Welcome to another Mathologer video. This is a special mathematical ferris wheel, a
geometrical gem that has only recently been discovered. Notice how the
cabins of the ferris wheel always just touch and brush along their neighbours.
Pretty mesmerizing an impossible looking isn't it?
Here's another example of the same kind of mathematical magic. These newly
discovered miracles use what are called shapes of constant widths which are
already very famous for their repertoire fantastic geometrical feats. So let's
begin with a crash course on shapes of constant widths and a quick survey of
their most famous and impressive tricks. Here we go. First question: How wide is
Homer? ... Very! Yep, but more precisely, we can trap him between vertical bars and
he's that wide. Right? Actually for mathematicians Homer is a man of many
widths. In fact, there are infinitely many widths, one for each possible direction
of the parallel bars. Here the bars point in a different direction and here's the
corresponding width of Homer. And here's another one and another one. Of course
there is this one special mathematical shape that is famous for having the same
width in all directions. What shape is that? Too easy :) Right? The circle of
course. So there's a circle, there's the width and in all direction you see
it's always the same. Now it turns out and this surprises almost everyone when
they first hear it, the circle is not the only shape with just one width. The
simplest of these non-circular shapes of constant width are the cabins of our
ferris wheel, these guys there. To construct our cabins start with three
corners of an equilateral triangle and draw just the right sized circles
centered at the corners, like this. One two, three, three corners and three
circles and the cabin shape we are after is the curvy triangle in the middle. This
shape is called the Reuleaux triangle, named after the 19th century german engineer
Franz Reuleux, that guy there. The constant width of the Reuleaux triangle is just the
common radius of the three circles we began with. Let's make sure that this is
really really true. So, okay, the widths here is just the radius of the circle
around this corner and then it's clear that the width stays the same as we
change directions. Just move it around like this and you can see the width
is always the same. It turns out there are many other shapes of constant width.
For example, just as we began with an equilateral triangle to construct Reuleaux
triangle, we can similarly begin with any regular polygon with an odd number of
corners. Here are the corners of a regular pentagon. Now draw a circle centered at
each of the five corners and passing through two opposite corners. Here's the
first such circle and here are the other ones.
And, as for Reuleaux, the curvy pentagon we've just captured in the middle has
constant widths. And this is as easy to show as we did for Reuleaux triangle and
here's the constant width shape obtained from a regular 7-gon, and so on. A
first challenge for you: What goes wrong if we try this construction with an even
sided polygon. Share your thoughts in the comments. But Reuleauish polygons are not
even close to all there is. Here's another very simple and very powerful method of
construction. Highlight the pentagram inside this constant width pentagon. Each
side of the pentagram is a radius of one of the five generating circles of the
pentagon and so the length of these pentagram sides is exactly the constant
width of the curvy pentagon. Now at the risk of incurring the wrath of some
monster summoned by our demonic shape, let's deform the pentagon. Here we only
have to make sure that overall all the five sides lengths stay the same. Now draw
in the five circles again one centered at each corner like this.
Magic :) Can you see what happened? Arguing exactly as before
whatever the wonky pentagram, the resulting wonky Pentagon in the middle
must be a shape of constant width. Pretty, hmm. Here are a few more (music) And you can do the
same starting with all those devilish stars hiding in their other polygons (music) Now
if you like your maths tactile, you can also build physical models of these
flexing stars. Have a look. Now, just in case you're wondering, I
built this star from hacksaw blades.
These constant width shapes are pretty incredible, right? But hold your applause.
Wait until you've seen some of their amazing tricks. First trick. Pin a
constant width shape between two pairs of parallel bars forming a square. Now
rotate the shape. Then the rotating shape touches all four sides of the square at
all times. Simply magical, isn't it. This setup can be used to build a physical
machine that can carve these kinds of rounded squares. I've linked to some super
nice movies in the description. Second trick: There actually exist coins in the
shape of the Reuleaux triangles and it's regular 7-gon relative. Why of
course they look very cool. But there are also practical considerations.
Coin-operated machines identify coins by looking at them edge on and measuring
their widths. So to work properly these machines require coins of constant
widths. Also it turns out that the non-circular coins over there have less area
than the circular one and therefore, everything else being equal, these coins
are cheaper to produce than the circular one, which can't be bad, right? Third trick.
Here's a very famous question, supposedly a standard in high stakes job interviews:
Why are manhole covers round? Well, obviously, to cover the round holes of
course :) That's the funniest answer though maybe not the one that will score
you the job. So what answer are they expecting? The standard answer and a very
good one is "because the circlar lids can never fall into the slightly smaller
circular holes" So just have a look, right. You can turn around this way, that way,
any way you want, they will never fall in. That is not true
for some other shapes such as squares. Right, so let's have a look. These shapes can
definitely fall into the corresponding holes. In this position they just fall in.
And this would be an obvious danger to the workers below. Okay
so now you are an expert on manholes and you can ace that next job interview, right?
Wrong!!! Because the candidate who comes right after you gets the same
question and smugly replies: Well, actually, manholes don't have to be round.
