D players including myself are typing*
Heard him speak in New York a few weeks ago. Really entertaining and insightful speaker.
Casinos HATE This Man: Find Out Why
Yeah, right, because Casinos make so much less money before him than after him. What the heck are you talking about? I'm not sure you know how Casinos work. Either that, or you haven't thought the statement through. The point where the rubber meets the road is the point where the rubber meets the road, and I can prove that mathematically.
You know he is a genius as he closes his eyes while explaining because he is visualizing it.
is this the klein bottle guys brother?
Ummm.... no.
Well you username checks out. I trust you.
Persi Diaconis.
Cliff Stoll.
Maybe half brothers...
It's the same guy, people.
Geez, I hope you're all joking...
I have a D120 I use for D&D random tables, I wonder if that's considered fair
i dont think so. do you use the official PHB? cuz if you do there are only d100 random tables and to trow a fair d100 just use two d10.
Well you could determine a fair number in 1-120 with 2d10, 1d20 and 1d4.
Roll the d4:
at [1,2,3] it's up to 1-100
at [4] its 101-120
In case of [1,2,3] simply roll 2d10 to get the digits for a d100
_or_
In case of [4] just roll the d20 for your 101-120
Is there a beholder listed on the table? ... that's not fair!
Pro-fair-sor Die-cone-is...
:)
I'll never forget his name again!
I took discrete probability at SUNY Albany with Professor Martin Hildebrand, whom I think had this Professor Diaconis as his PhD advisor. I imagine this professor has advised many PhD candidates, but Hildebrand seemed to stand out as pretty brilliant (Harvard PhD after all). Any of you guys take classes with Professor Diaconis or any of his "descendants"? I do believe my prof at UAlbany has (obviously) published with Prof Diaconis as well....
Nomen est omen!
Fun bit! The dice they are tossing in the examples, the ones with no paint in the numbers and very sharp edges? Those are from Gamescience, a company that prides itself on making them that way because the rounding of the edges can cause flaws with their balance and make them less random.
4:45 as you can see, there is a small triangle at each corner of the tetrahedron, which means there is a chance, an astronomically small chance, that the die can land exactly on the small triangle.
Exactly. It's just an unfair 8 sided die.
I love how often I come up with an issue with something someone is saying, and the question is brought up in the video.
I guess the sphere is the fairest of them all, but then again it can sit in the middle of multiple answers
Make it transparent and have a multi-directional laser in the centre that fires out vertically so you can better see what number it's landed on. Cheap and perfectly safe. Problem solved.
In my opinion its not because the way it rolls. The sphere if you look in terms of phisycs it only rolls at 1 direction, by the other way cubes turns on all directions. I don't know, i'm just giving my thoughts
now that i think about it , a sphere has only one side...
Of course, it depends on the rolling substrate. If it's lumpy, the sphere is super fair (if you've just labeled sections as sides), but as Jorge said, you completely control the one axis of rotation if you roll it on something flat.
Always a critical failure.
The rhombic triacontahedron HAS a transitive symmetry group on the edges. 5:53
0:08 "Tetrahedron" That ones got 8 faces. I had a dice with rounded edges, and it managed to stop on an edge. Only once though.
For an algorythm/symetry nerd like me, this is as entertaining as watching a movie in a cinema for normal people
A fair dice may exist, but a fair throw does not. You could make it very close to a fair throw, but conceptually it's impossible I think.
A fair throw is where you didn't intentionally manipulate the chances. The matter of the dice throwing to get a result from a fix pool that you didn't know before the throw.
But, it's needed to make it "unfair" to make it possible to determine a value. In a theoretically perfect world a theoretically "perfect" throw would cause the dice infinitely rolling without stopping, because that perfect you throwed. If not, and we assume that the dice is not mathematically perfect object, it would stand on an edge, like a coin. Throw a coin. Heads or tails, but you throwed so perfectly it didn't lean to either side.
this brings back memories of crystallography. Good o'l mineral symmetry groups.
This is how Cliff Stoll would look and act like if he wasn't constantly high
yeah
Hahahaha! "Caught me there... Let me AMEND the statement of my theorem...." - Love it...
7:20 That audio editing though...
I saw a 7-sided die. It was a pentagonal prism, so two sides were pentagons and there were 5 sides that were connecting them. The guy who made it was talking about how if you look at it, the pentagons look so much larger than the smaller connecting sides, and you'd think there was a higher chance of the pentagons landing up, but there actually wasn't. He made bets with people where if it landed on a pentagon, he'd give them one dollar, and if it landed on the others, they'd give him two. Of course, he ended up winning because it was a fair die.
