You Won’t Believe How These Shapes Roll! New Discovery in Math

this episode is sponsored by brilliant what if I told you that for almost any line you could draw there's a shape that could follow it in a new discovery a team of mathematicians have come up with an algorithm that creates a shape that can follow almost any path you can come up with then 3D print them they've called them trajecto and in a surprising twist these trajecto could help physicists solve problems in quantum mechanics welcome to up and Adam I'm Jade my first thought when I heard this was how does that work and my second thought was I want to make a trajecto so that's what we're doing in this video discovering how trajecto work and then making one of our own the trajecto mathematicians actually made their code open to anyone so if you have access to a 3D printer you can make your own too so the basic idea is very simple if I wrap this ball in some Play-Doh and roll it along a path while pressing down on it the grooves that form on the base of this sphere result in it following that exact path the harder I press down the more stable the rolling path the shape construction algorithm that produces trajecto shapes essentially mimics this pressing down harder behavior and calculates the exact indentations needed to trace any path but this method will only work with one period the shape will just stop rolling after one period of the line or it will just start doing something else what makes a trajecto special is that it can Roll Along an infinite itely periodic path a path where you see the same pattern repeat itself over and over again this is the difficult part but fortunately for us it involves a lot of cool math the first thing to know is that a trajecto is essentially made of two parts the plastic that gives it its shape and an inner sphere all of the math that makes trajecto work applies to just the inner Sphere not the plastic around it this is good news it greatly simplifies the math rather than working with infinitely different funky shapes we can just work with a sphere so if we want the trajecto to roll multiple periods we need to make sure that after its first period it ends up in the exact right position to begin its second period for that to happen the path needs to draw out a closed loop on the sphere when I roll this ball along my path you can see that it does not make a closed loop but when I roll this small it does so you can see that the size of the trajecto sphere needs to be scaled to match the path but that's not enough not only must the trajecto be in the right position by the time it gets to its next period it also has to be in the correct orientation an easy way to understand this is using a globe let's say my trajecto path starts here in Sydney by the time the first period of my trajecto roll is finished it's not enough for me just to end up back in Sydney the globe also has to be the right way up just like before I started rolling it mathematically this is extremely difficult to achieve to see why look at these two Rubik's Cubes I'm going to apply the same rotations to each Cube but in a different order this Cube I'll twist up then left left then up I end up with a different color on both cubes even though I applied the exact same rotations this effect is called the non-commutativity of 3D rotation groups it basically just means that when you twist 3D objects around the order that you do it in matters when you roll an object along a path you're essentially rotating it in all sorts of directions so the chance that you'll end up in the exact correct orientation by the time you finish is extremely low let's recap our first condition is that our path needs to form a closed loop on the inner sphere of our trajecto and our second condition is that the trajecto needs to end up in the same orientation as when it started by the end of each period due to a theorem in differential geometry called the gaus Bonet theorem these two conditions can only happen when the trajecto path cuts the surface area of the sphere in exactly half but according to the trajecto mathematicians these kinds of pods are infinitely rare the probability of stumbling across a path like this by pure chance is pretty much zero it kind of makes sense if you think about it I mean what are the odds that some random line you draw one even forms a closed loop you can see here that the sphere is the right size but the path still doesn't close on itself and two even if you did find a line that makes a closed loop what are the odds that it just happens to cut the sphere in exactly half we could cheat by modifying the path both to make sure the loop closes and to make sure it cuts the sphere in half but well that's not as cool a shape

