Nothing is moving in a circle. Each one of these dots is moving in a straight line - Nice Talk about burying the lead. That's the best bit. Is that the best bit? I want to show you an animation that I like and I want to give too much away I would like to know your opinion of this animation, so There are two yellow blobs, I'll give you that for free I'm curious what you see happening. It's kind of like they're rotating around each other.
- Go on say more. So if I talk about this dot, yeah.
- It's like it's drawing a circle You know, it's it's looping around the other one or they are the ones looping around you.
- Yeah, it goes either way, doesn't it? What I like about this animations, I've shown lots of people is animation, but they see different things. So some people see this orbiting thing when one is orbit in the other, but the other one keeps moving. Is that a fair description of what you saw?
- Yeah. Other people see straight lines.
- Oh, I definitely see straight lines. Okay So there's definitely a straight liney motion and is obviously there and there but at the same time other people have insisted that there's some sort of circular orbit-y flavor which is a first of all I like because Straight lines and circles feel opposites in some sense and though there's a mathematical sense where there might be the same too But both of them are seen by people in a naive description of what's happening I ended up building an animation on this when I was at school because I was curious about what this motion actually wasn't that's how I learned to program in Visual Basic I have re-built this one in geogebra I want to show you how I built it and also why I built it. You were right to talk about Orbiting things and also right about the straight lines If I put these lines on it's really obvious that one of them is going up and down one is going left to right...but they always dodge each other and I didn't make them move like this by putting their straight lines on. What I did was I made the circle around the outside exist. It's still not obvious to me how they works, but I made this point also exist So this blue point moving around is how I built this file. That blue dot controls the yellow. In fact the one going up and down is precisely just the vertical coordinate of the blue dot It's just tracking wherever the blue dot is in a vertical axis It's like the projection of it onto a vertical thing and the horizontal one is just the horizontal projection The reason why we see sort of circular motion going on is because it's driven by a circular motion. It's possible that you saw something else which was the connector between these dots is a natural thing to sort of see because it's actually because it is always the same length is equidistant and that's a fluke and some people in their head are kind of Averaging the two dots and they're seeing the midpoint of the two dots. Literally the average position that you dress and that is moving in a circle, which is really nice There's a bunch of mechanical constructions called a trammel of Archimedes where you can draw a circle From two things moving in straight lines and it's a mechanical way of drawing a perfect circle You just constrain these two dots to move in straight lines I didn't say that.
- A lot of people don't see it but they they feel there's something intuitively circular going on So let me go back to the reason why I built the thing. The yellow points are the coordinates of the blue point Let's just focus on the vertical one What do you think would happen if I track the vertical position of that over time? Essentially draw a graph. What will that graph be? Next time we start on the right side I'm gonna track the y coordinate and you'll just see a graph which maybe is familiar. Any...any thoughts, Brady? It's like a sine curve or something.
- It is precisely a sine curve. In fact, it's not just any sine curve. It is...is THE sine curve. So I was a teacher for a long time I was teaching people trigonometry. Sine and cosine turn up and Every single time I ever had a class learning that they will... "Sir, what is sine? What is it..." and There are lots of answers. People think of it as a ratio, opposite and hypotenuse. That might be familiar words. Other people think of as a function. It's got an input and an output. These things are true But what I like about a third answer, I'm gonna give you now, is that it captures Both of them. Sine is the y-coordinate of a point moving in a circle. Interestingly nothing to do with triangles Despite the name trigonometry coming from Trigon metry: measuring triangles I think was the worst named topic in mathematics. Sine is a circular function. In fact It's one of the circular functions because there's another coordinate we haven't tracked yet. So let's just do that one the horizontal One moving left and right there. If I track that going upwards you're gonna see the beginning of it It's not gonna be a huge surprise when you realize the trace is out the beginning of a cosine wave And if you flip that back down you see the cosine wave which is precisely the same as a sine wave that's shifted along. It's just like the circles going around a bit So sine and cosine as functions or as mathematical objects are precisely the y coordinate and x coordinate of a point moving in a circle That's why they're important because almost everything that goes in a cycle or repeats ever ie most things are described by circles and therefore described by trigonometry.
