we have spent the day here with a square bridge that can roll because despite the
fact it is a square cross-section it's on a very specific mathematical path
which means it can roll without Center of mass going up and down this is an all new
path this has not been done before so this is the mathematics of the Cody do rolling
[Music]
bridge this is Cody Dock in the East End of London which is traditionally a very industrial
kind of area but over time the kind of water-based industry has been gradually leaving and this has
been you know revitalized and there's more Community coming in and so they decided to
renovate this disused bit of water here to provide more um large boat docking where there's not space
for on the main river so over there that's like the main river that joins onto the temps however
at the moment there's no way to get a boat from out there into the moring over there so if they're
going to fix this up and they want to get boats in this big they have to get rid of this dam which is
the current pedestrian bridge and this is a very pedestrian heavy area so if they want to get rid
of this they have to put in a new bridge that can move out of the way to let the boats in and
out and originally they got planning permission for like an off-the-shelf you know caner lever
up and down Bridge you can basically buy it in a box from somewhere in the Netherlands but the
artists who've moved in down here were like hang on that's a bit boring we can do so much better
than that and one day one of them was talking to an architect design friend of theirs named
Thomas Randall page who thought they had a much better idea so I knew that you could have um cogs
which weren't circular that could interface as long as you had the kind of ins and outs on the
cogs um that corresponded with one another um and then I also had seen this uh Square wheeled
bicycle um that this mathematician Stan wagon had demonstrated could run on a track made of a series
of conary arches um and it was it was from kind of combining those two ideas that the the idea of a
rolling Square came about there were an enormous amount of technological challenges I mean I I was
very naive and thinking that it was quite an easy thing to do because I'd made a model that worked
and then uh as soon as we kind of up the scale and all the forces get larger all the weights
get bigger and you you know friction and wind and all these things start to really play quite major
roles and then there was also um some unexpected uh mathematical uh complexity uh because although
a a pure square rolling with sharp Corners rolls on a series of perfect conary arches um in order
to make this work we rounded the corners so you're not rolling on the point of the square
at any point it kind of there's a there's a fillet uh a radius fillet between the straight sections
and this produced two geometries the caner arch geometry which rolls along the straight edge of
the square and then another geometry where you're rolling around this fillet which turned out to be
um an elliptical integral which I didn't know anything about that's all news to me um and
working with the engineers luckily we had a very uh young graduate who took it upon himself to try
and solve and um the the this this geometrical conundrum at the corner it was important for this
project to be hand powered and although often uh when I show it to people and I tell them it takes
20 minutes to open uh and they say why don't you just motorize it why don't you just put an
electric motor on it and for me there's a poetry in the fact that it's hand powerered and there's
work to be done in opening it it doesn't need to open very regularly maybe once a week maximum so
um there's a kind of ritual aspect there's a kind of like you know you turn this handle for
20 minutes to do it and it's very low energy in that way in terms of I mean there's a lot of human
energy but you're not using um electricity or an engine and there's also a lot less to go wrong
when you have an engine or you have Motors they don't know when something's going wrong whereas
when you're turning a handle it suddenly gets harder or there's a weird squeaking noise you
stop um and you kind of so you can do away with all the sensors and all of the tech complexities
um and and for me there's poetry in that because as the shape rotates It Center of mass doesn't
go up and down that means we can roll that way and we're technically not doing any work as in
we're not lifting the mass up or lowering it down carefully of course there's friction it's reality
but the point is in Theory it should roll nice and smoothly the problem is we need a sensible Center
of mass for that to work and the actual platform of the bridge is a lot heavier than the rest of
it so it means the top of these this is full of concrete and left over bits of Steel rebar to try
and get enough Mass up there this is a hollow I think and then you got the platform so the center
of mass ends up somewhere sensible and we can roll it that way and here it goes they made a special
exception because we were there at the same time as an engineering open day which meant first
of all they rolled the bridge for no practical purpose other than the beauty of the mathematics
and the engineering and it also meant that as well as Tom the designer we got to meet the
structural engineer Alfred a bit of a Nuance that we only realized quite late in the fabrication
processes is when um you add the rounded Corners to the square hoops and that was a bit of a new
mathematical challenge that we hadn't anticipated and the issue with that is that the equation that
you use um to derive the cery um is usually based on the idea of having four straight lines Define
your square and that describes the movement that the square then has to take Roll Along this track
but as soon as you add a rounded Corner over that region your equation changes you're no longer
dealing with a straight line you're dealing with effectively a small offset Circle rotating about
a center of gravity which is not in the geometric Center of that Circle and um when you plug it all
through the differential equation you end up with some quite nasty integrals and so we've
got elliptic uh elliptic integral um result and um we used a script to numerically integrate
that and understand exactly what the rotation of the bridge was going to be over that portion and
