It goes crazy drawing all these circles. Hey I'm Matt Henderson; I'm a research scientist working on speech and language processing, but in my spare time I like making mathematical animations. This is slicing a cylinder, and this is a way to introduce the sine curve. I've set up in Mathematica something with a few sliders. If I lift this one here, you know, you can see we have a cylinder, that's the establishing shot. And here we're bringing in a plane to intersect the cylinder. So then the question is if we're to remove the top and unroll the this part of the cylinder what shape would you expect to get? And I obviously spoiled it by saying we're getting a sine wave. Originally I'd seen someone who'd taken a toilet roll and they'd sliced it in this way and then unrolled the the whole thing and it just created this sort of repeating sine wave. For me I was a little bit surprised to see, oh yeah that would be a sine wave - and it's sort of more of a natural way that that that crops up. Imagine that we have a point here and this is going to be a bit like a point in a pond with with water rippling out from it in concentric circles; and we'll also have these lines that are coming across from left to right. As we get the circles emanating from that point they're going to meet with the lines going from left to right and trace out a curve. So you can see
that it's tracing out this curve which is actually a parabola curve. So here we're still in a two-dimensional world where the circles are growing and the lines are moving; all of a sudden the camera has moved and we're revealing that there's a 3D perspective of the same thing. In the 3D world time is now just another dimension going down through the screen. Time is basically this dimension going down through the centre of the cone, or what was originally like into the screen. For example as the lines were moving left to right they're actually also moving down and they're intersecting the circles and the circles growing at a constant rate through time is the same as a cone in this 3D picture here. So what does it mean to say that concentric circles in the 2D world meeting with lines forming- is forming a parabola? What that means in the 3D picture is that if you slice a cone at the right angle then you'll get a parabola. So parabola is a special conic section, that's what you call that. This one is titled 'A method to draw infinitely many touching circles.' (Brady: Catchy name)
- Yeah. It starts with a rectangle that we're keeping at a constant area and we're interested in the patterns that that draws as you move one of the corners of the rectangle around. You can see that we're keeping this rectangle- for example here we're going to keep this point of the rectangle fixed here and we're going to move this one here. It starts off as a square and if we move this one up you can see that this length has got longer, and since we're keeping the area constant at 1, this length has to get shorter. And where- if we move this one around this one's sort of forced to be in a particular location. The first thing that we'll do is we'll cast this point off infinitely off to one side and off to the other side and we'll see what happens to this point here. So now we're going to cast it off to the right and we'll start drawing; so you can see this is moving along and drawing. And what's happening here, well actually it's drawing out a circle. What I want to do is get a nice pattern with lots and lots of circles inside this original circle here. So if I was to draw- instead of drawing a straight line I'm going to draw a circle. I'm going to start drawing circles in this grid here and see what happens to the point down here. So the the first circle we're going to draw here; and you can see that it has corresponded to a circle down here. And it's touching the big circle because this one is touching as well, right? Now let's draw another circle, drawing another circle at this point. Now this one is touching the original big circle which is this line and this one here, so this new circle is touching as you can see these two. So it's kind of obvious how to draw two circles that touch a line up in this world; less less obvious the exact sort of radius that you would need to for them to fit snugly like that in that world. So you know just continue to keep drawing circles up in this world up here. And now we're getting smaller circles because they're more distant. We have the establishing shots of what's happening, we draw the straight line which corresponds to the circle, and now we just start drawing the circles above the line. It goes crazy drawing all these circles, and zooming into the picture below we get this intricate pattern of infinitely many circles underneath. What this really is is a is a geometrical construction called circle inversion. Amazing - you'll never be able to look at circles the same way.
- The way that this would be normally presented I think is that you'd start with a circle. Let's simplify and say that the radius is 1. I'm wondering where does this point get mapped to? So this point will get mapped to somewhere along the line that connects the centre of the circle to that point; imagine that this distance is r. It will get mapped to the point inside the circle that's at distance 1 over r. And so, you know, there's a special point here- here r equals 1 over r because the radius is is 1 and anywhere inside here will get mapped you know way out here. Here it gets mapped like here, and so on and this has this property that you know if I draw- if I draw a circle in that picture then it turns into a circle outside. And and and also if you have a if you have a straight line, you know like we had here, then this can be thought of as a circle with an infinite radius so it would correspond to a circle down here that goes through the centre of that circle. The centre of the circle gets mapped out to infinity. I like it because, you know, what I've done is I've thought of a geometrical reason for why- how we could keep this constraint of you know mapping from r to 1 over r and that is, you know, in this picture if this is r if this the long side of the rectangle's r then this has to be 1 over r and that's just because we're saying that the area has to equal 1 at all times. So it's kind of like a sneaky way to say- to get all the nice properties of circle inversion and, you know, one nice thing you can do a circle inversion is create these pretty pictures of infinitely many circles touching each other. And really it's a trick, I didn't really do all the maths to figure out you know what radius and where the centres of all these complicated circles should be, I just did this much more easy grid structure on top. This episode was supported by KiwiCo; providers of these great subscription crates, an amazing hands-on way to get excited about science, technology, engineering, the arts and mathematics. You've seen in the past that I love making them but now I've started sending them to my nephew Sebastian on the other side of the world, and what a great monthly gift to get from his uncle. He even sends me back videos so I can watch the whole process. After last month's domino machine, today's task is manufacturing a walking robot. Everything he needs is already in the box. Now KiwiCo has all sorts of boxes and crates aimed at different age groups - see the website for more details about that - and they're already shipping to 40 countries. You can get 50% off your first box by using the url
KiwiCo.com/Numberphile. It's in the description if you want to click on that. Like it kind of sets the
human motion- It's awesome! There you go folks, officially awesome.