- Is a cosine wave very different to a sine wave then? It's exactly the same. Literally sine wave Is there in a cosine wave is the same way shifted is the sine of the other angle this there's no obvious angles But triangles have angles and we use trigonometry with angle so the angle comes from, if I draw the radius on, from the center of this thing the angle gives me a way of measuring where the blue Dot is at any point. So angle of zero there. It's the angle the radius makes and that's where you do sine of an angle and it gives you coordinate and so the Cosine is actually sine of the other angle, is the angle between the vertical and this radius instead of the horizontal and that's why sine and cosine Are basically the same. I love the fact that seeing this move makes me understand trigonometry There's not a triangle in sight until you put that radius back on and then you can see there's a right angle triangle That's kind of sweeping around inside here. And that's why trigonometry is to do with right angle triangles It's because x and y coordinates are right angles and the radius makes the hypotenuse But that means sine and cosine are like two aspects of the same motion. There's just a different perspective And when someone pointed this out. Could you draw it in three dimension? That sounds like and I got this on geogebra at the inspiration of a student I was teaching at the time. So let me show you a three-dimensional version of this So there's my circle I had originally that's the view we had before. That's the same motion I just had. five here from a three dimensional mode. I've got this axis going off into the distance We're gonna track the position of a and time is now gonna go sort of back in into that direction And we're gonna project the point along that axis. I'm gonna start it now and you see that point going off into space It's kind of hard to see without moving your head around in a three-dimensional world so I can put the path on and you can see, maybe not surprisingly, you get sort of spiral coming out and I reckon Most people will predict that but that means the sine and cosine are precisely somehow in that spiral because they're just aspects of that spiral If you look from the side There's the sine wave. Which means if you want to see the cosine wave you should probably look from the top or the bottom There's the cosine wave. You can see the same way if it starts in a different place and if you go back to the front again They're all just a projection of a circle and so sine and cosine are useful because they are the sort of Compressions of circular motion in within one dimension and I thought this 3d Diagonal that really helped me understand where this comes from. It does beg the question There's a third trig function that most people know about and I haven't shown you that yet Tan!
- Tan. Do you know what tan is short for, Brady? Isn't it tangent?
- Yeah, what's a tangent? I mean, I'm in full teacher mode now, but I think you know what tangent is Yeah, it's a line like kind of like grazing a circle.
- Yeah, so tangent it from Latin for touching So tangent is something just touches the curve doesn't have to be a circle. The word comes from precisely that set up This is a slightly more advanced file I can show you all of the things first will reminder a sine function and there's the graph we're building and cosine is the other Direction and you can see the graphs arriving tracing their way out. Tangent comes from drawing a tangent but if I drew a tangent Vertically to the place where we start all this motion and I track where the radius line would hit the tangent You'll get a situation that looks like this. There's the tangent line and the radius line Will always cut the tangent line somewhere and then this green length is precisely the tangent function. It is also the gradient of the radius because tangent is defined to be sine of a cosine It was kind of a huge relief to me personally. The word tangent wasn't a coincidence with the other definition of tangent Which is a line that just touches the circle these three graphs are all related to a circle. They're all really geometrical things they're not really to do with triangles except by accident because of the coordinate axes and There are three more functions you learn at A level at school which are one over these functions. So sec, cosec and cot are one over these three and they're also on this diagram. If I turn on cosec, cosec otherwise known as one over sine Just like sine, which is a vertical lengthwise diagram, cosec is where the tangent cuts the vertical axis and you can see a sine gets bigger cosec gets smaller and then the same at one point and then it Goes back the other way and so this U-shaped graph here is the cosec function otherwise known as one over sine and the same thing happens with sec is where the tangent intersects the x axis and cot, one over tan, ends up being The other tangent across the top here is where the radius intersects that all of the trig functions that we learn at school Are actually altered with circles which is why they are called the circular functions And I kind of wish we call trigonometry the circular functions from the word go Even if the use we most make of it is for finding missing sides in a right-angled triangle There's one more thing that I wanted to you I had this diagram happening before where these two dots are moving, but you now know how I did it I put those two lines on there, kind of the projection of that outer point, the projection vertically and horizontally Now once I realize I can do that. I don't have to restrict it to two lines. I could put three lines on You get a really nice effect of these three points moving And they always dodge each other. They're moving in what they might call simple harmonic motion, but then they're nicely sort of lined up So they don't ever hit each other. Well, I just really like this. So I'm going to crank up the number of lines Here we go Now what I love about this is that everybody can see the yellow circle and... Mind blown!
- The dots are in a circle definitely but nothing, nothing is moving in a circle each one of these dots is moving in a Straight line. So if I grab one with my finger this one Is just moving backwards forwards in a sinusoidal motion as for they call this with this wave-like motion But all of them because they're lined up in a certain way create There's a lovely illusion of a circle rolling around the inside of thing. To program this you just needed some knowledge of how to make A sine wave just shift around each time and it's just a really lovely optical illusion I like.
- Nice! Talk about burying the lede. That's the best bit.
- Is that the best bit?
- Yeah. That's the bit that's got no use But then I'm a sucker for things that've got no use. The thing is that I love Maths 'cause it's beautiful It turns out it's also useful but mathematicians usually do stuff because it's nice and then they're like, "Oh yeah, it's also useful But I would have done it anyway." All right You made me wait for it You know one of the things I love about brilliant aside from their like deep dive courses are their daily challenges They remind me a bit of the morning crossword a great way to kickstart your brain in the morning in fact Why not test yourself do a hundred of them in a hundred days? All the questions and puzzles on brilliant are carefully crafted to get the very best out of your mind They often include little moments of interaction like these cool sliders So here's a question: Can this line be positioned to cut these five circles in half? both in terms of area and perimeter What do you reckon? Where could you put it? After you've had a go? Why not Go check out this whole related geometry course Full of questions and lessons and proofs to see more go to brilliant.org/numberphile. The /numberphile will let them know you came from here, but more importantly it will give you 20% off a Premium subscription. It'll unlock everything on the site That's brilliant.org/numberphile. Our thanks to them for supporting this episode