um and from that we were able to sort of stitch together two results so one from the cery result
for the straight section and then one from the rounded corners and that's how you end
up with the overall shape that you have here to help understand why the rounded Corners were so
difficult mathematically I made my own animation here of a square wheel rolling on a cenery Surface
the animation is not great but you get the idea the point is the center of mass is not going up or
down as the square rotates because it's a cenery underneath and we know the equation of a cenery
easy to solve but those pointy Corners are a bit of an issue in terms of the engineering and so
they decided to round them off using sections of circles so now when the square bridge is rolling
on the flats its ceries and once it hits the corners that should be easy now it's just rolling
on a circular wheel we understand that except that circular wheel at the bottom is the bit that's
rolling but the center of mass is still up here so you got to kind of Imagine The Wheel as having a
sticky out bit with the bulk of its mass way over there and now it's got a roll on a Surface such
that that Center of mass doesn't go up and down even though it's all the way over there and that's
what Alfred had to solve numerically cuz there's no nice neat equation for that J remember earlier
when Thomas mentioned Stan wagon who was the first person to make a square wield trike well Stan saw
what Alfred was doing numerically here and thought well hang on what would the rest of that curve
look like and so Stan actually worked out the complete curve that a circular wheel can roll
on such that a displaced Center of mass does not go up or down and yeah roll is a bit generous here
because it's kind of adhered to the track but the wheel can go through the track but that red dot
is the contact point and while this is lovely in general all Alfred had to do was numerically work
out a little bit of the bottom sections of the curve and then put that in the gaps when the wheel
section of the bridge needed to roll an incredible solution and that wasn't even the only bit of
mathematics that often had to do everything's uh interacting here geometrically and you've got
you know the length of the Trap which um needs to match the perimeter of the bridge that rolls on it
and then obviously the teeth need to intersect so once you have um a set of equations that describe
the movement of your Bridge what you can do which was quite a nice result you realized is
um take the track and effectively roll it around the bridge and the same way the bridge would roll
around the track you do an inverse transformation of the track around the bridge and every time the
track intersects with the outside perimeter of the bridge you can you can make a cut out so you
know that when the bridge then rolls it's going to miss uh the pins and you're going to have cutouts
in exactly the correct positions um so we took all our translations ations applied them to the
pipe sections and then sort of wrapped it around the bridge and that's what gave us the the tooth
profile of the bridge just before fabrication when the fabricator David who works for a company
called cake was comparing the total length of the track which as I said should match up with the
length of the bridge that rolls on that track and he noticed that there was 17 mm difference between
the two and he um he kind of flagged this I think it was boxing day or day after boxing day while I
was watching TV with uh my family at Christmas and um and he kind of asked where where's this where's
this difference coming from and so we had to do a bit of digging around into the math to understand
where where that discrepancy was coming from and it was it was because of that different equation
around the corners that we spoke about but that kind of yeah took us a little bit by surprise and
and you kind of had to trust the maths go through the process derive the new shape and then when we
finally matched it up with the new equations there was only A.1 mm difference which we put
down to the tolerance of the numerical integ ation effectively at the end of the day after
everyone had left we had the pleasure of putting the bridge back into its original walkway position
I can say that this thing is uh very much true to its Victorian era ancestors and that it can
be powered just by a human it's definitely a nonzero amount of effort for the record you
can come to visit the bridge you can walk across the bridge you can Marvel at the bridge but you
can't wind the bridge you can't just show up and start cranking it you've got to contact people
in advance and in theory once it's actually an open bridge and ships are coming through ships
have to phone ahead and people will come out and do this properly but do visit the bridge it's very
cool not normally an applied mathematician [Applause]
before you go a quick announcement so this bridge is a fantastic example of taking an
interesting bit of mathematics and finding an unusual use for it and there's a competition
on the moment to take an unusual bit of mathematics and find interesting uses or
applications of it and the bit of mathematics specifically is the Hat the AP periodic Monti
that was discovered recently in fact it's the entire uncountably infinitely many family of
aoic mon tiles you need to find an unusual use or application for them so just recently there
was a thing in Oxford called hat Fest which was to celebrate this discovery and they announced
this competition which is sponsored by Jane Street they're my fantastic Financial friends
who are also sponsoring this video yes it's a sponsor message so as well as sponsoring
my videos and my channel there is now prize money in two different categories so if you
can find a use be it tactical artistic or just downright crafty for the Hat tile or any other
member of the aioic tiling family there is one category for school students who are in the US
or the UK and a separate open category for anybody I'm sure there are people watching
this video and come up with some fantastic and unusual applications of the Hat tile so thank
you so much to Jane Street for sponsoring the competition and my videos and thank you to all
of you for watching right to the end and some of you I'm sure will enter that competition
[Applause]