Rolls damage: 40k6
You're more than welcome Brady! Best channel ever!
Спасибо за видео очень интересно и полезно
What about a case for the three sided die... you can see three sided dice shapes within the old pieces for risk which symbolized 10 men. The 1 man pieces were squares, and the 10 man pieces aka cannons were the shape of what seemed a fair three sided dice.
The long story short is that as long as the individual panels of the polyhedrons are made of "equilateral" shapes (All sides AND angles are the same) then the die should be fair in terms of symmetry.
Maybe, there's an other parameter. Which side do you recognize as the "result"? We usually use the upper face of the dice, but for example the dodekahedron doesn't have two sides parallel.
Isn't it necessary to have a "result face" looking upwards?
How do you roll a die 3500000 times?
You build a machine to do it, with maybe 20 or 50 dice being tossed at a time, incorporating a camera to record them after each toss, and recognition software to count each outcome from each die, at each toss.
At the beginning of Part 2 he shows how a student did this as a project; he used 12 dice at a time.
Or you vedge out with dice while watching TV for 15 years.
how long have you been on youtube
You roll it once, then you wait until it comes to rest, then you pick it up and do it again. Do that over and over until you reach the number you are after. I hope this helps.
Your watch as much content as a typical 12-year-old and spend that time rolling, rolling, rolling.
Rolling a dice every 3 seconds. No resting. No sleeping. That's about a year.
Here's a fun question...
I often think about "true/pseudo" random numbers and how random certain things may be.
So, i know that pseudo random generators can be pretty bad and there are some that are fairly good at even histogram distributions.
So, what if we had a "perfect" die rendered in 3D and thrown in a physics engine.
If the physics engine dropped a die exactly the same every time (using some sort of pseudo random starting face), would this create a better distrobution than just a psRNG on it's own (assuming the physics engine produced different results when dropped)?
Whatever you decide to make random (the starting position, the way to throw, etc.) will depend on the RNG, so the result couldn't go any further than that
"same specific gravity"... Oh boy.
There's a clever trick to ensure you are flipping a fair coin (the logic can be generalised to dice).
The idea is that you start with a H/T coin with roughly 50-50 ratio for landing: and by 'roughly', I mean that 0.01
SO HOW CAN I WIN A THE CASINO
Don't play.
own a casino
@@Xormac2
That also works.
You can start by proofreading your comments before posting, so that you don't come off sounding like an ignorant moron. Morons don't fare too well in casinos. Next, go to videos by 'Dangerous Arm Craps' and watch and listen. Then put it all, everything you've got, across the numbers as soon as you get the dice and don't work them on the Come Out roll. Hit 4 numbers and pull the bets down. Send what you came with home and play only with the profit. You're welcome.
The best way to win the game is to not play
In more simple terms, it's easier to "game" a dice that was dihedral like at 8:45 than a simple cubic dice. So in the dihedral dice, it's easier to do things like unfairly rolling a simple cubic dice carefully so that 2 of the faces don't appear.
Gamescience dice! They're the best.
I thought i recognized those immediately, shame I still don't own a pair.
First thought that popped to mind when the video started. =)
Absolutely fascinating - I'd love to see a big venn diagram of face transitive, edge transitive and vertex transitive solids (maybe I should get to work on it myself...)
Rohan is really suspicious of dice now
Goddamit, is this a jojo reference?
@@lilysdong1457 xD yes, yes it is, and I hate how it got even here
What about a 'cylinder shape' with rounded ends and an 3 or more flattened faces? This is how d5 and d7 are made, it can't land on the ends because they're rounded and each side around the axis is equal chance. Right?
Have an awesome day everyone! :)
no
I am in pain
+NonTwinBrothers im also sick....:(
you to buddy :D
Don't tell me what to do.
One question for this brilliant video:
So, I naturally had the question whether the simple differentiation between our symmetry group objects and the "real" platonic solids is simply that for the platonic solids their moments of inertia are the same for all coordinate axis around the center of mass of the object.
Our symmetry group object f.e. is shaped such a way, that there is a specific "long" (or "short") axis, which would suggest different moments of inertia for rotation around that axis and in the plane perpendicular to it. That would mean that the object would prefer to rotate either around the axis or perpendicular to it, both making some faces more likely than others.
