Epic Circles
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The man who loved circles (Objectivity):
More links & stuff in full description below ↓↓↓ Pappus chains, circle inversion and a whole lot more in this EPIC video with Simon Pampena. Support us on Patreon: http://www.patreon.com/numberphileNUMBERPHILE Website: http://www.numberphile.com/Numberphile on Facebook: http://www.facebook.com/numberphileNumberphile tweets: https://twitter.com/numberphileSubscribe: http://bit.ly/Numberphile_SubNumberphile is supported by the Mathematical Sciences Research Institute (MSRI): http://bit.ly/MSRINumberphileVideos by Brady Haran Brady's videos subreddit: http://www.reddit.com/r/BradyHaran/Brady's latest videos across all channels: http://www.bradyharanblog.com/Sign up for (occasional) emails: http://eepurl.com/YdjL9Numberphile T-Shirts: teespring.com/stores/numberphileOther merchandise: https://store.dftba.com/collections/n...
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amazing you'll never be able to look at
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circles the same way after I show you
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the way to work out the radius of this
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circle I've got something special for
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you today I have a sequence of numbers
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that I want you to be able to work out
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what the next number in the sequence is
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these are fractions so it's going to be
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one on 15 one on 23 one on 39 one on 63
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and then something there are lots of
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ways I could show you how to solve this
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but today I'm going to show you a very
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special way how to solve it okay so
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we're just going to put this over here I
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need to get some equipment okay I got to
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start drawing some circles my goodness
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me have you ever have you ever done a
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video where it's like the paper itself
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has to be precise now I can use my other
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precision instrument here my second
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circle circle number to look at that
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okay so these ones are exactly the same
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and they're half the diameter of the big
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circle this is going to be fun once we
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actually get there that was hard to get
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these ones to kiss so they're now
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kissing these are kissing circles so
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mathematically we'd say that they're
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touching tangentially though they're
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kissing now oh okay Brady you know hard
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it is to get four circles to kiss do you
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know how hard that is you know I have to
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I have to confess you I've done this a
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few times this is the best time I've
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done it this blue circle here just so
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happens that the radius of this blue
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circle here is one fifteenth of this big
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circle it's 1:15 hang on
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okay let me keep going so this is called
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a papist chain what I'm doing here so
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having one circle touching the inside of
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another circle kissing and then having
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circles fitting in and touching all one
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another it's a big touch fest today so
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these things have been around for a long
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time okay so this one here touching
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touching touching so this one here this
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is this new blue circle this one happens
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to be one twenty third of the original
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radius one twenty third 115-120 third
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this one's good look at that beautiful
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it is very zen the next one here this
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little guy happens to be one thirty
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ninth of the original large circles
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there you go I don't know how much more
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of this compass I can use okay so this
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guy here his radius is 160 third the
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original radius of the big circle 163rd
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and so what's the next one going to be
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well let's draw it in shall we
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this here this here now belongs to these
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circles and so the question is what is
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the radius of this circle in reference
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to our first circle now there's plenty
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of ways we could actually work that out
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we could get busy with our rulers and
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start drawing triangles and maybe start
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getting angles and maybe start using
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different kind of theorems and whatnot
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to try and get there but there is one
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way to solve this problem which is
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absolutely amazing absolutely amazing
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great
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it is the way to solve this problem what
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I'm going to show you is actually
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amazingly hard core is a huge amount of
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information dealing with this so what
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you have to do is I'm going to try and
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show you just what we need for this
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problem and hopefully you can go and
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have a look at all the other stuff in
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your own time but this is something that
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we just go to Brady F trust me I'm going
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to show you exactly what we need for
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this problem and no more but if your
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imagination is fired up that's because
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you're thinking that's good the way to
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solve this problem is using a method
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called circle inversion there's a quick
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introduction I think most people have
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seen a reflection of themselves correct
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so if you think about that you've got
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your eyeball here so here's your eyeball
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so now you know how reflections work
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right so what you see is the mirror
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opposite but what actually is going on
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is it's this distance here 22 we're
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going to do this quite rough Brady if
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that's all right okay we know how
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mirrors work don't we so over this it's
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the eyeball it's your eyeball looking
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back at you right this is called a
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reflection so we're going to do a
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special type of reflection to solve this
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problem it's called a circle inversion
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okay now this is how it works first off
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you got to start with a circle so this
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circle here we're going to be it's our
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circle of inversion so first of all what
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we need to do is we need to what we need
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to know the radius that's really
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important Brady okay
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so in this case here the radius is 60
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say we put a point out here okay so say
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we put a point here okay and that point
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there is going to be a and this point
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here is o 4 origin so now we did a
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reflection what happens in inversion is
