Platonic solids, the things you can make

out of equilateral triangles. So here

I've got tetrahedron, four triangles, octahedron, eight triangles, and

the icosahedron, 20 triangles, what's changing

as you go from here to here to here is

how many triangles there are around each

corner or on each vertex so the tetrahedron

right has three triangles around the

corner and the octahedron has four and the

icosahedron has five, what comes next

well, six triangles around a vertex do

you get a platonic solid that has six

triangles around a vertex? no no no what

do you get?

alright so here's six, here's you know

there's one triangle, two, three, four, five,

six, so six triangles around vertex that

tiles the plane right it goes on forever

so it doesn't have any curve to it

so here's a way to think about it so

your triangle has 60 degrees at this

corner and so like 60 plus 60 plus 60

those the three around here that adds up

180 which is well less than 360 so it's

sort of curves and makes a cone shape

you do this with the octahedron you get

60 60 60 60 which is what 240? still less

than 30 60 with five of them you got 300

when you get 6 it adds up to 360 but

there's no there's no lack of space

there's no there's nothing missing so it

doesn't make a comment just makes it

flat 3 4 5 6

what happens if you put even if you try

and put seven triangles on each vertex?

actually you know what would be good to

try drawing it. I'm not going to be able

to make them all the equalateral but

let's see how far we can get. so I got to

fit seven, so there's going to be one

here and then I got to fit three in

here, so i got to do this and all right

so I fit seven around a vertex so

let's keep going so I got to here

already so maybe I'll do that that's

five six seven. 3 i gotta fit

for and here you can go for quite a

while actually before you start running

into trouble 1234 you can speed this up

right?

okay, one two three four, i got to fit three

in here you see it starts getting

difficult

there's too many of them

they don't really fit very nicely and

you start... now I'm in real trouble look

at this one two three four I gotta fit

three in here

1234 in here so you can sort of see

that this is... well try it at home,

how far can you get drawing triangles

seven triangles on each vertex. What if i

don't do it on paper what if i do it with 3d

printed stuff then you get this thing so

right well you can see there

1234567 triangles on each vertex you

know the first thing you try and do you

try and lay it flat and it doesn't work.

You can make bits of it be kind of

flat and there's some given the hinges

which sort of helps out but then it

bunches up somewhere else. There's

just no way to make it flat everywhere

it doesn't work. right what's going on

right? these are sort of like they're

closed around this or like spheres right?

three four and five six is the flat

plane. sphere, plane

what is this thing trying to be it

doesn't look like it's some nice surface

is just some horrible thing we don't

want seven years maybe it will be better

no it is not better right so this is a

eight equilateral triangles around each

vertex. I mean you can count them

to like it is horribly crinkly a sort of

like it's like call was like lettuce. so

these are positively curved things which

like a hill or Bowl sort of curves

towards you in all directions, whereas

like something that's flat it's not it's coming at all or it can

curve in one direction but then the

other direction is is flat right doesn't

does that make sense?

like its curved this way but it has to

be flat this where you can curve in both

directions so one of the directions

plants really just flat and these are

like it wants to be like a saddle. if you

imagine my hand is a horse right and

like the hand is like awesome like your

legs go either side of here so so right

so what does that mean like sometimes

it's coming towards you, and sometimes

it's curving away from you, so when

you've got both of those that's negative

coach or negative gassing curvature so

what are these things these are sort of

like so these are models of hyperbolic

space in a way they're sort of too much

of it. it doesn't want to fit in

ordinary euclidean space and kind of

wrinkles up and gets out of control

so these these ones here right you you

run out of triangles, right. once you're

done with your 20 triangles on the icons

evening you're done this one I could

have kept going forever right is it's

like I could just keep trying this out

words and it was just you know for the

table and go off around the world this

one

well so again i've only done like a

little bit of its it seems clear that I

could keep going

I mean I tried doing it over here with

the pen but it was sort of you know it's

difficult but it seems like you know if

you allowed to sort of wrinkle them up

out of the plane and like make this sort

of wrinkly thing that I could just keep

going well you kind of might run into

trouble there's actually an open

question nobody knows the answer if you

keep adding these things

how far can you go out into space

without it crashing into itself

I mean for this physical thing there's a

definite limit and and here's here's the

argument right so as you sort of go out

from like the central ring to the next

ring to the next ring like the amount

of space you've got to play with his

only going up like that the cube of that

distance because you're making it a

sphere of possible places you could put

triangles but the number of triangles

that you need actually goes up

exponentially so exponential vs cube you

lose right you if you go like a few

layers out you'll just get incredible

numbers of triangles in a solid wall

because these are made of plastic the

house thickness each one of the house

volume and then you just run out but the problem that's open is

like suppose there two-dimensional

triangles, like connected at the hinges

perfectly,

how far can you go? is there

actually limit? because like two

dimensional triangles you could stack

like tons of the next to each other and

maybe you'd somehow be able to fit them

in without crashing into each other we

don't think so but there's no proof like

nobody knows what's the worst that the

father so you can go

I mean it's sort of like really quickly

and nasty and not really smooth and in

the same in same sort of way like the

icosahedron is not that smooth it's got

these sort of corners so what you can do

to an icosahedron to make it smoother is

like make a geodesic dome so here's one

way to do it right you take each

triangle and you subdivide it into for

you know like a triforce cut it into

four you got these extra vertices on in

the middle of the edges and you

come out of the sphere and that makes a

smoother polyhedron it's no longer

regular but you know it's a smooth the

thing that's more like it so I was like

all right let's do that with this this

it's got this crinkly thing that's to like

here like the angles add up to

360 plus 60 is 420 too many

there's like a lot of angle at each

vertex, so if you do this sort of like

geodesic dome thing with this thing then

it turns into this thing, so this is like

it's all hard to see but so

I'm going to see this right so these

four triangles the the three on the

outside or slightly isosceles and the

one in the middle is actually equal at

all

so this is subdivided and then and then

the vertices to the activities just

squished around a little bit to make it

make it a bit smoother and then it's

it's nice to write it sort of you can

get quite large bits of it to be flat although

you can get all of it to be flat always

sort of bunches up somewhere

no matter how hard you try.

yeah it's a hyperbolic doily that's what

is really likes to be in sort of like

saddles or crinkly start for ya fun to

play with the short side so long sides n

the short side is n minus 1 so each

rectangle is n times n minus 1 this

three of them so it's three lots of that

and then there's one in the middle

well right so that's a really hard

question

okay all right here we go