Okay so just- well I meant kind of that
way but you know.
So we're going to talk about shapes of constant
width, this is what I want to show you,
and to show you I've built a sort of
diorama. Well what I've done, I've stuck a
lego man to a ruler so there it is. And I
want you to imagine that you're this
Lego man; it's annoying,
you've been stuck to a ruler so you
can't move around. But you have the idea,
maybe if I put some shapes between the
ruler and the ground I'll be able to
move around and so you experiment in
different ways. The first thing you try
maybe is some hexagons. You see the
hexagons, they don't work very well,
because as the hexagons rotate the
height changes and it's a bumpy ride. So
hexagons are not shapes of constant
width or height, they're shapes of
non-constant width, non constant height. So
Brady what shape would I have to put in
there to make this work? - (Brady: Well I'm)
(obviously- like a circle?) - Oh yeah greatest
invention of all time, there is. Because
the reason that works is as a circle
rotates the height stays the same, the
width stays the same. And that's the only
shape that can work right? That's why
circles are round. It wouldn't work with a
square, it wouldn't work with a triangle.
Would it work with a triangle but if we
sort of rounded the faces a little bit?
Probably not; well let's try that. So
these these are called Reuleaux triangles,
I might not be pronouncing that right
someone correct me in the comments, I'm
sure you will. So here it is right. So
they're not circles but they're the same
height all the way around and the same
width all the way around. No matter how
you turn these shapes they are the same
width even though they're not circles.
Yeah, when I first saw that that kind of
blew my mind a little bit. It's cool
isn't it? - (Skipping past this,)
(why don't we make wheels like that?) - Okay so actually the
reason we don't make wheels like this is
because if you put a central axis in
there you would bump bump up and down
because the distance from the centre to
the edge does change. It's the distance from the
top to the bottom that doesn't change.
Actually someone has built a bike, it was
to commemorate the Beijing Olympics I
think a few years ago. And the the frame
of the bike
rested on top of the wheels and so this
shape of wheel could then work in that
situation. It's also used in something
called a Wankel engine but we won't
talk about that. I'll show you how to to
make one of these: you take an odd sided
shape, say a triangle like this and I'm
using an equilateral triangle, and you
just round off the faces as if you've
got a compass, you put the compass point
on on one of the corners and then sort
of go like that and then like that and
then like that. So let's just do kind of
that, like that, like that. If you started
with a seven sided shape then you end up
with 50p coins. So 50p coins are shapes of
constant width. If you get a hold of two
50p coins and a couple of books for
example, and roll the books around,
they will stay the same distance apart
because even though they're not circles
50p coins are shapes of constant width.
Well they make them like that because
they can and it's an interesting thing;
but coins have to be shapes of constant
width to work in vending machines,
vending machines measure the width of a
coin as it goes through, so if that were
to change the vending machine wouldn't
know necessarily what coin you're
putting in. So you can make coins this
way, so they do. So that's the that's the
Reuleaux triangle, or Reuleaux triangle - I
think I've drawn that quite well look.
(That's amazing!) - I don't believe that! Very impressed.
You can take any triangle and turn it into a
shape of constant width; this one is a 3,
4, 5 triangle and so it shows you
actually that- how you have to lengthen
the edges here. So that's a- that's a
single unit extended there and two units
extended out there.
I actually got quite obsessed about this,
when I started to think about is there a
three-dimensional equivalent
to this? So the three-dimensional
equivalent of a circle is a sphere. So if
you take three now spheres I can roll
this around on here like that, the plane
of the table and the plane of this board
stay the same distance apart because
spheres are solids of constant width
we call them. But there are three-dimensional
equivalents of shapes of constant width
that are non spherical. The way to make
them is you take like one of these
and you make a solid of rotation like
that, so you just spin it around and it
sweeps out a three-dimensional shape. So
this is a shape that you can turn on a
lathe. So I went online and I had a look
around, does anyone- does anyone have these
things? I eventually came across a book
called 'How Round is your Circle' by
John Bryant and Chris Sangwin. They had
some of these shapes on the front cover
and I said, you know, can I get some of
these? I sent Chris an email.
It was quite a funny email actually, I said
something like, you know, I love your
shapes of constant width, where did you
get them? And he said well I had
thousands made with an educational grant,
how many would you like?
So he sent me a whole load, he sent about 50
and here's three of them here. It's a shape
of constant width that been rotated into
a solid of constant width and so they
work as well. So there you go. Now I was
very happy with that but then if you
look at the front cover of Chris
Sangwin's book, I recommend you check it
out, his solids of constant width are
metal and and they're pointy as well;
these have been cleverly rounded to make
them wear a bit better but I wanted some,
I wanted some metal ones, I wanted them to
be pointy. So I searched around on the
internet and eventually found a forum
for lathe enthusiasts. And they all said
it can't be done you can't do this, it's
not something you can turn on a lathe.
And then I got- brilliantly I got a
private message from one guy saying I'll
do it for you. So these are my metallic
solids of constant width. - (Wow)
So what I do with these, I put them- I
line them up and I stand on this and I
roll around on them. Not just for fun, I
mean for shows - and for fun.
(You can't tell us that and not-)
(You can't say that - can't just drop that and)
(then not do it!) - Really?
Okay. Now we are on a bit of carpet so it
might not work as well but- well I might
need someone to give me a push, I don't know how
you feel about that?
Well I meant kind of that way but, you know, it's
So push me towards these extra
ones here, yeah that's it, yeah, that's it.
Yeah there's an infinite number of those
as well, because again I could take that
the pentagon and rotate that and that
would become one; I could take, you know,
the 50p coin and rotate and that would be
one; or the nine sided shape, 11 sided
shape whatever. But there is a
more interesting example which is,
instead of just doing a solid of
rotation which- well it feels a bit like
cheating. So if you take the
three-dimensional equivalent of a
triangle, which is a tetrahedron, and
round off the faces of that then it's
actually not quite a solid of constant
width, you have to round off three of the
edges as well. So this doesn't have- it's
not a solid of rotation anymore, it has
less symmetry, it's even more kind of
counterintuitive. It seems less like it
could be a solid of constant width and
it's called a Meissner tetrahedron. You
can't get these turned on a lathe but
here they are made in plastic. You can
see we've got three slightly rounded
edges, which is something you have to do
to get it to work, three straight edges
like that - or sharp edges should I say -
and then basically rounded off faces
using the- again using the opposite
corner as a guide. So that point there is
the centre of this sphere here. And so
these are, believe it or not, solids of
constant width.
(Nice. And that's a nice oddity. Are)
(these useful? Do engineers and)
(people use these for anything?)
Engineers use the the Reuleaux triangle or
Reuleaux triangle in the Wankel engine and
that's that's about- oh you can you
there's a drill as well with- it's got it
uses that triangle in there somewhere
and you can use it to cut a square hole,
you can drill a square hole with it yeah.
One thing I thought is maybe ball
bearings that won't roll away. You can
put it on a slope and it doesn't- well
it's going to slide because it's
slippy but it's it's not going to roll
because the centre of mass is really low.
That could be used for something but- - (Dragon's Den?)
Yeah alright, I'll give it a whirl.
So I know what you're all thinking -
where can I get my very own shape or
solid of constant width? And the answer
is: Steve and the guys at Maths Gear have
had them made for you, check it out. That
is the ultimate geeky Christmas present
right there, what a great little stocking
filler. I get no Commission from this I'm
just saying, that's a great present, I'd
love to have one of them under the tree.
I think the shapes here come in wood or
acrylic and here's the solids. I've put a
link in the video description - you're
crazy if you don't check it out.