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first of all we're going to measure oh a
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so we know how a equals 30 so this point
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here we're actually going to invert it
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and this is how you do it
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you take this length here right which we
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know is 30 and you multiply by the point
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we're going to we don't know where it is
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just yet and it equals the circle radius
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squared so this equals 60 squared so in
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this case this is 30 times where we want
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to go equals 60 squared so if we just
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divide through and do a quick little bit
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of calculation then that goes twice so
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that means equals 120 and so what that
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what we've done there is we've actually
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taken this point all the way out on
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exactly the same line to here so this
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length here is Oh a dash and it equals
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120 and this length here is okay
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and that equals 30 so 30 times 120
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equals the radius squared so this point
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has been inverted to this point so
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whether you're inside the circle you go
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to the outside and if you're outside you
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go to the inside and it's all based on
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this these formulas so that's it that
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circle inversion you want to see what
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circle inversion can do in please right
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so I've just drawn a circle within my
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circle of inversion so we've only
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inverted one point what happens to all
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the points the infinite number of points
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in a circle when you do the same process
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okay so we're going to take that circle
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and invert it we're going to invert the
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circle that's right so what do you think
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is going to happen to all these points
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tell me well my instinct yeah is that
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you draw another circle out out here
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somewhere out in the in the great beyond
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is going to be circular that would be my
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instinct not an oval my instinct is will
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be another circle G you've got good
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instincts
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so this point here has been inverted to
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that point there so you can see the
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closer it is to the circle of inversion
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the closer it is on the other side so
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cave divide that O'Brady so we haven't
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planned this very well this is huge
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so cos this one is here this inverts out
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all the way to here so this has already
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got some very unusual properties right
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Wow look at this we've actually gone
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over the original and we're all over the
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place with this one this is huge so it's
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different to a reflection where this
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where it's not scaled but this in a
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sense is scaled this inverts to this and
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what you'll find is that you'll get
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these points on the edge you see that
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where they're both touching there'll be
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one point where they touch both here and
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here at one point where they touch over
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here and here okay if we measure this
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length and times it by that length
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we'll get 60 squared okay so what
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happens is a circle will invert to
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another circle okay and we don't have to
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do this because we showed before the
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points go back so if we started with
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this circle on the outside it had come
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back to this one now the other thing
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that you need to know is that this is
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true as well if the circle happens to
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overlap the circle of inversion so you
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could think what happens if the circle
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that we're inverting what happens to
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this circle Brady well we still get a
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circle well we will we definitely still
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get a circle this point if this is a
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little important of invert to a point
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there is only a fraction in what yeah
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exactly that's going to start there and
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come out big yep but what's the two
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points we've already seen on the circle
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can you see two points that we already
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know are going to be on this inverted
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circle the ones that touch the line the
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ones that intersect the line exactly so
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you can you see there Brady we've done a
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inversion circle and it's crossed here
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it's crossed here kind of
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a little bit there but you can see that
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it still makes another circle now is one
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more I'm going to show you is one more I
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got to show you I can I just say it's
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important that you prove these things to
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yourself but I'm not going to prove them
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today I'll show you this because this
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one this one's weird and what about a
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circle that actually passes through the
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origin well think about it well if
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there's a point on the origin right
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what's the length from the origin if
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you're at the origin zero so in really
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it's going to blow up so points that get
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very very close to the origin are going
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to get huge towards infinity so we're
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really talking about a massive circle
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here and you know how massive it just
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looks like a line so any circle that
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passes through the origin of the circle
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inversion will invert to a line and lo
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and behold check it out look at that so
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circle inversion amazing look at the
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properties circles invert two circles
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okay
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big goes to small small goes to big if a
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circle crosses the circle of inversion
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it still looks like a circle and then I
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think most stunning of all a circle that
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passes through the origin will invert to
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a straight line so this is all we need
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these two facts will kind of fact 1a and
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1b and then fact two okay that's all we
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need so now with our information about
