- This puzzle was invented
by Joel Langer in the '80s and it's unlike any puzzle
I've ever seen before. The challenge is to take this
loop of wire outta the bag and then put it back in again. How can this thing be folded
back down into a circle that fits inside the bag? It seems impossible, but I know it isn't because I've seen the solved state. It's possible to flatten the puzzle, but it's too big to get
back inside the bag, so you try fold it over,
but then bits of it pop out and well, it doesn't look
anything like the solved state. I was eventually able to solve this puzzle thanks to Mark Pauley, Michele
Vidulis and their colleagues. They realized that objects
like this are fascinating for all sorts of reasons, and
they wanted to find out more, and that's really what
this video is about, these weird objects. So this is an example of a
knot in the mathematical sense. Here's the difference
between a regular knot and a mathematical knot. This is a regular knot, but
without cutting the rope, I can continuously deform it
until the knot disappears. That's because of these loose ends. To create a mathematical knot, you just have to trap the knot by joining the two ends together. In other words, a mathematical
knot is a loop of string that can't be untangled. Knot theory is an interesting
subject in itself, but this video is about what happens when you create a physical realization of a mathematical knot
using elastic material. Here we are using nitinol wire, and the result is some
really weird geometry. So in their paper, Mark
and Michele went searching for other elastic knots that had interesting properties. Luckily for them, there
already exists something akin to the periodic table, but for knots. By the way, there's a really
nice theorem about knots that says it doesn't matter
how complicated your knot is, it can always be arranged
into a braided loop. For example, this knot is
equivalent to this braided loop. In other words, it's been
proven mathematically that this can be arranged
into something like this. Our job is to figure out how. These braided loops
aren't stable, by the way, when they're made of nitinol or anything elastic, which
is actually really cool because it means they pop up like this. They're springy. Actually, some knots are stable in that braided loop configuration. For example, the trefoil
knot is only stable in that configuration when
made of an elastic material, but it turns out it's actually quite rare among the periodic table of knots. The simplest knots after the trefoil is called the figure 8 knot, and there seems to be three
different stable configurations when constructed from nitinol. There's one that's basically flat that looks like Marvin the Martian. There's this one that
looks weirdly asymmetrical. And then there's this arrangement that's actually really nice. I like the fact that it's
got the same symmetry as a tetrahedron. Well, not quite as much
symmetry as a tetrahedron. So okay, it has the same
symmetry as a tetrahedron where you've colored the
edges in a certain way. Anyway, you also get this
nice flower arrangement. When you look at it straight down. You'll see that a lot
with these knots actually. When I say that a certain
knot configuration is stable, I mean that it would be stable even if there weren't any friction. Like there's a decent amount of friction where the knot is self touching, but it's still pretty slippery. But Mark and Michele were able to show that these three
configurations are stable, even for a frictionless knot
by modeling it in a computer. Interesting quick
tangent, it's been proven that all elastic knots are self touching. In other words, there's no way to make a knot from
elastic wire in such a way that no part of the knot touches
any other part of the knot. Like I can force, for example, the trefoil knot into an arrangement that is non-self-touching,
but it's not stable, and if I let go, it ends
up touching itself again. That might seem completely obvious, but actually it is possible to create a non-self touching
knot with elastic ribbon that has two fixed points. So the case for elastic
wire with no fixed points really did need proving. So how did they go about looking for these stable configurations? Well, they would take
a knot from the table, like the figure 8 knot, for example, and they would draw the knot
in 3D simulation software as a bunch of straight lines on a grid. A bit like the Windows' pipe screensaver. The simulation software knows about the elasticity of the wire. So when you hit play on the simulation, it quickly settles down
into a stable configuration. A stable configuration is
a local energy minimum. A bit like rolling a ball down a hill. The ball won't move if it's
stationary at a local minimum. But what is this energy
that's being minimized in the case of elastic knots? Well, it's the energy that's stored in the bending of the wire, like I have to expend
energy to bend this wire, and that energy is being stored
as elastic potential energy, which can then be released back into kinetic energy, for example. So in this case, low energy
means low total curvature. So I know this is stable
because any way I try to move, it causes the total curvature to increase. It's like trying to push
a ball up out of a valley. So that's how you find one stable state. But how do you find all the others? Well, you draw out the knot again in your simulation software, and then you randomly vary the path. Then you press play again
and you see if it relaxes into a new different stable state that you hadn't seen before. You then repeat that thousands of times and see what you get. So interestingly, this is
not an exhaustive approach. We can't be sure that we've
found all the stable states. Like if you had three stable states that were really close to each other and you are randomly
picking a starting point in that energy landscape, you may never pick the starting point that's gonna get you to that
middle stable configuration. I'm showing a 1D energy landscape here, but it would actually
