5-Sided Die

>>8757851

-original
-math-related
-not troll bullshit
-directly relates geometry and probability

This is the best post I've seen on /sci/ in a couple months OP, just so you know.

Conjecture: the rectangular faces and the triangular faces must have roughly equal area.

>>8757851

great question. first you'd have to parameterize the probability of each side, which would likely involve the area as well as the "stability" given by how top-heavy the die is when resting on that size.

>>8757851
Inscribe the shape in a sphere. Project the edges of the shape into the surface of the sphere such that they are in the directions at which the weight of the shape is balanced. Set the areas of the parts of the surface demarcated by the projected edges equal. The resulting shape is your die.

How it lands would depend heavily on the throwing style, I don't think you can say anything definitive about it.

>>8758350
>Conjecture: the rectangular faces and the triangular faces must have roughly equal area.

equal area, but probably also equal center of gravity.

having a non-square, rectangular side might make the die less sensitive to initial conditions since it's much more stable along the longer side.

>>8758383
yes, i was just thinking that too

>>8758383

Pushback: sufficient bias toward one of the two types of side will very obviously predispose the die to land on that type of side.

A very "long" die, (having long, thin rectangular faces, with areas much larger than the triangular faces, so that the whole thing looks more-and-more like a stick) will obviously almost always land on one of its rectangular faces. Similarly, a very "short" die (having very small rectangular areas relative to the triangular areas, so that the thing looks more and more like a triangular coin) will obviously almost always land on one of its triangular faces.

Your thing (I suppose) becomes more relevant in the happy medium territory, which is exactly where the discussion has gone. If the thing is held still by a robot arm at such-an-orientation and dropped straight down (for example), I could see how a bias toward one side or the other might crop up. But if the robot arm holds one of the above "extreme" dies and does the same thing, the biases of each shape will of course be vindicated.

>>8758375
>that they are in the directions at which the weight of the shape is balanced

i don't quite understand what you mean by this.

>>8757851

you mean prism though

>>8757851
Why wouldn't you just use a ten sided die with only five numbers on it?

>>8758402
In other words, if you were to balance the shape on that edge, the direction of gravity would be pointing through the edge in that direction.

H=sqrt(3)*b ?
that would provide equal area for the rectangular and triangular faces. I'm not sure about the center of gravity though...

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>>8758375
>such that they are in the directions at which the weight of the shape is balanced
what did he mean by this, /sci/?

>>8758437
See >>8758426

>>8758426
Draw a free-body diagram?

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this is what google proposes

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>>8757851
Already exists man
https://youtu.be/tSQIir5xxWc?t=152

To find an analytic solution, you would first need to define what "throwing a die" is so that you can actually calculate the relevant probabilities. Since "throwing" a die by holding 1mm above some flat surface and then letting go will almost always have a deterministic result, a throw should have some minimum "intensity" (e.g. nonzero initial velocity and angular momentum).
Personally I would suggest modelling the die as a discrete-time 5-state Markov chain. It would be aperiodic and thus have a stationary distribution which we interpret as the probability of landing on a face. The transition probabilities are functions of b and H. Finding an appropriate expression for these seems difficult, however.

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>>8757851
Hows about a football shape instead? but with 5 pronounced edges and 5 surfaces

>>8758447
>https://youtu.be/tSQIir5xxWc?t=152

wow, what a gentleman and a scholar

>>8758466
You can also watch him talk about inking dice for 40 minutes

https://www.youtube.com/watch?v=zaDzqYZufIQ&t=571s

>>8758428

[math]H*b = b^2*\sqrt{3}/4 [/math]

[math]H = b*\sqrt{3}/4 [/math]

did you lose the 4 or am i fucking it up?

center of gravity for rectangle side is H/2 (triangle side down)

for triangle side it's going to be h/3

[math]h = b*\sqrt(3)/2 [/math]

and
[math]b*\sqrt(3)/6 \neq b*\sqrt(3)/8 [/math]

someone check this. i don't think you can have equal area with equal center of gravity

>>8758471
awesome guy, but unfortunately I'm not autistic enough to watch the whole thing

>>8758513
Just speed it up

>>8758516
I did, I watched some parts further up, but it's 40 minutes, I need to shitpost on other boards too

actually, i think that having equal center of gravity would be more important than equal area. setting equal center of gravity might be all you need to do. any physics majors want to chime in?

>>8758484
idk man im hungover as shit. I think you're right

>>8758449

it isn't even easy to define the states. a die doesn't have to land flatly on one side to keep rolling. it can bounce off of its corners too.

The angles between the normal vectors of each surface need to be equal. Each surface must also have the same area.

>>8758587
>The angles between the normal vectors of each surface need to be equal

this is impossible with the shape in the OP. it's also not true, since the probability of landing on a given side is a smooth function of b and H

>>8758565
I'm not actually considering the die as a physical object anymore. I chose a mathematical approach that makes it easy to define the probabilities we're interested in; the difficulty is now in finding the transition matrix, and this should ideally include "strange" transitions (i.e. what you mentioned).

>>8758612

any uncomplicated definition of a "state" would involve some very complicated math to find the transition function. a markov chain is a poor model for this problem. it's a poor shape for a die too.

you'd probably have better luck just simulating it. even then, look at the five-sided die someone posted. it has heavily beveled edges and isn't really five sided at all.

>>8758350
Physics guy here, I don't think that conjecture works and I can prove it with energy.