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circle inversion how can we actually
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solve this problem well what we need to
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do first we need to make a decision
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about what sort of circle we're actually
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going to use to invert the way to solve
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this is to get this bad boy again and
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what we're going to do is we're actually
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going to do a circle of inversion that
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is exactly the same as our original
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circle okay so this is our original
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circle we're going to put the origin of
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our circle of inversion right at the end
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so this this green dotted line is our
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Circle of inversion and this is what
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we're going to do all our work with okay
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so this is our circle of inversion and
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remember it's got exactly the same
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radius as our original circle that's
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important now
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remember what I said Brady what are the
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what are the things that we need to know
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about inversions we need to know circles
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invert two circles if they go over the
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same over the circle of inversion they
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still in virtuous circles unless part of
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the circle goes through the center of
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origin of the circle of inversion then
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it turns into straight wrong bingo now
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this is the reason why I came because
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now I am able to invert these circles
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and just using a ruler right I'm able to
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Mabel to show you how beautiful how
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simple it is now to interpret this
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problem this big circle here Brady this
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big circle here what does it invert -
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well first of all it passes through the
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center of inversion so we know it coz is
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going to turn into a straight line now
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where is that straight line going to be
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well you know what I don't have to
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calculate anything because I already
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know that it intersects the circle of
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inversion there so this is the radius so
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this point goes back to itself so we
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know that this point is on the straight
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line because it intersects the circle of
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inversion I know what the other one is -
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where exactly so straight away you only
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need two points to the final line yeah
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so that we we know we can actually just
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really simply draw this in we've
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just put a our first inverted circle
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next what about this circle here so this
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circle here also passes through the
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center of origin but it kisses the
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inside it kisses the inside of the
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inversion circle now that means it's
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going to invert to a straight line
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exactly but where is it going to be is
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it going to be a straight line like that
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or like that well let's have a look this
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circle here okay and this circle here
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are both going to invert two straight
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lines but they only touch at the point
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of origin and that point of origin in
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effect is infinity so if two lines only
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touch at infinity well this is kind of
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tricky stuff here that's parallel that's
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the only that's the only explanation
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really isn't it so you see the logic
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that you can draw out of just these
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simple rules of circles go to circles
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the logic here is that well if it goes
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anything else it will eventually hit
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which means that point must be an
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intersecting point back here but we know
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that doesn't happen because there's
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origin point so you can deductively
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reason what happens here okay Brady so
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this big circle is this one and this
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circle is this one now what happens to
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this circle here well again we don't
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have to measure anything this circle
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here touches the big circle and touches
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this circle and we already know that
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this circle is that line and this circle
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is that line so this has to be touching
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both this and this and so really there's
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only one thing it can be if a circle
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inverts to a circle it has to be a
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circle that fits between these parallel
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lines well let's draw that in Brady
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this has happened to there this point
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hasn't moved because it's on the circle
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of inversion but a circle has inverted
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to a circle now what about this guy here
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how do we invert this one well straight
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away we know that two points have to be
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here and here because they invert well
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they're on the circle of inversion
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where's another point well this one here
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is on this circle here and even though
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it actually in in this diagram it
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actually is on it an inverted circle we
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have to remember that that's not where
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it's going it's actually going to here
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okay this guy here these two touch right
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there so this is touching this circle
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this circle of this circle I think you
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should be able to see what's going on
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now if this circle is touching this
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circle this circle and this circle then
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in the inversion these two circles
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aligns this circle is here this has to
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touch that that and that
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which means it has to be another a
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circle exactly the same radius exactly
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the same diameter see how much symmetry
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there is how beautiful this is how
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awesome this is Brady I mean this the
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symmetry of this is just breathtaking in
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the inverted world we still get kissing
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okay now Brady there's another circle
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this blue one here this is the guy here
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which is one fifteenth of the original
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radius is what I told you now we
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actually have to invert this guy so this
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circle is that circle this circle is
