[Music]
hey Vsauce Michael here if every single
one of us held hands together in a chain
of unity around Earth would there be
enough of us to go all the way around
the planet there are about seven and a
half billion of us and that's a lot but
remember that that many human bodies
thrown together into one big pile would
barely fill the Grand Canyon this is all
of us in one place the physical bulk of
all living human flesh on earth today
would only make a cone about 7,000
people tall and 2,000 across that's it
but that's a three-dimensional shape
what if we made a one-dimensional
single-file line of people each with say
1 meter of room and we stretched that
around the planet well we would make it
all the way around and still have 99.5
percent of the human population left if
we made a circle that included everyone
the ring of people would be more than
2.4 million kilometers in diameter
dwarfing the orbit of our own moon now
that's not just a circle that is a sir
cool let's talk about circles today
specifically something that they do roll
but what is rolling well rolling occurs
whenever something moves with respect to
something else and is always in contact
with that something else and the contact
points have instantaneous velocities of
zero that is there's no sliding in
mathematics the path traced by a point
on a rolling object is called a roulette
French four little wheel the centre of a
disk produces a roulette that's just a
smooth straight line while rolling on
one
and this is why disks are good wheels a
square on the other hand would be bumpy
but square centers can make straight
smooth Roulettes
across the right surfaces this is the
principle behind square wheels which I
recently had the pleasure of enjoying at
the Museum of mathematics in New York
City Stan wagon wrote a fantastic and
famous article about wheels which I've
linked down in the description below
it's a great read he also contributed a
fantastic interactive tool to the
Wolfram demonstration project that
allows you to build a wheel and then
find the corresponding road shape that
allows it to roll smoothly the roulette
traced by a point on a disk as it rolls
on a straight line has a special name
it's called a trochoidal Greek word for
real TRO Coase now trochoidal prolate
depending on whether the tracing point
is inside or outside the circumference
if the point is on the circumference the
resulting curve is called a cycloid
cycloids are very special and they are
the star of this episode now I've been
working with Adam Savage a lot lately as
we gear up for brain candy live our 40
city tour that I hope to see you at
recently I asked Adam for some help with
rillettes Adam give a favorite polygon
you can't they're like your children you
got a favorite eye I don't have a
favorite either I do have a favorite
thing that you can do with the polygon
okay make a cyclic on a cyclic cyclic on
is an actual actual thing it's an actual
thing do have a polygon around here
here this this here's a squid right
right say this is a square yeah
if I take one of the vertexes of this
squid and I start rolling this square
and you follow where that vertex oh
shoot
let me roll this better are you ready
yeah all right the actual path
it describes a curve described here yeah
that curve is called a cyclic on and as
you pick polygons with more and more
sides you get closer and closer to what
I want from you today okay a cycloid a
cycloid now I I could have just done
this in Photoshop and done like an
animation like I normally do but you
have a shop we can actually build and
you can build the bit okay why did I
want to build a cycloid well let me ask
you this with only gravity to move you
what's the fastest way to roll or slide
from a point to some other point below
but not directly below would it be a
straight line well that would certainly
be the shortest path but when you fall
gravity accelerates you and falling
vertically a lot right away would mean
having a higher top speed during more of
the journey and that can more than make
up for the fact that a path like this is
much longer than a straight line but of
these two considerations accelerate
quickly and don't have too long of a
path what's the optimal combination
finding the answer the path of least
time is called the brick east a chrome
problem and it's been around for a while
Galileo thought the answer was a path
that's just a piece of a circle but he
was wrong there's a better one and in
1697 Johann Bernoulli came up with the
answer using a very clever approach to
see how he solved it let's start with a
similar problem you are standing in some
mud and you want to run to a ball in the
street as quickly as possible now a
straight line would be the path of
shortest distance but if you can run on
pavement much faster than you can run in
mud the path of shortest time would be
one in which you spend less time moving
slowly in mud and more time
on the surface your bast on the angle
you should enter the pavement at depends
on how fast you move on both surfaces as
it turns out when the ratio between the
signs of these two angles equals the
ratio between your speeds across both
surfaces the resulting path will be the
optimal quickest route this is called
Snell's law light always obeys Snell's
law when it changes speeds like when it
leaves a material in which it moves more
slowly like water and enters one in
which it moves more quickly like air it
always refract according to Snell's law
in other words light always follows the
route that is the fastest for it to take
Bernoulli used this fact to tackle the
brick ista chrome problem light changing
speed is analogous to a falling object
changing speed but of course a falling
object doesn't just speed up once its
speed is always increasing its
accelerating to mimic this using light
which Bernoulli knew would always show
him the fastest possible route all he
had to do was add more and more thinner
and thinner layers in which the speed of
light was faster and faster and faster
and well what do you know there it is
the brick East a chrome curve the path
of least time roll down a track like
this and you will beat anything rolling
down any other path every time Bernoulli
was clever enough to realize that this
curve can be described in another way as
a roulette specifically he noticed that