have many dimensions. What's really cool is that
Mark and Michele were able to find three stable
states of the figure 8 knot that no one had ever seen before, and for the more complicated knots, they discovered loads of new ones. The next simplest 3D knot
after the figure 8 is this one. It doesn't have a name, I don't think, but its designation is 9-40 because it's arbitrarily the
40th knot with nine crossings. This one comes in butterfly
and Marvin the Martian, but with three feet, this
knot is more complicated, and as a result it's more springy. That's quite a jump, isn't it? This is a simpler one that
creates a sort of bow tie. If I try to pull this into
a 3D shape in the same way that I would with the others
by pulling these lobes in, you can see why the
arrangement isn't stable. You see, because this
isn't locked in position, this knot simply doesn't
have the right arrangement of overs and unders to keep that in place. This one is a really good puzzle knot because it's also got
a stable configuration that looks like this,
and getting from that back to the bow tie is really hard. So this one is a flat
knot and always will be. It does still have a Marvin though. Let's go back to the original
puzzle knot from Langer then, because I think I have what I
need now to solve the puzzle. For a while, I didn't think it even had a stable flat configuration because like if I put it into the butterfly shape, this part of the rod here
isn't locked in place, so when I let go, it switches back to the three-sided container thingy. But actually, if I go back
to the forced butterfly and do my usual trick
to turn it into Marvin, actually, there you go. There is a stable flat configuration. So what is the trick to
get it back in the bag? Well, I noticed with this
knot, the way you do it is to look end on and squish it like that, and it kind of flips
into that arrangement. In other words, I need to line
up these two little triangles and squish them together. The puzzle knot doesn't
have little triangles, it has big triangles,
and that's the problem. But look, if I squeeze it
to make little triangles and then push those
little triangles together, that solves the puzzle, and I can get it back in
the bag without letting go, otherwise, it'll pop up again. If you look at their paper, they've got this brilliant
image on the front, and I really wanted to recreate that, a person-sized elastic knot. So I bought 20 meters
of camping tent poles. You know how putting up a
tent can be quite challenging? Well, this was like an
order of magnitude harder. If all this is making you
think about those tents that put themselves up, well, me too. In fact, it's such an interesting topic that I made a whole video
about it linked to that in the card and the description. But look, I did get it done in the end. Surprisingly, at this scale, the weight of the rods is enough to keep it in that unstable configuration
of a braided loop. But if I begin to lift some of the bars up just a little bit, it
pops out quite quickly. But if I give it a little
guidance on the way up, I can form that brilliant
three-dimensional knot. These tempos, definitely
deformed plastically a little bit during the
construction, unlike the nitinol, which just has incredible
elastic properties. I wanted to see if I could
get it to pop up on its own. So I made the whole thing much tighter by removing a few of the rods. In the end, it was still
held down by its own weight, but it didn't have to be lifted up as much before it took off, and when
it did, it was more violent. So William Thompson hypothesized that atoms were knotted
vortices in the ether, and he tried to create an actual
periodic table of elements by creating a table of all possible knots, like hydrogen is the figure 8 knot, helium is the cinquefoil knot, and so on. It's such a beautiful idea. Part of me wishes it
was true. But it's not. And the truth is important. Like in science, we have
a pretty robust system for getting at the truth, but the scientific method
isn't how most people learn new science. Most people get their science
news from, well, the news. Even if the facts are accurate,
you have to worry about like how is the story being spun
and what's being left out? That's not just true for science. It's true for politics,
economics, culture, everything. How do you know if your news
source is politically biased? If their reporting is serving an agenda? One solution is the sponsor
of this video, Ground News. Ground News was created
by a former NASA engineer to empower readers to navigate
the complex media landscape. They process about 60,000 articles a day from news sources around the world and group related stories together so that readers can compare
news coverage in real time. And you get all this extra
information about sources like do they have a political bias? How reliable is their reporting
practices? Who owns them? For example, this science
story about climate change, you can see a summary of how a news story is being reported differently
across the political spectrum, or you can click in and read for yourself. As someone who reads
a lot of science news, it's been eyeopening
to see how a news story that really should be neutral
is being reported differently on either side of the political spectrum. Seeing science news and all
news contextualize in this way I think is really important. And wherever you sit on
the political spectrum, perhaps you'll agree with
me that polarization is bad, and maybe a bit of understanding and empathy for the other
side could really help. If you're interested, the offer
on this one is really good. If you go to ground.news/stevemould, you'll get 40% off the Vantage plan, which gives you unlimited access to every single Ground News feature. The link is also in the description, so check out ground news today. I hope you enjoyed this video. If you did, don't forget to hit subscribe, and the algorithm thinks
you'll enjoy this video next. (upbeat music)
