Welcome to another Mathologer video. You are
here for the mysterious IRIS in the thumbnail, aren’t you? Peaked your interest, eh? Well, let’s
chase it down. Start with any old triangle. At the corner opposite the red side grow two copies of
that red side, like this. Then opposite the blue, the same thing grow two copies of the blue side
and there, two greens. A triangle with whiskers, cute :) Now, doesn’t matter what triangle we
start with, those six end points they wait what’s special about those six points? Can you
guess? Those six points are … you got it? … Yep, those six points lie on a circle. Really? Sure if
it was going to be anything, then it was a circle but it’s still pretty surprising, isn’t it? Three
random points usually determine a circle. On the other hand, four random points usually don’t lie
on a circle and of course it’s even less likely for random five or six points to lie on a circle.
Anyway, this miraculous six-point circle is called Conway’s circle named after its discoverer,
the legendary mathematician John Conway. Conway was a mathematics professor at Princeton
university, a certified genius, and the inventor of many ingenious mathematical games that also
became incredibly popular outside mathematics. For example, chances are, even if you are not a
mathematician that you are familiar with Conway’s bizarre Game of Life. There you’ve seen this sort
of alien 2d life before, right? That’s due to Conway. Anyway, little miracles like his circle
would constantly pop up in conversations with John Conway and if he deemed you worthy, you could
expect to be challenged to come up with a proof on the spot. I’ll show you the ultimate super pretty
proof next, a proof which involves the IRIS in the title of this video. I’ll also tell you about
some exciting new life beyond Conway’s circle: in particular the beautiful windscreen wiper
theorem will have you smiling next time it rains :) However, if you consider yourself worthy,
then before you watch the rest of the video, accept Conway’s challenge and try to come up with
a proof of your own. Report on your efforts in the comments :) Here John Conway looks a little bit
like Saint John, doesn’t he ? :) Anyway did you come up with a proof? No? Not a problem, not
an easy one :) Want a hint? Okay, here’s one. Whenever you deal with a would-be circle, what’s
the first thing you should be looking for? Yep, its heart, of course, the center of the circle
:) And how do we locate the center? Well, for example by shrinking the circle. There
shrink. There that’s the center. But wait a minute. Before the circle collapsed into the
centre, did you notice something remarkable? No? Did you blink? Let me show you again. Ready? This
time watch closely :) You saw it this time, right? Just before the circle collapses to a point it
appears to simultaneously touch all the sides of the triangle. There rewind a bit. …That’s really
cool, isn’t it? And so it looks like the whiskered triangle actually gives birth to an IRIS and not
just the circle on the outside. Equipped with this extra clue does the triangle lover in you want to
give proving all this another go? Pause the video if you like, but we’ll keep going and chase down
a pretty proof ourselves. Let’s start with a ring and highlight one of its cords. Then it’s clear
that all such cords have the same length. obvious, right? Also, the point where a cord touches the
inner circle is the exact midpoint of the cord. There. Also obvious. Alright, what this means
is that if we start with a circle and three equal struts with highlighted centers. Then if we
arrange these struts around the circle like this we can be sure that the six ends are on a circle
that, together with the one we started with, forms an iris. Still all crystal clear, right?
So to prove that the whiskered triangle really extends to an iris all we have to show is two
things: First, we have to show these three struts are of equal length. Pretty damn obvious,
right? They are all made up of 1 red, 1 blue, and 1 green each. Oookay :) And the second
thing we have to show is that the points at which the struts touch the inner circle are the
centers of the struts. How do we do that? Well, here is a nice idea. Let’s swivel this highlighted
horizontal strut around one of the corners of the triangle. That orange corner there. Okay, go,
swivel. Since the two blue ends are of equal length. the two struts will end up in perfect
superposition. Nice. But did you notice something extra nice? There. Looks like the contact points
with the circle also land on top of each other. Why is that? Let’s have another look at the
starting configuration. Well these two distances here are clearly equal. This means that not only
do the two struts end up superimposed perfectly, but also the two points of contact do. Alright.
Now, color the strut like this. So what we want to show is that the magenta is exactly as long as the
aqua. Ready for a nice AHA moment in three stages? Here we go. Stage 1 swivel around the right
corner. Perfect coincidence. Stage two. Swivel around the top corner. Perfect coincidence again.
Stage three. Perfect coincidence yet again and, overall, the aqua and magenta parts have switched
places, which shows that the aqua is exactly as long as the magenta. You liked this? Very nice
isn’t it? And of course the same is true for the other two struts. Which then shows that the six
points are on a circle. And we’re done. We have confirmed the existence of Conway’s IRIS. But let
me show you a second way to see that the two parts of our strut are of equal length. There. Remember
that these two distances are equal? As are these and these. But then, remember two copies of the
bottom side of our triangle become the whiskers on top. Then those two become the whiskers on
the right. And finally those are the whiskers on the left. Now let’s highlight the horizontal
strut again. Here comes the magic :) Tada :) Same length again. Not bad either, don’t you think :)
Now let’s have a vote. Did you like the swivel proof best, that one? Or did you like the coloring
proof best? Before you continue with the video, record your vote in the comments. I actually
trialled a version of this presentation in one of my classes at uni and there most my students
voted for the coloring proof. Fair enough, the colouring proof is definitely quicker.