On the other hand, an object with all moments of inertia the same for any rotation around the center of mass would keep its initial rotations and such. No rotation axis would be more likely than another.
I don't know all platonic solid's moments of inertia, but I assume due to their high symmetry in vertex, edge and face, that they should have the same moments of inertia.
That way, only some dice would be fair in their theoretical way of flying through the air with a given angular momentum. So just maybe that might be another thing to consider.
Anyway, I liked the video. Good job :)
The casino dice are pretty
I want some.
Maybe I’ll get some on amazon tomorrow.
Anyone see where the D10 die fits into the classification scheme at 7:21? It looks a bit like the second one but D10 isn't symmetric about the horizontal plane. I'm thinking maybe it's second row from the bottom, second from the right.
D10 is not a fair die.
If you worry about the fairness of dice, you are either a math professor or a munchkin.
Or a D&D nerd.
What about if you consider the fourth dimension? Would they still be perfect dice?
old news to every dungeons and dragons player lol
Thanks for making great videos with smart people Brady! :)
It really bugs me that his 4 sided die have the corners shaved off !
Have you ever tired rolling a tetrahedron. The corners are very sharp abd make jt hard to pick up. Shavjng them off makings the die more weildy.
lots of time i'm an avid pen and paper player, playing pathfinder mostly now
That's to keep it from hurting when you step on it. Dice roll off the table more often than you think.
"turning it around three things"
I love this guy~
why am I still watching this at 1 am?
This video is insightful and delightful and I feel smarter having watched it.
D2 = a coin
Wrong.
lol no
you can create fair dice that have symmetrical edges, vertices and sides IF you allow the faces to have even a tiny amount of curvature. For example a toblerone that doesn't have flat ends, but points.
At the start, those 5 shapes would be called Platonian solids, technically speaking.
Yes, but it's, "Platonic."
ffggddss Close enough.
Ahhh, procrastination brings me back to my work, AND to E.T. Jaynes's book I'm in the process of reading!
it looks weird that his eyes seem constantly closed
old people HAHAHA
he's trying to image the geometrical shapes he's talking about, makes stuff easier for some people ;)
Just like Brock.
Ooooh, he got me all excited at the end there about the next Tadashi video!
only five in the 3rd dimension!
I'm interested in what types of polychoron dice would be fair, assuming a 4-dimensional space.
The same ones. and the hyper diamond.
There's two more (the triangular and pentagonal bipyramid) plus several other if you allow using non-regular faces.
@@robo3007
Regulars only, sorry.
There's only three in 4d.
One, Infinity, five, three, three, three...
When is Persi Diaconis coming back to Numberphile?
Any nerd could have told you what those original five dice are:
D4, D6, D8, D12, and D20
never thought i would be this engaged in a video about dice.
Poor kids, hahaha.
Hey, random relevant question: I've been getting into Dungeons and Dragons recently and I was faced with a decision. You are rolling a 20-sided die, trying to get the highest number possible. You have three options: you can either roll three times, roll twice with a +2 on each roll, or roll once with a +4 on the roll. Which do you pick?
The answer is either 3 or 2 rolls. Rolling twice has been mathematically analyzed by countless players and found to be worth about +5. I could analyze the other case for you, but I'm too tired for that right now.
You lost me at dice.
If you made a square die that had the numeric values of 0-5 and rolled that with a 1-6 die, could you get every number from 1 to 11? With rolling two 1-6 dies you won’t ever get 1, is there a number that can’t be made between 1-11 that can’t be made with the 0-5 and 1-6 dies?
Friendly suggestion, get to the point like your earlier videos.
Your edits have become boring. I used to get excited watching your videos now I'm watching them to support you.
to extrapolate the theory, BLOWING on the dice for luck, could influence the outcome of the dice roll !
one side of the die would be warmer than the other, influencing the dynamics of the roll ... great video
A bicone (two cones connected by their circles) would be also a fair die because the two contact regions (nappe segments) are the same shape.
You have made another video called "Wierd But Fair Dice". Is the die in that video the one at 7:24 in the 7th collumn, 3rd row?
Magic Mirror on the wall, what dice is the fairest one of all?
by only applying to the simple fairness definition, you can make a 7 sided dice, if you allow curved planes:
start with a 7-gon like they did with a 5-gon at 6:49.
now insted of connecting the two points with straight lines (like they did) use curved lines.
you end up with a somewhat football like looking 7 sided dice, that should just work fine.
of course this doesn't only work with 7 sides, but i picked that one because it's such an undicey number :-)