it was a cycloid the path traced by a
point on a circle rolling along a line a
cycloid satisfies Snell's law everywhere
to see why I highly recommend watching
this video on the brick East a chrome
problem this channel is fantastic by the
way I'm a huge fan the visuals and
explanations are top-notch anyway a
cycloid provides the perfect balance
between keeping the journey distance
short and picking up speed early now I
told Adam all of this
and I told him that it would be really
fun to have a cycloid curve we could
roll things down and he said clearly we
should start building we should just
start building it what we need is a
circle okay that's a the height that we
want mm-hmm and then we're going to use
that circle to trace a night cycloid
curve I want to do a race right so yeah
yeah look if we're gonna do like if you
have point a here and point B here and
you say that there's some kind of curve
that's better than a straight line in
terms of something traveling between
those two then I want to also make a
straight line from A to B yeah and maybe
also a really extreme curve like that
right okay got some protractors there
you have a compass extension I do I have
all sorts of I've never seen such thing
is that it yeah that connects to that
give or take I almost am I feel guilty
that this is like a dream of mine coming
true oh really but it's such a nerdy
dream too it's not like I want this you
know Red Ryder BB gun it's like I just
want to curve that things roll down
okay so then clearly when we're done
with this this is this is my Christmas
present to you
I think I'm currently like working out a
way to do this in my head that actually
makes it fairly compact and not super
not super complicated yeah take a blade
and cut out like an inch around the
whole thing
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how are you gonna do the finishing I'm
going to cut this unless you would like
to on the band's you can start okay
[Music]
okay now that's a little bit rough so
we're gonna finish it on my disc sander
oh wow
that's close enough it's pretty good
yeah so now you want to use this to draw
a psych psych Lloyd yeah so you'll need
a little hole that's what yeah so that
the point of our drawing implement is on
the rim not above it I'm doing this
right inside I don't think so it is
that's even better than having the rim
hole because we want the line to be
right on the edge okay so what we're
gonna do is we're gonna create a pattern
for the cycloid curve that I'm then
going to transfer to acrylic to clear
acrylic regular right allow us to see
things really clearly yeah this allows
you right roller sight because we're
just doing it it's literally like that
right oh yeah see this is perfect yeah
pushed against here I shouldn't slip
this is like a Ouija board for geometry
nerds
it is a Ouija board for geometry nerds
okay there we go
that's smooth out Ricci stack rone
that's that's that's the curve that
we're talking about yeah that's the
beginning that's the end and this is our
pattern yeah cool that's cool
okay so I'm gonna end up with let's say
3/4 of an inch thick but it's gonna have
channels table saw it out of it mm-hmm
travel its length and in those channels
will sit my clear acrylic acrylic curves
and it'll also have an upright
that also will have the channels milled
out of it on the table soft and that'll
allow the curve to sit and be supported
yeah a little backstop here easily at
the bottom that will allow us to hear
that they all hit at the same time
there's a couple things going on one is
is that we've got cycloid straight line
and then we've got extreme curve right
and are these you had mentioned bending
the acrylic so we could adjust the no
magically I have this is you like this
the acrylic will just be a thin of
acrylic just a thin sheet of it
traveling on that will be and I have
material for this some Delrin or acetyl
rollers that look like this so from the
side they'll look like this they'll look
like an H ah huh in which the acrylic
sits in there and the roller is
self-supporting on the acrylic but rolls
down
ah that is tolerance love it
I like those curves while Adam and I
build a real-life cycloid track let's
take some time to appreciate other kinds
of Roulettes as mentioned before
trochoidal anon straight lines but an
epitome oi is made when a disk rolls
around the outside of a circle roll a
disk inside a circle and what you've
made is a hypo trochoidal names for the
curves you make when using a spirograph
toy notice that the holes don't lie on
the circumference of the disks though
some do come close there's a special
name for epic rock woods and hypo
trochoidal
an algis to disks rolling on straight
lines they are epicycles and hypo
cycloids now if two circles have the
same radius a point on the rolling one
will touch the stationary one exactly
once always in the same spot creating a
cusp this cute heart-shaped epicycle ID
is also known as a cardioid if the
rolling circle has half the largers
radius you'll get a 2 cusped epicyclic
the shape of which is called a nephrite
because it apparently looks like a
kidney I guess 1/3 the radius gives you
three cusps 1/4 four cusps and so on as
for hypo cycloids if the inner circles
radius is 1/4 of the larger x' the
resulting Roulette curve is called an
asteroid because it looks like a star
which the ancients also thought about
asteroids 1/3 the radius and you've got
a deltoid named after its resemblance to
the Greek letter Delta 1/2 the radius
and well you get a straight line this
fun relationship is called a 2 C couple
rotational motion turned into linear
motion follow a number of points on the
rolling circle and you'll get the famous
illusion where every individual point
moves in a straight line but the whole
thing describes a rolling circle put a
handle on it
and you've built a trammel of Archimedes
aka an ellipse a graph when used to
ellipses AKA a hillbilly entertainment
center
when bought in Osceola Missouri anyway
let's get back to Adam and eyes curve
comparison build okay okay there's your
finish line you ready I'm ready
all right I'll count us down three two
one go okay here we go
3 2 1 go 1 2 3 with the asleep was
second straight was last straight was
last the shortest distance between two
points was last slowest way to get there
