Do you want to see my new favourite shape?
Yeah of course you do! It's this one here,
it's called the largest small hexagon.
And I just love it because I love its
name, the largest small hexagon. So let's
break down what the name means; so it's a
hexagon, which is a six sided shape. Small
means that all the diagonals, like corner
to corner, they're all less than and
equal to 1, that's what small means. And
then largest small hexagon means it has
the largest area with that rule. If all
the diagonals are less than or equal to
1 the idea is it could fit
in a
1x1 box like that.
In fact you could roll it around inside
that box and it would fit nicely in
there.
What's surprising is it's not the
regular hexagon, because that's what my
guess would have been. Okay what is the
largest area if you had all the
diagonals fitting inside a 1x1
box? And I would have guessed regular
hexagon.
But it's not. So it's this thing so this
is not a regular hexagon, it's irregular,
it has a larger area than the regular
hexagon by about 4% and this
was discovered in the 1970s by Ron
Graham. So let me show you a little bit
more about this. So I was surprised that
it wasn't a regular hexagon, and if we
did shapes with less sides it
actually is the regular shape.
So for a triangle
the largest small triangle
is
the regular equilateral triangle; these
have a side length 1
and that would be your answer.
For a quadrilateral
it is the square, the square has
diagonals of 1. So it's not side length
1, it's diagonals of length 1, that is
your largest small quadrilateral.
Although there's more than one answer
for the quadrilateral, because it could
be
any quadrilateral which has two
diagonals that are at right angles to
each other that have length 1; this has
length 1 and this has length 1 like
that. If I put them like that that would
be a square but I could have a
quadrilateral of any sort of kite shape
like this and those would all have the
same answer, they all have the same area
so they all count as the largest small
quadrilateral. If you're looking at the
pentagon it's the regular pentagon. So
here is the regular pentagon and then
your diagonals have length 1 and that
is the largest small
pentagon. So therefore you think, ah the
largest small hexagon
would be the regular hexagon,
here it is, but it turns out
it's not.
So to solve this what Ron Graham did is
he used some graph theory, which is a
little bit surprising. Graph theory is
the maths of points and networks, so dots
and lines; and he knew that some of these
diagonals had to have length 1. And he
just kind of looked at all the
possibilities and he drew them as graphs,
he knew that some of these possibilities
were definitely not going to be the
largest area, so he could get rid of
those. And he had about
10 possibilities to go through and he
got rid of them, he says no that doesn't
work, and then with the few that he had
left he sort of wiggled the angles and
said, well if I wiggle the angle does
that make it bit larger? He worked on it
a bit longer and then he found
this shape. So here it is again, so this
is our
largest small hexagon. But there is some
symmetry, it's not the regular hexagon,
but there is some symmetry to this.
(Brady: It looks like a pentagon.) - It does look like
a pentagon; there's a kind kind of a reason
why it looks like a pentagon, let me show
you. So the way it's been constructed
actually is it's this shape. So this is a
pentagon where all the diagonals are
equal to 1. And then for the hexagon
all we've done is we've taken a vertical
line down here of length 1 and we've
added an extra corner to the bottom of
this pentagon. So that's kind of the
shape
of the largest small hexagon. So then Ron
Graham said, are the other
shapes that have even number of sides, is
that the same kind of idea for that? That
was his conjecture.
Well we didn't prove that for a long
time; the octagon took another 30 years
to prove.
And we did, we proved it.
So the octagon isn't the regular octagon,
but it is this shape here. Here we have
our largest small octagon; and it was
made the same way. So started with a
seven-sided shape like this, all the
diagonals have length 1, add on one
more diagonal and a little extra corner
to the bottom there and that's why we
get our octagon. And then only recently,
2007, we we were able to prove that this
is true. So for the even sided shapes
this is how you construct the largest
small shape.
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