5-Sided Die

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

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...

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

See

Draw a free-body diagram?

this is what google proposes

Already exists man
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.

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

>youtu.be/tSQIir5xxWc?t=152

wow, what a gentleman and a scholar

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

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

[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

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

Just speed it up

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?

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

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.

>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

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).

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.

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 and

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

>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.

*Well-defined and analytically solvable.

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.

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.

2/2

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...

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

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

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

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

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.

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...

behold the engineer in a sea of scientists and autists

20 sided die
Mod 5 + 1
...
Profit

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

Go make your 5 sided dreidel somewhere else fucking jew

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.

Not sure if retarded

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

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?

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

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