Conway's IRIS and the windscreen wiper theorem

0:06 Welcome to another Mathologer video.

0:07 You are here for the mysterious IRIS in the thumbnail, aren’t you?

0:11 Peaked your interest, eh?

0:13 Well, let’s chase it down.

0:15 Start with any old triangle.

0:18 At the corner opposite the red side grow two copies of that red side, like this.

0:23 Then opposite the blue,

0:25 the same thing grow two copies of the blue side and there, two greens.

0:30 A triangle with whiskers, cute:) Now,

0:33 doesn’t matter what triangle we start with, those six

0:36 end points they wait what’s special about those six points?

0:40 Can you guess?

0:41 Those six points are… you got it?

0:44 Yep, those six points lie on a circle.

0:46 Really?

0:47 Sure if it was going to be anything,

0:50 then it was a circle but it’s still pretty surprising, isn’t it?

0:54 Three random points usually determine a circle.

0:57 On the other hand, four random points usually

1:00 don’t lie on a circle and of course

1:02 it’s even less likely for random five or six points to lie on a circle.

1:08 Anyway, this miraculous six-point circle is

1:10 called Conway’s circle named after its discoverer,

1:14 the legendary mathematician John Conway.

1:17 Conway was a mathematics professor at Princeton university, a certified genius,

1:21 and the inventor of many ingenious mathematical

1:24 games that also became incredibly popular outside mathematics.

1:28 For example, chances are, even if you are not a mathematician that you

1:32 are familiar with Conway’s bizarre Game of Life.

1:35 There you’ve seen this sort of alien 2d life before, right?

1:38 That’s due to Conway.

1:40 Anyway, little miracles like his circle would constantly pop up

1:44 in conversations with John Conway and if he deemed you worthy,

1:48 you could expect to be challenged to come up with a proof on the spot.

1:53 I’ll show you the ultimate super pretty proof next,

1:56 a proof which involves the IRIS in the title of this video.

2:00 I’ll also tell you about some exciting new life beyond Conway’s circle:

2:04 in particular the beautiful windscreen wiper theorem will

2:07 have you smiling next time it rains:) However,

2:11 if you consider yourself worthy, then before you watch the rest of the video,

2:16 accept Conway’s challenge and try to come up with a proof of your own.

2:20 Report on your efforts in the comments:) Here John

2:23 Conway looks a little bit like Saint John, doesn’t he?

2:27 Anyway did you come up with a proof?

2:30 No?

2:30 Not a problem, not an easy one:) Want a hint?

2:34 Okay, here’s one.

2:35 Whenever you deal with a would-be circle,

2:38 what’s the first thing you should be looking for?

2:41 Yep, its heart, of course,

2:43 the center of the circle:) And how do we locate the center?

2:47 Well, for example by shrinking the circle.

2:51 There shrink.

2:52 There that’s the center.

2:55 But wait a minute.

2:57 Before the circle collapsed into the centre,

3:00 did you notice something remarkable?

3:03 No?

3:03 Did you blink?

3:05 Let me show you again.

3:08 Ready?

3:08 This time watch closely:) You saw it this time, right?

3:15 Just before the circle collapses to a point it

3:18 appears to simultaneously touch all the sides of the triangle.

3:22 There rewind a bit.

3:24 …That’s really cool, isn’t it?

3:26 And so it looks like the whiskered triangle actually gives birth

3:30 to an IRIS and not just the circle on the outside.

3:35 Equipped with this extra clue does the triangle lover

3:38 in you want to give proving all this another go?

3:41 Pause the video if you like,

3:43 but we’ll keep going and chase down a pretty proof ourselves.

3:47 Let’s start with a ring and highlight one of its cords.

3:51 Then it’s clear that all such cords have the same length.

3:55 obvious, right?

3:55 Also, the point where a cord touches the inner

4:00 circle is the exact midpoint of the cord.

4:03 There.

4:04 Also obvious.

4:05 Alright, what this means is that if we start

4:09 with a circle and three equal struts with highlighted centers.

4:14 Then if we arrange these struts around the circle

4:18 like this we can be sure that the six

4:22 ends are on a circle that, together with the one we started with, forms an iris.

4:29 Still all crystal clear, right?

