How Knot To Hang A Painting

How Knot To Hang A Painting


– This week’s guest video
was pitched to me as a puzzle and I couldn’t solve it. So please welcome
physicist Jade Tan-Holmes who’s going to show you the same puzzle and then solve it with maths. Jade, take it away. – Here’s a puzzle for you. You have a painting, a piece of string,
and two pins stuck in a wall. Can you hang the painting in such a way so that if you remove
either one of the pins the painting falls? The thing to focus on here is
how to tie the string around the pins, not any of the physics of the objects. So if you tie the string in the normal way
one would hang a painting, this obviously won’t work. It’s pretty easy to think of ways we can tie the string so it falls
when one of this pins is removed. But the challenge is to make it fall
if either one is removed. Pause the video now if you’d
like to try it for yourself. So what we’re working towards is a way to tie the string around both pins
as a whole but neither one individually. So there are a few different
ways to solve this puzzle. But we’re going to do the one
which takes us through knot theory. Knot theory is the mathematical
study of tangled systems. As this is a problem about
tangling things together, chances are knot theory
will come in handy. With puzzles like these
it’s sometimes hard to know where to even begin. A common technique mathematicians use is to strip away everything but
the most important features. So, let’s think about
what we can get rid of. What parts of the problem
aren’t providing us with any useful information? Let’s look at the painting. It’s cool, but is it
really telling us anything? It’s just joining the
two ends of the string. If we get rid of it and
tie the string together we can reframe the question as can we hang the string
so that if either one of the pins is removed,
the string will fall? It’s just a simpler version
of the same question. Sorry, fashionable chicken. [chicken squawks] What else is unnecessary info? Let’s take a look at these pins. Basically nothing matters about them except how they interact with the string and the fact that they don’t
interact with each other. We don’t care about their
positions, size, shape or colour. We don’t even care that they’re pins. Mathematically, it makes
sense to want to deal with just one type of
object instead of two. Imagine reading a story that’s written in both English and Chinese. Sure, both languages
make up the whole story but it’d be much easier to work with if it was all the same language. This is exactly what
we’re going to do now. Translate our problem all
into the same language, the language of knot theory, by converting our pins into strings. All the properties we care
about are still there. If we tie them shut they
can’t interact with each other and they can trap our original string. In knot theory a closed string
is called a knot and a group of entangled
knots is called a link. Now we can reframe the question again as can you tie the three knots so that
when together they’re interlocked, but if you remove one,
the others fall apart? So far we’ve gotten rid
of a bunch of information, but we haven’t really
isolated what’s important. What should be focus on if
we want to solve the puzzle? Well, there’s really only one way a string
can interact with another. It can cross over them
or it can cross under. The crossings are the only
things we have control over and are the key to unlocking the puzzle. The crossings basically
determine one knot from another. But the thing is,
the same crossing can look very different. This makes knots hard to reason about
in an organised way. It’s like if you had all
the components of a story but had no idea what order
they were meant to be in. But if we organise the crossings we can reason much more
clearly about the knot. Let’s take this random link of two knots. If I start on a random string,
at a random point and start following the string,
drawing on my paper as I go, we can keep track of all the crossings
in an organised way. We keep going until
we come back to where we started. We’ve just created something
called a braid diagram. Knot theorists use these
to model all kinds of links. The point of braid diagrams is
to keep track of the crossings while getting rid of all the
unnecessary information distracting us. In a braid diagram, the lines representing our
strings are called strands. If we label the strand
positions one, two, and three, when the strand in position one
crosses over the strand in position two we can label this crossing X, and when the strand is position one
crosses under the strand in position two, we’ll call this inverse X, as it basically undoes what X just did. When the strand in position two
crosses over the strand in position three we’ll call this Y. And when the strand in position two
crosses under the strand in position three, we’ll call this inverse Y. This notation is called a braid group. And actually everything we need to know
about our problem is encoded in these letters. The way these letters work
is that the X and inverse X cancel and the Y and inverse Y cancel. But they only cancel out if
they are next to each other. So coming back to our problem we want the knots to be
interlocked as a whole but none of them locked
individually to another. So if we think about this
in terms of our braid group, removing a knot is like
removing all crossings involving that knot. If we remove the green string
all the Xs will be removed. And if we remove the pink string all the Ys will be removed. What we want is a configuration
where as a whole nothing cancels. But if we remove either term,
everything cancels. Let’s see what we can do. We’ve got three strings. Our system is symmetric,
so what we do first really doesn’t matter. Let’s just start with an X. Remember the pins can’t
interact with each other, so we have no choice but to
do another X or inverse X. We don’t want to have an X
and an inverse X next to each other because we don’t want
anything to cancel yet. So we’ll do another X crossing. This is just like wrapping
one string around the other. We could keep going like this with the Xs
but we need to include the other string, so let’s do some Ys. Now because we want our terms
to cancel out in the end, that means every X needs an inverse X and every Y needs an inverse Y. If we do two inverse Y crossings now they’ll immediately cancel
out the Ys we just did, which isn’t what we want. So let’s try two inverse Xs. Now we can do two inverse Ys. So let’s take a look at where we’re at. If you remove all the Y terms, the remaining X terms
cancel each other out. And if you remove all the X terms, the remaining Y terms
cancel each other out. But when everything is together,
nothing cancels. That’s exactly what we want. Now let’s work backwards
to the original puzzle, chicken and all. We can use the braid
diagram to model our link. Let’s see if it has the property we want. That when all three of them
are together they’re locked, but if you remove one
the others fall apart. Yup! Now let’s bring back the
chicken and the pins. Okay guys, this is it. So now we’re going to map
our braid group notation onto our system of pins. So, remember from the braid group diagram that basically a double X corresponded to
wrapping one string around the other. And a double inverse X corresponded to
undoing that move. So here we’re going to say that a double X corresponds to
wrapping around a pin clockwise and a double inverse X
corresponds to undoing that move which would just be wrapping
around the pin counterclockwise. So, double X. Double inverse X. Double Y. Double inverse Y. So let’s do it. Double X Double Y. Double inverse X, wrapping
around counterclockwise. Double inverse Y. So now the moment of truth. [drumroll] [thudding] BOTH: Mathematics! So this definitely wasn’t the easiest or most straightforward
way to solve this puzzle. So, why did we do it this way? Well, now we have this whole framework for solving this problem for
any number of pins easily. If we want to add
another pin to the puzzle this would be a nightmare to brute force. But we can just work backwards
from our braid group, and it’s simple. Today we used knot theory
to solve a fun puzzle, but it actually leads to some
very fundamental mathematics. We were searching for a
way to wrap the string around both pins as a whole,
but neither one individually. But more broadly we had to keep track
of the relationships between smaller parts of a bigger system. This kind of connection has led to
deep insights in many areas of science. Just recently, people have been exploring
the design of quantum computers using particles that behave just like
the braids that we saw in this video. So if you want to win
a Nobel prize someday better start tying some knots! – Thank you Jade!
Go subscribe to Up and Atom. I would recommend starting with her video
on quantum tunnelling. And next week, a video that
may leave you breathless.

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