Let "throwing a die" mean giving it some initial kinetic energy in a direction and letting it die down until it doesn't have enough left to flip to another face. If we consider the two different types of flip:

1. flipping from a square face to a triangular face
2. flipping from a triangular face to a square face

So if we look at the state of the equal-face-area triangular prisim resting on a square face versus resting on a triangular face, we see that the center of gravity is lower when resting on the square face. So, the act of rolling from a square face to a triangular face would result in a positive change in potential, consuming a lot of the initial kinetic energy, and from a triangular face to a square face results in a negative change in potential, leading to less of the initial potential being consumed. So, the die is more likely to stay in the square-face-down state, because it is more likely to run out of kinetic energy there because it needs more to roll again.

This implies H < h for the perfectly balanced die, I dunno by how much, but my gut says it has a factor of [math]\sqrt{3}[\math] in there, which gives a little bit of validation to >>8758447 and >>8758445

>>8758501
pretty sure you're right, and also because of >>8758636 I believe center of gravity is dominatingly important compared to face area

>>8758621
>a markov chain is a poor model for this problem.
I agree, but I couldn't think of a better one wherein the problem is well-defined.
An easier approach would be a simulation applying different forces at different angles to the centre of mass and seeing what face it lands on, and honestly an approximate solution is probably good enough for most purposes. I just think trying to find an analytic solution is more interesting.

>>8758621
>>8758644
*Well-defined and analytically solvable.

>>8757851
An equilateral pyramid seems really inefficient for this. Why not a pentagonal prism with rounded ends? Then every side can be the exact same, and have equal chance of being rolled. The die would need to have a heavy enough center in order to prevent it from staying on the rounded sides, though.

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OP here

I meant to write prism instead of pyramid but I figured that was obvious from the image

I thought about this for a while. The approach I took was to consider the energy required to flip the die from one face to another, of which there are three combinations (rectangular face down to rectangular face down, rectangular to triangular, and triangular to rectangular).

For the die to be fair the energy required to flip the die from its current face down to any other face must be equal. This is the case for a cubic die. I also assumed that the flip occurs along an edge (rather than at a point).

But for a triangular prism it is not possible. Equating minimum energy required to flip from R to T and the back from T to R enforces H = b/sqrt(3).

But then the energy required to go from R to R is necessarily greater (see working).

I'm not sure if this energy criterion is the best approach, because obviously there must be some ratio of H/b that in practice gives equal chance. Also, the effects of rounded edges must be significant.

I actually doubt it would be possible to determine optimum H/b without actually physically making and rolling the die.

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>>8759635
2/2

>>8759642
To continue with this approach, I do a simple random walk to simulate a tumbling 5-sided die

Again there are three possibilities: RR (rectangle to rectangle), RT, and TR

If T is down, then there is a 100% chance that the next time it tumbles R will be down

If R is down, there are two possibilities. I've taken the probability of RR and RT to be proportional to the energies required to cause those respective flips. So if the energy to go from RR is double that to go from RT, I've considered that p(RR) = 1/3 and p(RT) = 2/3, etc.

Then, I choose several different variations of H/b, work out the energies associated with each type of flip, and do a random simulation. In the program I wrote, R down = 1 and T down = 2. So in the long term you would expect the average value to be (3*1 + 2*2)/5 = 1.4.

Results coming now...

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>>8759800
After 100,000 simulations at H/b = 0.5, 0.6, 0.7, 0.8, 0.9, 1:

I've narrowed it down to about 0.65

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>>8759822
After a bit more refinement, I get it down to H/b = 0.645.

>>8759859

Interesting also there is a patent on 'fair' 5-sided dice (although this one has bevelled edges)

https://www.google.com/patents/US6926275

From what I can gather, the dimension given are:

H = 13.6 mm

h = 17.96 mm

So b = 20.74 mm and H/b = 0.656

Pretty close to H/b = 0.645 from the simulation

>>8757851
Make a D20 and mod5 the answer.

The answer is difficult because it is a 3d solid with physics so you'd have to calculate the force on the dice for each edge and corner and find a way to optimize it.

I'm not sure you could make it fair.

>>8759910
Finally, you can think of this as a Markov chain and figure out an exact solution based on the assumption that the probability of going from R-R or from R-T is proportional to the energy required to do so:

You get H/b = sqrt(15)/6 = 0.6455...

>>8758422
behold the engineer in a sea of scientists and autists

>>8757851
20 sided die
Mod 5 + 1
...
Profit

>>8757851
How is this even a die? You'd have a triangular prism with numbers on the faces.

>>8757851
Go make your 5 sided dreidel somewhere else fucking jew

>>8758513
I bet you haven't even watched all the numberphile vids either...baka what has this board come to

OK I got an idea.
There are two kinds of rest states:
#1 rectangle side down
#2 triangle side down
from the first state there are two ways to transition to state 2... kinda. You can tip it over either side.
From state 2 there are 3 ways to get to 1.
So how about the dimensions are set in a way that the energy it takes to tip the dice from state 1 to state 2 (the potential energy difference) when the center of mass is highest durng the tipping (on the edge) has the ratio 3:2 to the energy from state 2 to state 1.
It's really fucking arbitrary but the idea is that you have less ways to go in one direction so it should take more energy to perform that turn so the total energy is the same for both directions. It's a shitty model but it could be a start.

>>8760547
Not sure if retarded

Holy shit, this is probably the most civil, intelligent thread I've seen on /sci/. Good work anons

>>8759642
Brainlet interloper here, stupid question time: does arriving at a contradiction when trying to solve for a ratio imply that the result is an irrational number?

>>8761926
Possibly there is a complex solution, I'm not sure, but if there was I don't think it would have any meaning in this context

>>8761972
Sorry to add to this... I misread your post and thought you wrote complex instead of irrational

There's no reason why it couldn't be irrational