We got these weird sort of skew dice.
Like usual dice are cubes, right?
And these are not, these are kind of squished somehow.
Well they are a little bit sorta eldritch,
or like occult or distorted from the fourth dimension or something.
And what's the first question?
[Well the first question is are they fair?]
Are they fair?
Well let's roll them and find out shall we, I don't know.
Uhh, ahh that looks fair to me.
Uhh, yeah well so you know they work just like ordinary dice,
and right, are they fair? Well ok.
This is an ordinary dice.
So that's your, you know, your usual cube-shaped dice.
So why do you think this is fair?
And what is it about it to make it fair?
[It seems very symmetrical and all the sides seem the same area.]
Right, right yes, so really the reason why you know this thing,
I mean it's not actually fair
cause you know there's different like the six dents taken out of this,
but there isn't and you know what, forgetting all that.
Suppose it is an ordinary, you know just the perfect mathematical cube.
Physics sort of messes things up
and it, and it makes things, you know, bounce around in some crazy way.
And you want to know that there's no bias, right?
So, so here is a way to think about like what is it that makes this fair.
So if I sort of take this away and erase all the faces
and and write new ones on them and give it back to you,
you wouldn't be able to tell that I've done anything
because it's got these sort of symmetries that I can like move this around.
So it's about the symmetry.
It's about like, you know, I guess the three is the same as the five,
because you can move the thing to change the five into three
but it doesn't change the shape of it.
Ok so now I got these things,
so all I have to do is convince you that, that all the faces are the same,
that the symmetry can take any to any others.
So there you go, like so you can see the three-fold rotational symmetry here.
Like if I rotate it, like that, it's the same,
and then the three on the back: if I do this they're also the same.
So they're all the same as well, it's got the same...
So the property's called "isohedral".
"Isohedral" is the property of a polyhedron
which is like the symmetries move the faces to,
you can move any face to any other face.
I can rotate 120 degrees. I can rotate 120 degrees again, whoops.
And I can do this switch around thing here.
So these have like just enough symmetry.
These have the minimum amount of symmetry you can have to be fair,
whereas your ordinary cube has excessive amounts of symmetry, you could say, right?
I mean there is way, there's extra symmetries like this thing is the same as...
It's the same shape, but like I haven't... It's still face two.
Right so this has way more symmetry than you need
than this has just enough to be fair.
This is the symmetry of the cube, that doesn't make it any more fair, right?
The two is still the two.
Whereas this thing is nothing you can do that doesn't sort of
mean, right to anything I can do to this symmetry.
It means that, you know, it moves some face to some other face,
showing that you know and you need that to be fair,
all the things are like necessary.
Ok so we have like six-sided dice
and people have seen like 10-sided dice and 20-sided dice.
How far can you go?
Right so, well if you want a fair die?
Right so this is a 60- sided die,
which has the same sort of symmetry properties as we were talking about before.
Every face is the same as every other one.
So here's face 24, right,
and I guess I've got face, what's the opposite, opposite of 24?
'Cause they have to add to up 61, right?
'Cause that's what you do.
So that's 24, that's 37.
You turn it around and I guess maybe after turn it upside down again
and then it makes it the same, the same shape after you're done.
So 24 is the same as 37.
You do that with any of the other ones here.
One problem with this is that it's really round,
so kind of rolls forever,
so I can try and do it like, here we go.
[You rolled a one!]
That that's a critical fail I think. I wanna do 60.
[This one you're showing me now looks kind of fair again.]
[Does this one have excessive symmetry or this one is as bare minimum?]
Well it's getting tough like when you get to sixty sides, like,
like you run out of kind of symmetries that you can do.
There's one extra symmetry this has that isn't necessary in some sense,
which is the reflection.
Right, so like you can see a line of a mirror symmetry there.
So there is a way that, there is an isohedron, err…
a polyhedron that has the symmetry thing with sixty sides.
The only problem like I really really wish we could have done this,
the only problem is that it doesn't have opposite sides parallel.
So when it lands on the table there isn't an obvious face that's up.
Its kinda, we're really disappointed actually that we can't do that.
But yeah so that there are a couple of other candidates out there with sixty
which have a little bit less symmetry but they don't work for other reasons.
[Can you make bigger?]
Can you go bigger? Yeah yeah.
So, so there is one more thing you can do with a hundred and twenty sided die.
I can actually sort of explain how you do it just looking at this.
So this is got sixty
and as you said it's got too much symmetry, right?
This is reflection.
So what you do is you take each of these sort of kite-shaped faces,
and you divided it into two.
You have two little triangles, one on either side,
and then you've gone from 60 to 120.
So we're working on it.
Do you know about Numberphile2?
That's our second channel where there's bonus footage and out-takes.
Make sure you see the extras from this video. They're really interesting.
So this is a skewed version of the D12 which is similarly weird.
And while you're there, why not subscribe to Numberphile2?
That way you know all about the extra stuff
I don't always mention here on the main channel.
…, which means that of course we had to do both of them, right?
So we had to make two mirror images.
'Cause... I don't know, that's the right thing to do, right?