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  • Now let's draw three circles.

  • Generate circles are separated like this in space.

  • Given the pair of circles, for example, large in the middle, you can draw a common tangent to those two circles, like so like.

  • So to be precise, we're drawing too outta tangents because even cross the's like this attention contingent to the inner tensions that cross.

  • But let's talk about the out of vengeance on those two.

  • Other intentions be These circles are usually of different sizes will meet at the Vertex, or tip, which l marking blacked out.

  • Okay on, because there three pairs dodge on middle, middle and small and small large there, three pairs of tangents and the serum is that if you take the tip off the first pair keep of the second pair of tensions on top of the third pair of tangents, they're aligned along the street.

  • Nine.

  • What the chances are three arbitrary points on the plane lined along a straight line zero, but in this case they do ally very, very beautifully along a straight line.

  • This serum goes back as far as I know to.

  • French mathematician engineer called March 1 of the founders of a corporate technique in Paris, and he is very famous for descriptive geometry, Onda humbly and deeply.

  • So the help move us a pile of stuff from here to there, minimising energy.

  • And, of course, then that kind of research.

  • But he also has a few contributions in elementary geometry like this on this is unprovable by different methods.

  • For example, Minute house and so forth will do it in the classical method.

  • But days are very, very beautiful approach, which again has to do with lifting the whole picture to the three dimensions instead of those three circles.

  • Let's imagine that Visa, actually, pictures of three spheres are seen from above.

  • Please imagine that those three spheres are sitting on the same table.

  • Another was.

  • Their centers are different heights.

  • Large round will be lifted a lot.

  • On this small one will be lifted only a small amount of middle one a little middle ish, but they're all sitting on the same tables, given our say, a large bowl on the middle Bull, you can draw common tangent cone, so calm like an ice cream cone.

  • So something that a surface like this, which sort of expands on which so the shrink shrink strings to keep on.

  • If you look at that picture from above, you see what we saw before.

  • That is two circles on a pair of common constant lines on the tip so you can interpret that picture as actually a three dimensional picture.

  • But seeing into the mission appears okay on because they're three balls, lash on middle will have a comb and tip.

  • Middle and small will have a comb tip and small and large common tip.

  • You have three tips of those three cones.

  • On the other hand, please imagine again those three spheres sitting on a common floor on now.

  • If you please take a transparent seeding from above, just tilt, tilt it and just low it so that it touches those three spheres at the same time from above.

  • You have to tell you you can't be heard until you have Children.

  • It touches those three spheres at the same time, so you have now a seeding, which touches those three fears from above on the floor, which touches those three spheres from below.

  • Each of those cones that we talked about, for example, this cone between large and middle sphere.

  • Well, it's touching the seating on the floor at the same time.

  • This call.

  • So the tip of this cone is east side, the seating on inside the floor at the same kind.

  • Okay, on the same goes for the other two tips.

  • So all those tips off the cones in the top plane that is sitting on the bottom plane that is floor at the same time.

  • But you know, those two planes in space intersect along a straight line.

  • So we have just shown that those tips of cones are longer, straight line.

  • And what what is that straight line?

  • It's the intersection of the ceiling that we put from above on the floor, on which with portables.

  • And if you look at that picture from above, you see this picture so that dotted line is the intersection off the seeding, which is still a thing like that on then the floor, which is supporting the thing from you.

  • So you can also see a cartoon.

  • You just live to descend, and that's deceiving by prime.

  • But they didn't did.

  • Floyd spy on in the common cone on a tangent cone to the two spheres is like that tip is lying in the intersection intersection between the seeding on the floor.

  • Okay, so by lifting the this complicated problem to the third dimension, the problem goes away.

  • In fact, if you look at it the right way, you know, there used to be one of those interesting pictures where you look at it with some out of focus site 70 jumps out here into a three d on its a bit reckless.

  • If you look at it the right way, you can set seeing it like a three dimensional picture on.

  • Then the serum is clear.

  • There's no problem.

  • I mean, off course.

  • These three points must be on the same line because that's how I think is actually a sequel to a previous video with the Professor about earthquakes and spheres.

  • Hopefully, you've caught that one toe.

  • Or if you just want to see more videos with the professor, click on the link that's on the screen or in the video description.

Now let's draw three circles.

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ボールとコーン - Numberphile (Balls and Cones - Numberphile)

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    林宜悉 に公開 2021 年 01 月 14 日
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