And of course she is correct. Any manhole of constant width would be just as safe
so that's just a Reuleaux one, right. Turn it around, turn any way you want, you know,
it will never fall into the hole. Fourth trick: non-circular rollers can make for
a perfectly smooth ride :)
You can't be too blase with shapes of constant width. For example though we can make
nice Reuleaux rollers, making a car with Reuleaux of wheels would result in a pretty
bumpy ride. Having said that, there's also a very clever constant width trick that
circumvents this problem. I'll explain how this miracle car works a little
later. How do all these tricks work? Well, except for the final crazy car, all of
the tricks can be easily explained with the defining property of shapes of
constant width. But as well as this defining property, shapes of constant
widths have a number of other non-obvious and very interesting
properties. My absolute favourite among these properties is Barbier's theorem. This
theorem says that all, absolutely all shapes of the same constant width also
have the same perimeter as the circle. So all those guys over there have the same
perimeter. Amazing, amazing, amazing. Second challenge for you: explain or prove
why this is so for the special case of the star based shapes that I showed you.
Here is a hint: the key ingredient of the proof is to show that in all of these
stars the sum of the spiky angles is the same, which is a very beautiful result in
itself don't think? So now it's time for another ride on our ferris wheel. Let's
figure out what it is about constant width shapes that leads to the cabins of
the ferris wheel smoothly touching as they do. To sneak up on the question
imagine that in between those two bars there is a shape of constant width.
Right there. Imagine! :) Now slide the bars together until they touch the shape on
either side. Let's mark a point of contact on the left side, there. What
about the right side? Of course there must be some point of contact on the
right since otherwise we could bring in the right wall a little. It turns out
that there is exactly one point of contact on the right and the second
point is located directly opposite the contact point on the left, like this. Why?
Why can there not be a contact point somewhere else down there?
Simply put because the green bar is longer than the distance between the
blue walls. Let me explain. Our mystery shape supposedly has
constant width. But the width in the direction of the green bar would have to
be at least as long as the green bar and so longer than the width determined by
the blue walls. Of course that's impossible
for our constant width shape. So, again, if you wedge a shape of constant width
between two parallel bars, the two bars touch the shape in exactly one point
on each side and these points are located directly opposite each other.
Okay now let's now use this property to explain our ferris wheel. First let's
look again at the Reuleaux triangle spinning inside the square, there. Pay
attention to the touching points on the left and right walls and notice that
they really are always directly opposite each other.
Let's duplicate the pictures so we have two squares side by
side. Now rotate our Reuleaux triangles in unison. Because of the opposite points
property it is now clear that the two shapes must always touch at the common
point where they contact the middle wall. Instant magic :)
Next here are the two Reuleaux triangles again this time in blue. Why in blue?
Well because I ran out of green :) I've also marked two corresponding points.
let's rotate the triangles again focusing on the two points. The points
move around but of course are always in the same position relative to each other.
So the line connecting the points is always horizontal and always has the
same length. So now imagine that as the animation is running you visually cancel
out the moving of the points by moving the whole screen around in just the
right way. What do you see? Well, got it? Well the points stay fixed and the
triangles now wrote it in unison around them. How neat is that? The same also
clearly works no matter how the two shapes start out touching. Very pretty.
Alright, let's do some more cloning. We can assemble multiple copies of our
shape into any touching configuration and when we rotate all the shapes in
sync all the touching will be maintained. And that's super, super duper pretty.
Almost there. To construct our ferris wheel, we simply relocate the rotation
points and we arrange a suitable number of our triangles around a suitably sized
circle. Perfect! Well, the ferris wheel actually is not
moving and the passengers will be tipped out as the cabins rotate, but almost
perfect. So to satisfy these fussy passengers, to make the cabins move
around the wheel and to have them always point downward we make one final
adjustment. As the cabins are spinning in the counterclockwise direction we make
the ferris wheel rotate in the opposite, the clockwise direction.
Exactly the same type of rotation combo produces the second effect that i
showed at the beginning. Also super duper nice, isn't it? And of course this does not
only work for Reuleaux triangles. Everything I've shown just works as well for any
shape of constant width. But there's still one more layer of trickery and it
has to do with the crazy car I showed you earlier. Let's focus on one of the wheels.
There's a familiar Reuleaux triangle in the middle, doing its now familiar
rotation within the little square. But there's also a second larger constant
width shape that rotates in unison with the Reuleaux triangle. In turn this dark
gray shape is rotating inside a larger square. But there's more. The squares have
the same center and contact points of the two constant width shapes with their
squares are always aligned. There aligned aligned, aligned and aligned, always aligned.
This makes it clear how the larger shape is constructed from the Reuleaux
triangle. Just take any of its diameters and extend it on both sides by segments
of equal lengths, right all very pretty. So you see how the orange bits swipe
over the gray area and that makes up the other shape. But this means that if the
frame of the car is attached to the inner square, then because the blue
distance from the square to the ground is always the same the car will move
without any ups and downs just like on proper circular wheels. Very ingenious
isn't it. But that's not all. Because the contact points of the two shapes are
always aligned we can also pull things apart like this. And if we now rotate the
two shapes in unison they will always touch and this means that we can play
our touching dance games with a combination of Reuleaux triangles and
their parallel shapes and this opens up all sorts of possibilities for
serious constant width fun. I'll finish with some animations based
on this observation and the other things we talked about today. But before that
let me tidy up with some acknowledgments. The idea for the ferris wheel is due to
the Japanese computer scientist Kenichi Miura. In fact when I saw him talk about
this idea at the 2010 Gathering for Gardner he talked about a waterwheel and
not about a ferris wheel. The idea for the crazy car is due to Claudia Masferrer
Leon and Sebastian von Wuthenau Mayer who I think were still mad students in
Mexico when they made their discovery and yours truly the Mathologer put all
the pieces together and nutted out the general theory. In the description I've
linked to my write-up, as well as to some other nice articles
and videos about shapes of constant widths. And that's it from me for today.
Enjoy the rest of the animations. (Music)