yeah it certainly was let's try it one
more time yep because it was super close
the brick ista Crone curve was by far
the winner yeah what a mouthful of a
name by the way barista drunk East uh
Crone but Keith's brick east across not
related to the brachiosaur looked at our
that's I once looked up the difference
between ingenious and ingenuity and they
don't have them and they're not related
no all right now ready I'm ready
3 2 1 go
yeah same results same result
1 2 these are so close it's hard to tell
[Music]
well we're answering the main question
which is that the barista chrome curve
is the fastest way to get there well a
brick easter chrome curve is also known
as a tata chrome curve that has another
property that's that we should test
what's it and that is it's all you know
hold up before you get to this we've
established between these three curves
that the cycloid made brick easter
chrome curve is by far the fastest it's
by far the fast okay great you get steep
and unfortunately you pick up a lot of
speed right away but then you've got a
lot of this with little acceleration yep
you go straight down and you know funny
enough you what you want is that perfect
balance of gravities acceleration but
also moving to where you need to be and
that is asked that's fascinating to me
that a geometry which is the cycloid
curve would yield such an efficient
exploitation of the forces involved yeah
exactly
because if there was no acceleration if
there was just one force in the
beginning straight-line would probably
be the fastest right okay what is the
other quality of the Tata chrome curve
you just said it
Tata chrome means same time so as the
geometry and math tells us no matter
where you start an object yeah
what whoops the clamp city it was my
fault but also come on clamp no it
sector you know how to do these tables
are a pain in the ass because they have
these lips on them and they actually
that's like it's it's actually my fault
so I'm gonna remove the straight line
guy as well so we're left with a cycloid
which is called a brick East a Crone
curve but it also has a bizarre property
where no matter where I start an object
on it when I let go they always reach
the bottom in the same amount of time
wait a second so if I started here the
amount of time it takes for this to get
to the end
is the same amount of time it takes for
it to get to the end from here
yeah and the same from right here well
Wow okay so starting from here it's
gonna be tough because of friction yeah
now if you do this in software it's
perfect but that's boring yeah this is
the real world and maybe we won't start
the network I mean if we do one two and
three yeah are we actually I feel like
we could probably we could probably get
one to work from here like that there
might be a lot of friction involved but
look at always see we can always see
we've got enough of them that's why we
made three curves so we can test this
property and these were all cut and
sanded clamped together so they are
incredibly similar I will have to tamper
the edge as I did on this one yeah a
little known fact that once you've torn
paper off of Plexiglas it crumples into
a really nice wall to be thrown long
distances a Plexiglas in the first place
I've never seen it happen because it
doesn't really feel glued I think this
is I hope that's cool because in theory
and practice theory in theory theory and
practice are the same thing but in
practice I went to the University of
Chicago where one of their sings
well that sounds good in practice how is
it in theory hahahaha life of the mind
there so let us see if you do if I do
one up here and you do one there and
okay yes so this has got to be without
this I think the same person should
release them because then okay I'm it
better so here put that one here okay
all right so three different positions
three this one has longer to go yep this
one has the shortest path so who's gonna
reach the bottom first okay here we go
let's see would you wish it would say
the one in might this one yeah go first
ready yeah good pretty good that was
awesome that was so they all ended up
they lined right up alright let's do
let's let's switch it up okay even I
think you're gonna be the one to release
them yeah otherwise it won't be easy to
time here we go three two one I'd be
garnish them lining up yeah they line up
like they're waiting for each other I'm
gonna stand further away cuz I want to
see the full path okay ready yeah three
two one how cool is that they totally
line up to hit at about the same
periodicity they went for here so given
the vagaries of some extra frictions
here and there there they're actually
hitting this at the same sort of
periodicity that they are when they
start at the same point so there's that
and then
[Music]
is this some is this the Tata chrome
curved demonstration ring of your dreams
it really is it is it's also the brick
Easter Kron rig of my dreams and the
cycloid rig of my dreams so check that I
keep doing that yeah that is really cool
another one yeah that was yeah exactly
on point Wow
raesha see in this game no matter where
you start no matter who you are you're
always a winner there's always a tie so
this is brain candy for me Adam toasties
this is something that was previously
abstract and only seen in animations and
in text made real now I can put these
wherever I want I'm not stuck with what
someone else did I can physically hold
it and that's that's what makes brain
candy exciting for this it is like that
I love taking the theoretical and making
it physical and actually honestly we've
always had what seemed to me like sister
enterprises yeah and it's nice to join
them together yeah this is this is a
little child of ours isn't it maybe
maybe one of us wasn't so good at
sanding and maybe the other of us kind
of saved the day but it's real now yeah
and it's alive and I think it's loving
its life that was fun it was really fun
true Adam thank you so much for your
help
working with you is always amazing I
hope to see all of you out there
watching at brain candy live it's going
to be incredible and in your daily lives
may you always find the Tata chrome the
solution that brings you and others
together even if you started in
different places
and as always thanks for watching
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