A needlessly complicated but awesome bridge.

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]

Channel: Stand-up Maths

Views: 777,640

Keywords: maths, math, mathematics, comedy, stand-up

Id: SsGEcLwjgEg

Length: 13min 24sec (804 seconds)

Published: Thu Aug 31 2023