4:31 So to prove that the whiskered triangle really extends

4:36 to an iris all we have to show is two things:

4:39 First, we have to show these three struts are of equal length.

4:44 Pretty damn obvious, right?

4:45 They are all made up of 1 red, 1 blue, and 1 green each.

4:50 Oookay:) And the second thing we have to show is that the points

4:55 at which the struts touch the inner circle are the centers of the struts.

5:00 How do we do that?

5:02 Well, here is a nice idea.

5:04 Let’s swivel this highlighted horizontal strut around

5:08 one of the corners of the triangle.

5:11 That orange corner there.

5:13 Okay, go, swivel.

5:15 Since the two blue ends are of equal length.

5:19 the two struts will end up in perfect superposition.

5:23 Nice.

5:23 But did you notice something extra nice?

5:26 There.

5:27 Looks like the contact points with the circle also land on top of each other.

5:32 Why is that?

5:34 Let’s have another look at the starting configuration.

5:38 Well these two distances here are clearly equal.

5:41 This means that not only do the two struts end up superimposed perfectly,

5:47 but also the two points of contact do.

5:51 Alright.

5:51 Now, color the strut like this.

5:54 So what we want to show is that the magenta is exactly as long as the aqua.

6:01 Ready for a nice AHA moment in three stages?

6:05 Here we go.

6:07 Stage 1 swivel around the right corner.

6:11 Perfect coincidence.

6:12 Stage two.

6:13 Swivel around the top corner.

6:16 Perfect coincidence again.

6:18 Stage three.

6:19 Perfect coincidence yet again and, overall,

6:22 the aqua and magenta parts have switched places,

6:25 which shows that the aqua is exactly as long as the magenta.

6:30 You liked this?

6:31 Very nice isn’t it?

6:33 And of course the same is true for the other two struts.

6:37 Which then shows that the six points are on a circle.

6:40 And we’re done.

6:41 We have confirmed the existence of Conway’s IRIS.

6:44 But let me show you a second way to see

6:48 that the two parts of our strut are of equal length.

6:52 There.

6:53 Remember that these two distances are equal?

6:56 As are these and these.

6:59 But then, remember two copies of the bottom

7:02 side of our triangle become the whiskers on top.

7:06 Then those two become the whiskers on the right.

7:11 And finally those are the whiskers on the left.

7:16 Now let’s highlight the horizontal strut again.

7:19 Here comes the magic:) Tada:) Same length again.

7:23 Not bad either, don’t you think:) Now let’s have a vote.

7:27 Did you like the swivel proof best, that one?

7:31 Or did you like the coloring proof best?

7:34 Before you continue with the video, record your vote in the comments.

7:39 I actually trialled a version of this presentation in one of my classes

7:43 at uni and there most my students voted for the coloring proof.

7:48 Fair enough, the colouring proof is definitely quicker.

7:52 However, apart from ALSO being very pretty,

7:55 the swivelling proof has another major thing going for it.

7:59 The swivelling proof actually spawns a beautiful new way to generate

8:03 the iris that even the most hardcore triangle lovers among you will not

8:08 be aware of, a new way that makes Conway’s static IRIS contract

8:12 and expand like a real-life iris:) That’s the next chapter of our video.

8:24 Let’s go for a second round of swivelling.

8:27 This time focus on the magenta endpoint of the swivelling strut.

8:35 That one there.

8:37 Aha, as you can see, as we swivel,

8:40 the magenta endpoint cycles through all six endpoints of the whiskers,

8:44 our special six points.

8:46 And when you look at the swivelling strut,

8:48 aren’t you reminded of windscreen wipers in action?

8:54 And so, just focussing on the windscreen wipers,

9:01 gives a second very intriguing way of generating our six points.

9:05 For easy comparison here again is Conway’s first way.

9:08 Triangle, then whiskered triangle.

9:10 And there are the six points.

9:13 Now second way.

9:14 Triangle.

9:15 Then infinite whiskers place a copy of the red side to locate the first point.

9:22 Then find all the remaining points by this beautiful windscreen wiper action.

9:32 And there is the circle again.

9:37 and its smaller incircle baby sister.

9:39 But now the super nifty extra aspect of this second

9:43 way of generating the IRIS is that it turns

9:47 out to be a special case of the windscreen

9:50 wiper theorem which gives birth to infinitely many more IRISES.