However, apart from ALSO being very pretty, the swivelling proof has another major thing going
for it. The swivelling proof actually spawns a beautiful new way to generate the iris that even
the most hardcore triangle lovers among you will not be aware of, a new way that makes Conway’s
static IRIS contract and expand like a real-life iris :) That’s the next chapter of our video.
Let’s go for a second round of swivelling. This time focus on the magenta endpoint of
the swivelling strut. That one there. Aha, as you can see, as we swivel, the magenta endpoint
cycles through all six endpoints of the whiskers, our special six points. And when you look
at the swivelling strut, aren’t you reminded of windscreen wipers in action? And so,
just focussing on the windscreen wipers, gives a second very intriguing way of generating
our six points. For easy comparison here again is Conway’s first way. Triangle, then whiskered
triangle. And there are the six points. Now second way. Triangle. Then infinite whiskers place a copy
of the red side to locate the first point. Then find all the remaining points by this beautiful
windscreen wiper action. And there is the circle again. and its smaller incircle baby sister. But
now the super nifty extra aspect of this second way of generating the IRIS is that it turns out to
be a special case of the windscreen wiper theorem which gives birth to infinitely many more IRISES.
Here are a few of these irises. Intrigued? Well, then let me show you where those other irises come
from. Start again with the special point on the bottom strut. But now instead of this special
starting point let’s start with another point on the horizontal. This one here, a bit further
in. Go iiiin. Now unleash the windscreen wipers again. We get six new points that are also irised.
And all this actually works for any starting point whatsoever on the horizontal line. Even points
on the triangle’s sides itself work. Have a look. How wonderful is that? So Conway’s circle theorem
is really just a special case of the very general “windscreen wiper theorem” which says this: Start
with any point on the whiskers of a triangle. Then windscreen wipering the point gives six points
on a circle that together with the incircle of the triangle forms an iris. I should mention that
the full windscreen wiper theorem can be proved by swivelling exactly as we did for Conway’s special
case. So, my students got it wrong: the swivelling proof is in fact the superior proof. Yep, I
should really go back and fail them all :) Just kidding :)I have not been able to establish the
exact chronology of the discoveries of Conway’s circle theorem and the windscreen wiper theorem.
Conway’s circle theorem only achieved some fame after John Conway died of COVID in 2020, when Mat
Baker mentioned the circle in his tribute blog for John Conway. Since then a number of different
proofs for Conway’s circle theorem have surfaced and what I presented today was distilled from
proofs by Colin Beveridge and Paul Farrel. Then, it was only pointed out last year by Michael
de Villiers that Conway’s circle theorem is a special case of the windscreen wiper theorem
which has been know since at least 1994. Anyway, I’ll put links to whatever I’ve been able to find
chronologywise in the description of this video. If you happen to have more information about
who discovered what and when or some anecdotes involving either theorem, please let me know in
the comments. Actually, windscreen wiper theorem is my name for something that was originally known
by the not-terribly catchy name “side divider” theorem. I also made up the name Conway’s iris.
If YouTube has taught me anything, then it’s the fact that having a good title or name can be key
to your video or theorem being a success. Okay to finish off, let me show you a dramatic rendering
of the windscreen wiper theorem involving some raindrops and a very unexpected twist at the end.
What do you think of my rainy day rendering of the windscreen wiper theorem? Nice, right? But where
is the twist that I promised you? Well have a look at this. After all that wiping there is still
water in the IRIS. Not that it matters as far as anything I’ve said so far is concerned but
think about it for a second. Those windscreen wipers were of different lengths. This means
that the wiped area cannot be a circle. But if it’s not a circle, what sort of shape is the wiped
area? Well, let’s have a close look. Not a circle, but just like the circle this weird curve contains
the six special points. For comparison, here is the circle. Interesting isn’t it? What’s even more
interesting is that this weird curve turns out to be one of those curious curves of constant width
that I already reported on in an earlier video. What this means is that if you wedge the curve
in between two parallel lines. Then, no matter which way the parallel lines are oriented, their
distance is always the same. So, in a way this curve, just like a circle has the same diameter in
all directions. Among other things this means that when you rotate the curve inside a square of just
the right size it it will touch all sides of the square at all times. Really beautiful and totally
unexpected, don’t you think? And here is a little puzzle for you to finish things off for today.
Ready? Which is larger, the diameter of Conway’s circle or that of the weird curve. Let us know
what you think in the comments. Until next time :)
