ratio let's compute it over here so first I'm supposed to take the distance from here to here right now that's five point five C plus eleven C so there's going to be sixteen point five C now I'm supposed to compute this distance NUMBER_TOKEN C plus X C so I'll factor out the C and I'll get NUMBER_TOKEN plus X times C and now I'm supposed to divide this by the small distance in the middle which is NUMBER_TOKEN C and this big distance which is sixteen point five C plus XC and I'll factor out the C now one lovely thing that happens here is that all the C's cancel out like this cancels out with this this cancels out with this and so we just get something in terms of X this is the measured cross ratio this is the real cross ratio and then the magic here is that these two numbers are actually equal to each other so this is equal to this one NUMBER_TOKEN plus sixteen point five x is equal to one eighty one point five plus NUMBER_TOKEN x times one point two five one so that's equal to x one point two five one equals so through value of x all right all right so that's how far he is from the box so that's how far he is from the box exactly so X is equal to approximately NUMBER_TOKEN point sixty three which if you look at it it's actually pretty much the size of the box right so the the box is sixteen point five and he's sixteen point six away so he's basically two boxes away from the goal line and then if we add this so that's it that's how you do this it's pretty cool no I really thought this was a lot easier I actually had to I mean I'll tell you how this happened I was teaching a gym class in geometry and it came time to the projected to teach projective geometry and I learned about the conservation in that class that I was teaching for the first time and I thought wow this sounds actually really powerful I've come to love the cross ratio I think it's I think it's a beautiful thing and I think there's this really interesting quote by Robin Hartshorn who is one of the world's foremost geometers and in his textbook when he's talking about the cross ratio he says I must say frankly that I cannot visualize a cross ratio geometrically if you like it is magic you might say it is a triumph of algebra to invent this quantity that turns out to be so valuable and could not be imagined geometrically or if you were a geometry at heart you may say that it is an invention of the devil and hated your whole life so dramatic note I really like it and I think it tells the story of how in mathematics you don't get to choose what field you work on any you know you think you're doing geometry all of a sudden this very quantity comes in and we're actually relying on this pretty deep algebraic fact that at least I have no geometric intuition for and I think that's really beautiful the funny thing is he went for a show and not across yeah that's right hi everyone as usual we'd like to really thank our patreon supporters whose names you're seeing on the screen at the moment these people get special access to our patreon feed plus some of them get special traits such as well original Brown papers from the videos original postcards written here at this desk other weird bits and pieces like numberphile dice and
START_SELFPROMO_TOKEN everyone as usual we'd like to really thank our patreon supporters whose names you're seeing on the screen at the moment these people get special access to our patreon feed plus some of them get special traits such as well original Brown papers from the videos original postcards written here at this desk other weird bits and pieces like numberphile dice and END_SELFPROMO_TOKEN