Epic Circles - Numberphile

amazing you'll never be able to look at circles the same way after I show you the way to work out the radius of this circle I've got something special for you today I have a sequence of numbers that I want you to be able to work out what the next number in the sequence is these are fractions so it's going to be one on 15 one on 23 one on 39 one on 63 and then something there are lots of ways I could show you how to solve this but today I'm going to show you a very special way how to solve it okay so we're just going to put this over here I need to get some equipment okay I got to start drawing some circles my goodness me have you ever have you ever done a video where it's like the paper itself has to be precise now I can use my other precision instrument here my second circle circle number to look at that okay so these ones are exactly the same and they're half the diameter of the big circle this is going to be fun once we actually get there that was hard to get these ones to kiss so they're now kissing these are kissing circles so mathematically we'd say that they're touching tangentially though they're kissing now oh okay Brady you know hard it is to get four circles to kiss do you know how hard that is you know I have to I have to confess you I've done this a few times this is the best time I've done it this blue circle here just so happens that the radius of this blue circle here is one fifteenth of this big circle it's 1:15 hang on okay let me keep going so this is called a papist chain what I'm doing here so having one circle touching the inside of another circle kissing and then having circles fitting in and touching all one another it's a big touch fest today so these things have been around for a long time okay so this one here touching touching touching so this one here this is this new blue circle this one happens to be one twenty third of the original radius one twenty third 115-120 third this one's good look at that beautiful it is very zen the next one here this little guy happens to be one thirty ninth of the original large circles there you go I don't know how much more of this compass I can use okay so this guy here his radius is 160 third the original radius of the big circle 163rd and so what's the next one going to be well let's draw it in shall we this here this here now belongs to these circles and so the question is what is the radius of this circle in reference to our first circle now there's plenty of ways we could actually work that out we could get busy with our rulers and start drawing triangles and maybe start getting angles and maybe start using different kind of theorems and whatnot to try and get there but there is one way to solve this problem which is absolutely amazing absolutely amazing great it is the way to solve this problem what I'm going to show you is actually amazingly hard core is a huge amount of information dealing with this so what you have to do is I'm going to try and show you just what we need for this problem and hopefully you can go and have a look at all the other stuff in your own time but this is something that we just go to Brady F trust me I'm going to show you exactly what we need for this problem and no more but if your imagination is fired up that's because you're thinking that's good the way to solve this problem is using a method called circle inversion there's a quick introduction I think most people have seen a reflection of themselves correct so if you think about that you've got your eyeball here so here's your eyeball so now you know how reflections work right so what you see is the mirror opposite but what actually is going on is it's this distance here 22 we're going to do this quite rough Brady if that's all right okay we know how mirrors work don't we so over this it's the eyeball it's your eyeball looking back at you right this is called a reflection so we're going to do a special type of reflection to solve this problem it's called a circle inversion okay now this is how it works first off you got to start with a circle so this circle here we're going to be it's our circle of inversion so first of all what we need to do is we need to what we need to know the radius that's really important Brady okay so in this case here the radius is 60 say we put a point out here okay so say we put a point here okay and that point there is going to be a and this point here is o 4 origin so now we did a reflection what happens in inversion is first of all we're going to measure oh a so we know how a equals 30 so this point here we're actually going to invert it and this is how you do it you take this length here right which we know is 30 and you multiply by the point we're going to we don't know where it is just yet and it equals the circle radius squared so this equals 60 squared so in this case this is 30 times where we want to go equals 60 squared so if we just divide through and do a quick little bit of calculation then that goes twice so that means equals 120 and so what that what we've done there is we've actually taken this point all the way out on exactly the same line to here so this length here is Oh a dash and it equals 120 and this length here is okay and that equals 30 so 30 times 120 equals the radius squared so this point has been inverted to this point so whether you're inside the circle you go to the outside and if you're outside you go to the inside and it's all based on this these formulas so that's it that circle inversion you want to see what circle inversion can do in please right so I've just drawn a circle within my circle of inversion so we've only inverted one point what happens to all the points the infinite number of points in a circle when you do the same process okay so we're going to take that circle and invert it we're going to invert the circle that's right so what do you think is going to happen to all these points tell me well my instinct yeah is that you draw another circle out out here somewhere out in the in the great beyond is going to be circular that would be my instinct not an oval my instinct is will be another circle G you've got good instincts so this point here has been inverted to that point there so you can see the closer it is to the circle of inversion the closer it is on the other side so cave divide that O'Brady so we haven't planned this very well this is huge so cos this one is here this inverts out all the way to here so this has already got some very unusual properties right Wow look at this we've actually gone over the original and we're all over the place with this one this is huge so it's different to a reflection where this where it's not scaled but this in a sense is scaled this inverts to this and what you'll find is that you'll get these points on the edge you see that where they're both touching there'll be one point where they touch both here and here at one point where they touch over here and here okay if we measure this length and times it by that length we'll get 60 squared okay so what happens is a circle will invert to another circle okay and we don't have to do this because we showed before the points go back so if we started with this circle on the outside it had come back to this one now the other thing that you need to know is that this is true as well if the circle happens to overlap the circle of inversion so you could think what happens if the circle that we're inverting what happens to this circle Brady well we still get a circle well we will we definitely still get a circle this point if this is a little important of invert to a point there is only a fraction in what yeah exactly that's going to start there and come out big yep but what's the two points we've already seen on the circle can you see two points that we already know are going to be on this inverted circle the ones that touch the line the ones that intersect the line exactly so you can you see there Brady we've done a inversion circle and it's crossed here it's crossed here kind of a little bit there but you can see that it still makes another circle now is one more I'm going to show you is one more I got to show you I can I just say it's important that you prove these things to yourself but I'm not going to prove them today I'll show you this because this one this one's weird and what about a circle that actually passes through the origin well think about it well if there's a point on the origin right what's the length from the origin if you're at the origin zero so in really it's going to blow up so points that get very very close to the origin are going to get huge towards infinity so we're really talking about a massive circle here and you know how massive it just looks like a line so any circle that passes through the origin of the circle inversion will invert to a line and lo and behold check it out look at that so circle inversion amazing look at the properties circles invert two circles okay big goes to small small goes to big if a circle crosses the circle of inversion it still looks like a circle and then I think most stunning of all a circle that passes through the origin will invert to a straight line so this is all we need these two facts will kind of fact 1a and 1b and then fact two okay that's all we need so now with our information about circle inversion how can we actually solve this problem well what we need to do first we need to make a decision about what sort of circle we're actually going to use to invert the way to solve this is to get this bad boy again