We've got triangles, but these are special triangles. They're called hero triangles, you'll like that.
"Heronian triangles" and these are particular types of hero triangles.
A hero triangle is a triangle that has integer sides, whole number sides, and an integer area.
Okay, so it's just something that mathematicians like to play with and you should think of these a lot like...
perfect numbers, and things like that!
It's just something that mathematicians like to have fun with, we've we played with these for a long time,
even now we're still coming up with new things about them, but these are particularly special.
(Brady: They're superhero triangles!)
These are our superhero triangles. Absolutely!
I've now given you permission to put clickbait thumbnails on this video. Yeah, Captain America over here!
I can't believe it! Super hero triangles! So my first one, it's a right-angled triangle.
So the idea is they've got integer sides.
And, so, this has sides 5, 12 and 13, which is called a
Pythagorean triangle as well,
a Pythagorean triple.
But, this is what I wanted to point out, look at the perimeter. The perimeter is adding up the three sides
and so I reckon the perimeter is 30
So 12 plus 13 plus 5 and the area,
if you work out the area, well it's half the base times the height,
if you remember the formula for an area of a triangle, which will be 30 as well.
So they've got the same perimeter, they've got the same area.
That's what these five triangles have in common.
So if I do my next one, you can see this one's a right-angled triangle as well
Perimeter is 24, area is 24.
And let's have a look at these next three. This one, perimeter. What is the perimeter?
36. And, you won't be surprised, area is 36 as well. - (Brady: These ones aren't right angles are they, if you look?)
So these, yeah these last three I'm drawing out are not right-angled triangles. We look at the perimeter, oh, perimeter 42;
area 42, everyone's favorites.
And the last one, this really long one then, so the perimeter is 60,
Area is 60, right.
They do remind me of perfect numbers, the way that they kind of refer back to itself
A perfect number is when you look at the factors of the number and it adds up to itself, and it reminds me of that
so the perimeter and the area are the same, and even though there are an infinity of
possible triangles we could have with integer sides,
integer areas,
these are the only five
that have the same perimeter and the same area. So there is five and five only.
(Brady: presumably you can,)
(upsize these same five?)
That is a good point. But the problem with that is, if we took say, this triangle if we double it,
then we would double the perimeter, and the area gets multiplied by four.
So when you double the perimeter, area is multiplied by four.
So even including scaled up versions of triangles, uh, this is the five.
The only five we can find. So let me give you, give you some facts about - these things go back thousands of years,
it's just one of those things mathematicians study. So these are facts about the Heronian
triangles which means they have integer sides, and integer area. So, they are named after a mathematician
from way back, 2,000 years ago, Hero of Alexandria. They're not that hard to find, we saw that we can do it with
Pythagorean
triangle - Pythagorean triples. That's a well-known thing, and there are infinitely many of these, and they're quite easy to make. I mean the
simplest is the 3, 4, 5 triangle, perimeter 12 and the area, half the base times the height is
6. So you can do this, but any
Pythagorean triangle would work, as another example 17, 15, 8 would work as well,
so it's got an integer perimeter, which I think is 40, and the area is a
integer as well, 60, that's going to be guaranteed to be an integer. - (Brady: So they're heroes but not superheroes?)
That's right. Guaranteed every
Pythagorean triangle is going to have this property because you're going to have an integer perimeter, and yes, you're guaranteed to have
integer area as well.
I'm afraid what, what I mean is you're guaranteed - because it's half the base times the height,
it means you are guaranteed that one of these sides is
an even number. So out of all those infinite
Pythagorean triangles, only two of them have this property, where the area is the same as the perimeter and then the other three are not
right-angled triangles.
So what else could we have? so I can tell you straight away, if you wanted an equilateral triangle like this,
with sides all the same length, whatever that is. I'm afraid that's never going to have an integer
area. So you can't do it with equilateral triangles. They don't exist. What else could we do, what about an isosceles triangle?
Yes, we can do that. I'll show you an example, a 5, a 5, and a 6.
Perimeter is 16, and the area is going to be 12. So we've got
integer perimeter, integer area. It's a bit of a cheat though.