9:58 Here are a few of these irises.

10:04 Intrigued?

10:05 Well, then let me show you where those other irises come from.

10:08 Start again with the special point on the bottom strut.

10:12 But now instead of this special starting point

10:16 let’s start with another point on the horizontal.

10:20 This one here, a bit further in.

10:23 Go iiiin.

10:24 Now unleash the windscreen wipers again.

10:27 We get six new points that are also irised.

10:34 And all this actually works for any

10:39 starting point whatsoever on the horizontal line.

10:42 Even points on the triangle’s sides itself work.

10:46 Have a look.

10:57 How wonderful is that?

10:59 So Conway’s circle theorem is really just

11:01 a special case of the very general “windscreen

11:04 wiper theorem” which says this: Start with any

11:07 point on the whiskers of a triangle.

11:10 Then windscreen wipering the point gives six points on a circle

11:13 that together with the incircle of the triangle forms an iris.

11:17 I should mention that the full windscreen wiper theorem can be

11:21 proved by swivelling exactly as we did for Conway’s special case.

11:24 So, my students got it wrong:

11:27 the swivelling proof is in fact the superior proof.

11:29 Yep, I should really go back and fail

11:32 them all:) Just kidding :)I have not been able

11:35 to establish the exact chronology of the discoveries

11:37 of Conway’s circle theorem and the windscreen wiper theorem.

11:41 Conway’s circle theorem only achieved some fame

11:44 after John Conway died of COVID in 2020,

11:46 when Mat Baker mentioned the circle in his tribute blog for John Conway.

11:51 Since then a number of different

11:53 proofs for Conway’s circle theorem have surfaced

11:56 and what I presented today was distilled

11:59 from proofs by Colin Beveridge and Paul Farrel.

12:02 Then, it was only pointed out last year

12:05 by Michael de Villiers that Conway’s circle theorem

12:07 is a special case of the windscreen wiper

12:10 theorem which has been know since at least 1994.

12:14 Anyway, I’ll put links to whatever I’ve been able

12:16 to find chronologywise in the description of this video.

12:19 If you happen to have more information about who

12:22 discovered what and when or some anecdotes involving either theorem,

12:26 please let me know in the comments.

12:29 Actually, windscreen wiper theorem is my name for something that was

12:34 originally known by the not-terribly catchy name “side divider” theorem.

12:37 I also made up the name Conway’s iris.

12:40 If YouTube has taught me anything,

12:42 then it’s the fact that having a good title or name

12:46 can be key to your video or theorem being a success.

12:49 Okay to finish off, let me show you a dramatic rendering of the windscreen

12:53 wiper theorem involving some raindrops and a very unexpected twist at the end.

13:11 What do you think of my rainy day rendering of the windscreen wiper theorem?

13:13 Nice, right?

13:14 But where is the twist that I promised you?

13:16 Well have a look at this.

13:18 After all that wiping there is still water in the IRIS.

13:22 Not that it matters as far as anything I’ve said

13:26 so far is concerned but think about it for a second.

13:29 Those windscreen wipers were of different lengths.

13:32 This means that the wiped area cannot be a circle.

13:36 But if it’s not a circle, what sort of shape is the wiped area?

13:41 Well, let’s have a close look.

13:43 Not a circle, but just like the circle

13:46 this weird curve contains the six special points.

13:49 For comparison, here is the circle.

13:51 Interesting isn’t it?

13:52 What’s even more interesting is that this weird curve turns out to be one

13:58 of those curious curves of constant width

14:00 that I already reported on in an earlier video.

14:03 What this means is that if you wedge the curve in between two parallel lines.

14:09 Then, no matter which way the parallel lines are oriented,

14:14 their distance is always the same.

14:18 So, in a way this curve,

14:21 just like a circle has the same diameter in all directions.

14:26 Among other things this means that when you

14:28 rotate the curve inside a square of just

14:31 the right size it it will touch all sides of the square at all times.

14:36 Really beautiful and totally unexpected, don’t you think?

14:42 And here is a little puzzle for you to finish things off for today.

14:49 Ready?

14:49 Which is larger, the diameter of Conway’s circle or that of the weird curve.

14:54 Let us know what you think in the comments.

14:57 Until next time:)