and what we're going to do is we're actually going to do a circle of inversion that is exactly the same as our original circle okay so this is our original circle we're going to put the origin of our circle of inversion right at the end so this this green dotted line is our Circle of inversion and this is what we're going to do all our work with okay so this is our circle of inversion and remember it's got exactly the same radius as our original circle that's important now remember what I said Brady what are the what are the things that we need to know about inversions we need to know circles invert two circles if they go over the same over the circle of inversion they still in virtuous circles unless part of the circle goes through the center of origin of the circle of inversion then it turns into straight wrong bingo now this is the reason why I came because now I am able to invert these circles and just using a ruler right I'm able to Mabel to show you how beautiful how simple it is now to interpret this problem this big circle here Brady this big circle here what does it invert - well first of all it passes through the center of inversion so we know it coz is going to turn into a straight line now where is that straight line going to be well you know what I don't have to calculate anything because I already know that it intersects the circle of inversion there so this is the radius so this point goes back to itself so we know that this point is on the straight line because it intersects the circle of inversion I know what the other one is - where exactly so straight away you only need two points to the final line yeah so that we we know we can actually just really simply draw this in we've just put a our first inverted circle next what about this circle here so this circle here also passes through the center of origin but it kisses the inside it kisses the inside of the inversion circle now that means it's going to invert to a straight line exactly but where is it going to be is it going to be a straight line like that or like that well let's have a look this circle here okay and this circle here are both going to invert two straight lines but they only touch at the point of origin and that point of origin in effect is infinity so if two lines only touch at infinity well this is kind of tricky stuff here that's parallel that's the only that's the only explanation really isn't it so you see the logic that you can draw out of just these simple rules of circles go to circles the logic here is that well if it goes anything else it will eventually hit which means that point must be an intersecting point back here but we know that doesn't happen because there's origin point so you can deductively reason what happens here okay Brady so this big circle is this one and this circle is this one now what happens to this circle here well again we don't have to measure anything this circle here touches the big circle and touches this circle and we already know that this circle is that line and this circle is that line so this has to be touching both this and this and so really there's only one thing it can be if a circle inverts to a circle it has to be a circle that fits between these parallel lines well let's draw that in Brady this has happened to there this point hasn't moved because it's on the circle of inversion but a circle has inverted to a circle now what about this guy here how do we invert this one well straight away we know that two points have to be here and here because they invert well they're on the circle of inversion where's another point well this one here is on this circle here and even though it actually in in this diagram it actually is on it an inverted circle we have to remember that that's not where it's going it's actually going to here okay this guy here these two touch right there so this is touching this circle this circle of this circle I think you should be able to see what's going on now if this circle is touching this circle this circle and this circle then in the inversion these two circles aligns this circle is here this has to touch that that and that which means it has to be another a circle exactly the same radius exactly the same diameter see how much symmetry there is how beautiful this is how awesome this is Brady I mean this the symmetry of this is just breathtaking in the inverted world we still get kissing okay now Brady there's another circle this blue one here this is the guy here which is one fifteenth of the original radius is what I told you now we actually have to invert this guy so this circle is that circle this circle is that circle this circle is actually over here so that circles there so this guy in here is kissing one two three and of these purples it's one two three so we know that this blue guy has to actually be in here has to touch all three so is it a rule that if you kiss in real life you have to kiss exactly now this little circle is the inversion of the blue circle what about the one that we really want this one down here how do we get there well we've inverted one two we're going to invert this guy now so this guy inverts and you know what it's becoming so simple to do because we know that it's a circle that has to touch this guy here so this is touching this and this is this so this circle has to touch here is touching the big circle which we know is that line you know it's such in this circle it's that line so it's just a straightforward process where we now have to just draw circles in one on top of another this Pappas chain of circles getting progressively smaller and smaller and smaller under inversion ends up being circles stacked on top of each other exactly the same radius exactly the same diameter because they're fitting between two parallel lines so this is absolutely beautiful this is this one this is this one this is this one this is this one this is this one so now if we think about it in the Pappas chain the blue guys are touching these circles and touching this circle here so in the inversion okay we've already seen that this blue circle inverts here so the pattern is going to stay the same there's going to be a circle there it's going to be a circle there they're going to circle the other than T so for their circle bit and all going to be exactly the same size okay so if we can work out the diameter of this we can work out the diameter of this and then we've got the radius so how can we do that this is epic this is seriously epic all right here we go so we've got now this is a right angle and we know that this circle here even though that's so tiny the center of this circle will pass through the center of this circle up here we now have a massive right angle triangle if we know this length here we can invert that length and it will give us the length through this circle and because it's passing through the origin of this little circle it will tell us the diameter and if we divide that by two it will tell us the radius so shall we do it let's bring this home I tell you what this is absolutely epic it's very pretty though this whole this whole length here is our it's the circle of inversion so this half here is going to be our on to right and this length here well this circle here is a half of that so right from here to here that's our own four and this this little blue inverted circle here is a quarter of that there so that's actually our on 16 so just here so what we're going to do is we know that this whole length here all the way across our take our on 16 which is equal to 15 are on 16 so that's what that length is equal to now what about this long guy here well again here this distance from here to here is going to be at our own 4 and all of these are on 2 are on 2 plus r2 plus r1 2 plus our own - so that's going to be R plus R so that's 2 R plus R on 4 so what we've got here then so this whole length here that's huge 9r on for now what we want to do is that we've got Pythagoras's theorem and we worked out this length and this length we can work out the hypotenuse from this o which is the center of the circle inversion all the way to this center here of this blue circle when you work that out that comes to 39 on sixteen are now we actually want to know the distance Oh to be dashed because we know that will invert right back to this small circle and we also want to know Oh - OH - but let's do this one first so we know that o to B dash is equal to 39 on 16 are plus the radius of the small circle which is are on 16 so that equals this Oh a so OAS over here we're going back from this by our on 16 we're going back by the radius so it looks like this 39 on 16 our take are on 16 same denominators so 39 so we end up getting 38 on 16 are now we want to invert these so we want to take o to be - back to be okay so let's do that so o to be when we invert this we end up getting 16 on 40 ah okay and we do the same thing with OH - a - we bring it back to the tiny little circle there and that means either the a is this inverted okay and then we end up getting 8 on 19 are now we've got out of the a out of the B we know they passes through the center so we know if we go oh a take o be we will get the diameter of this small circle and that happens to make it 2 so that's that take that which ends up being 2 on 95r because the radius is half the diameter so the radius here of this small circle so radius of small circle equals one on 95r so we know that the original circle if we divide by 95 will give us that radius which means this is 1 1 95

