A2 Basic 33 Folder Collection
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so here's three of only three of the
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
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Too Many Triangles - Numberphile

33 Folder Collection
林宜悉 published on March 28, 2020
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