that can follow any line you can draw as long as that line has been modified just doesn't have the same ring to it so while it might seem like all hope is lost trajecto mathematicians used more cool math to get around this problem see all the trajecto we've considered so far have been one period trajecto that means that by the time the trajecto has completed one period of its path it has completed one full Revolution but did you ever think about a two period trajecto has to roll through two repeating periods in one revolution you can see that with this trajecto here it completes a first period while on its pink side and then a second period while on its green side trajecto mathematicians found that while it's almost impossible to stumble across a one period trajecto that fulfills our mathematical conditions for a two period trajecto it's surprising ly easy the reason why it's so easy to make a two period trajecto boils down to some fascinating geometry roll a troid sphere along one period of a path this time we're not trying to get the path to form a closed loop on the sphere instead we can connect the two end points using a great Arc is an arc such that if you completed it around the sphere it would divide the sphere into two equal halves because of some fancy differential geometry tree you will always be able to find a sphere that's the right size such that this enclosed area is one4 of the surface area of a sphere now here's the clever bit if we rotate that one period path 180° on the surface of the sphere around our great Ark we end up with a closed loop two period path that encloses exactly half the surface area of the sphere in other words the mathematical conditions for a trajecto have been fulfilled now let's make some to make a trajecto you just draw a path in the software and it generates the shape then you 3D print them before I show you the trajecto I made shout out to the guys who discovered these crazy shapes they were clearly masters of geometry and mathematical thinking but you know you don't need to be a professional academic or spend loads of money on formal schooling to develop insights about math in fact the best way I know of is free and easy practice curiosity and the right learning platform brilliant what sets brilliant apart from other learning platforms is their Interactive Learning style and heavy focus on problem solving I worked my way through this geometry course and the transformation I've seen in my problem solving skills is huge I'm not the most mathematically minded person which is why I'm pretty good at explaining it I need to break everything down to its most basic part to understand it so what fascinates me most about math is the way of thinking so logical and elegant with brilliant I feel like I have access to that inner world this might be weird but one of my favorite Parts is actually getting things wrong I love looking through the solution because it's often something I never would have thought of myself and I feel like I'm getting a deeper view into how to think mathematically as I got further through the course I noticed my own thinking start to change and before I knew it I was thinking more mathematically and solving harder problems it's really rewarding in to see your logic skills grow and evolve whatever level you're at brilliant has a course that will Challenge and excite you to try everything brilliant has to offer free for a full 30 days visit brilliant. org upin adom or click on the link in the description the first 200 of you will get 20% off brilliant's annual premium subscription our minds are our best asset and there's no better investment than expanding it all right now let me show you my trajecto first we went for a pretty generic squiggle then I wanted to try something more angular there was a bit of Bounce at the sharp turns but it still went pretty well it took a lot of trial and error to get the slope just right and we had to try a few different surfaces to deal with slipping thank God I had my friend Dan helping out for this one I was just feeling silly it started off well but started to smooth out as it picked up speed then we tried the haror bridge this one actually did better than I expected I knew from my research SE that trajecto have trouble with sharp turns theoretically they should work for nearly any path but in the real world the bounce can throw off the trajectory this is a pretty sharp turn so I'm actually s oh wait there it is yeah that's more what I was expecting M mhm trajecto also struggle to go uphill because of well gravity but it's sometimes possible if you have the right amount of inertia self intersections can be a struggle too in the real world

friction inertia the slope of the table and the mass of the trajecto all play a part in whether the trajecto rolls or not once again thwarted by reality I uploaded a YouTube short about trajecto last year and a lot of you suggested making one that traces out a signature or a name but unless it doesn't have self intersections go back uphill or have sharp turns it probably won't work in practice but Stan had the idea to do a heartbeat which I thought was really cool so here's a trajectory of my heartbeat we had to nudge it a bit to keep it rolling but we did our best and in the end that's all that matters I feel like now's a good time to talk about why anyone would want to do all of this the mathematicians that discovered trajecto were just following their curiosity and having fun but it turns out that trajecto math comes in handy in a bunch of fields in physics one application that I thought was really cool is in the world of quantum computers use units of quantum information called cubits because cubits are governed by the laws of quantum mechanics they can exist in a kind of mix of two states at the same time this is called a superp position and it's what makes quantum computers so powerful scientists represent cubits using a sphere with a vector inside it a block sphere we can influence cubits with an external electric field what two period troid tell us is that if we apply the same electric field twice in a row to a cubit it'll turned to the same state it started in this could give insight into how to control cubits and lead to advances in Quantum Computing thanks for watching bye