Channel: Numberphile

Views: 1,682,253

Rating: 4.8952956 out of 5

Keywords: numberphile, circles, circle, pappus chain

Id: sG_6nlMZ8f4

Length: 26min 35sec (1595 seconds)

Published: Sun Apr 13 2014

Reddit Comments

[u/eggsformeandyou]

He says it is simple, he says its easy. "this circle is that circle and this circle is that circle..." looks amazing but I dont understand.

[u/ruwisc]

Interestingly, those five values in the respective denominators (15, 23, 39, 63, 95) fit nicely onto a parabola - the radius of the n^th blue circle is 1/(4n² - 4n + 15). I'd assume that that continues on but have no real basis for it.

[u/OneWasAssaultedPeanu]

Please correct me if I'm wrong, but wouldn't the circle that intersects the origin invert with an undefined point? The point that intersects the origin on the circle would have to invert to a point r² / 0 millimeters away from the origin, right?

[u/wildgurularry]

When I took Intro to Geometry at university, we used some software (Cabri Geometry I believe) that could do circle inversions. Really fun stuff.

I remember one assignment we had where the last question asked us to show a certain property of something. I wasn't sure if it was asking us to prove the property or not, so I started trying to prove it. In the end, I determined that it was probably too difficult for us to prove given what we had learned so far.

However, I prettied up the sketch of a proof that I had (with certain gaps), and attached it to the back of my assignment. The assignment itself was one or two pages. My sketch of a proof for the last question was four pages.

When I got the assignment back, I saw that I had a mark of 120% on it. It's the highest mark I ever got on a math assignment. I also received a mark of 100% for that course... the highest mark I've ever received in a math course.

[u/ethanicles7]

Alternatively, 23=15+8*1

39=23+8*2

63=39+8*3

95=63+8*4

[u/unhOLINess]

Can anyone give a good reason why he reflected circle is still a circle? I can't think of an intuitive argument, especially since the reflected center of the original circle is not the center of the reflected circle.

[u/illyric]

http://i.imgur.com/EMrknJP.gif