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that circle this circle is actually over
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here so that circles there so this guy
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in here is kissing one two three and of
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these purples it's one two three so we
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know that this blue guy has to actually
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be in here has to touch all three so is
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it a rule that if you kiss in real life
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you have to kiss
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exactly now this little circle is the
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inversion of the blue circle what about
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the one that we really want this one
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down here how do we get there
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well we've inverted one two we're going
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to invert this guy now so this guy
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inverts and you know what it's becoming
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so simple to do because we know that
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it's a circle that has to touch this guy
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here so this is touching this and this
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is this so this circle has to touch here
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is touching the big circle which we know
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is that line you know it's such in this
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circle it's that line so it's just a
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straightforward process
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where we now have to just draw circles
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in one on top of another
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this Pappas chain of circles getting
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progressively smaller and smaller and
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smaller under inversion ends up being
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circles stacked on top of each other
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exactly the same radius exactly the same
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diameter because they're fitting between
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two parallel lines so this is absolutely
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beautiful this is this one this is this
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one this is this one this is this one
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this is this one so now if we think
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about it in the Pappas chain the blue
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guys are touching these circles and
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touching this circle here so in the
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inversion okay we've already seen that
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this blue circle inverts here so the
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pattern is going to stay the same
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there's going to be a circle there it's
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going to be a circle there they're going
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to circle the other than T so for their
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circle bit and all going to be exactly
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the same size okay so if we can work out
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the diameter of this we can work out the
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diameter of this and then we've got the
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radius so how can we do that this is
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epic this is seriously epic all right
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here we go so we've got now this is a
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right angle and we know that this circle
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here even though that's so tiny the
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center of this circle will pass through
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the center of this circle up here
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we now have a massive right angle
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triangle if we know this length here we
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can invert that length and it will give
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us the length through this circle and
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because it's passing through the origin
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of this little circle it will tell us
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the diameter and if we divide that by
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two it will tell us the radius so shall
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we do it let's bring this home I tell
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you what this is absolutely epic
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it's very pretty though this whole this
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whole length here is our it's the circle
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of inversion so this half here is going
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to be our on to right and this length
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here
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well this circle here is a half of that
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so right from here to here that's our
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own four and this this little blue
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inverted circle here is a quarter of
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that there so that's actually our on 16
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so just here so what we're going to do
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is we know that this whole length here
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all the way across our take our on 16
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which is equal to 15 are on 16 so that's
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what that length is equal to now what
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about this long guy here well again here
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this distance from here to here is going
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to be at our own 4 and all of these are
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on 2 are on 2 plus r2 plus r1 2 plus our
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own - so that's going to be R plus R so
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that's 2 R plus R on 4 so what we've got
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here then so this whole length here
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that's huge 9r on for now what we want
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to do is that we've got Pythagoras's
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theorem and we worked out this length
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and this length we can work out the
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hypotenuse from this o which is the
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center of the circle inversion all the
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way to this center here of this blue
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circle when you work that out that comes
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to 39 on
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sixteen are now we actually want to know
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the distance Oh to be dashed because we
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know that will invert right back to this
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small circle and we also want to know Oh
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- OH - but let's do this one first so we
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know that o to B dash is equal to 39 on
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16 are plus the radius of the small
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circle which is are on 16 so that equals
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this Oh a so OAS over here we're going
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back from this by our on 16 we're going
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back by the radius so it looks like this
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39 on 16 our take are on 16 same
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denominators so 39 so we end up getting
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38 on 16 are now we want to invert these
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so we want to take o to be - back to be
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okay so let's do that
24:50
so o to be when we invert this we end up
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getting 16 on 40 ah okay
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and we do the same thing with OH - a -
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we bring it back to the tiny little
25:05
circle there and that means either the a
25:08
is this inverted okay and then we end up
25:13
getting 8 on 19 are now we've got out of
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the a out of the B we know they passes
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through the center so we know if we go
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oh a take o be we will get the diameter
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of this small circle and that happens to
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make it 2 so that's that take that which
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ends up being 2 on 95r
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because the radius is half the diameter
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so the radius here of this small circle
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so radius of small circle equals one on
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95r so we know that the original circle
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if we divide by 95 will give us that
26:18
radius which means this is 1 1 95