I don't know if you noticed it was a bit of a cheat because if I drop a vertical line here, what it is
it's actually two
right-angled triangles stuck together. It's actually a 5, 4, 3
right-angled triangle, stuck together. So I'm afraid, yes you can do isosceles triangles
although they are a bit of a cheat. So, can we make triangles that are not a bit of a cheat?
Okay, so we'll go a bit more general now. Here's another triangle. It's going to be a 15, a
20, and a 25. So, oh, what's it going to be; perimeter is 60, area is
150, so this works, this is one of our hero triangles.
But I've cheated again, because you can drop a vertical line, you can split it into two right-angled triangles.
So the question is are they - are they all cheats? Are there any triangles that aren't just made out of
Pythagorean right-angled triangles? And so you can do triangles that aren't cheats. Here is a 5,
29, and 30. So here's a triangle with a perimeter of
64, and an area of 72. And that cannot be broken up into two right-angled triangles that are
Pythagorean. If you drop a
vertical line here, well, we know, what, it's half
the base times the height - we're trying to work out the height now. So area is 1/2 base
times height.
I'm trying to work out the height,
so the height is going to be,
well, twice the area
divided by my base
which is
144 divided by 30,
but it's not an integer, which means it's not going to be a right angled triangle with integer sides. When you get these,
these are called the
indecomposable ones, they're ones that you can't break up.
You can break them up into right-angled triangles. and they're going to be rational sided right-angled triangles,
but not always integer ones. So these are, like, indecomposable.
So why are these called Hero triangles, named after Hero of Alexandria, who - mathematician,
2,000 years ago, worked in the Library of Alexandria, right, this big famous thing. And there is a
formula for the area of a triangle that is credited to Hero of Alexandria, which I wasn't taught at school
and I wish I was. I was taught
area of a triangle is half the base times the height, and I've always used that. And
sometimes the height is difficult to work out and you have to use a bit of Pythagoras's theorem,
work out the height and then you can do half base times the height,
well, here's a formula where we don't need to know the height of your triangle. The area of this triangle is
going to use a, b and c.
Lovely. Now the perimeter is a + b + c.
I'm going to use something called a semi-perimeter,
which you might be able to guess is half the perimeter. And we're going to use that in the formula for area.
The area of this triangle is the semi-perimeter,
multiplied by semi-perimeter minus a -
side a -
multiplied by
semi-perimeter minus
b, multiplied by semi-perimeter minus c. And the final thing is all square rooted.
So, I wonder if I wanted to work out my previous example,
which was a
5, 29, 30 and I don't want to work out the height, then I can use this formula.
It's gonna be half the perimeter multiplied by: 32 minus,
- let's do side a's 5, so that's going to be 27 multiplied by 32 minus
29.
Which is easy, 3 minus, 32 minus 30,
which is just a 2, that is going to be square rooted so you can - we'll work it out, okay.
You multiply this out. It is 72. - (Brady: Not bad!) - It's a lovely formula, isn't it?
You get the formula by Pythagoras' theorem. Use Pythagoras' theorem over and over again,
rewrite it in terms of a, b and c.
But it turns out really neat, you get this lovely neat formula, without having to work out the height of the triangle.
But we were interested in triangles that have the same perimeter and the same area.
(Brady: The superheroes)
These are the superheroes. We've got a formula for area,
so let's do it. All right, we've got area is this - we want it to be the same as
2s, that's the perimeter, is equal to area, which is the square root thing. All right,
oh, well, let's just tidy it up, you know, square both sides,
4 s squared,
is this stuff, and then we can divide by s. We've got s on both sides. So we get 4 s
equals s minus a, s minus b,
s minus c - little formula - and, I might not go through the whole thing, but you can do this;
I recommend if you want to solve this now, and find those triangles,
let's call these three terms x, y and z.
The left-hand side: 4 times s will become 4 times x plus y plus z. So, oh, let's say x
is less than y, less than z. Let's put them in the order, and there's no reason why it shouldn't be in that order,
yeah, it doesn't take that much to work out.
Um, actually, let's do it. Let's do it. You do x equals 1.
Stick that into this, solve it. If you solve that,
you will find three solutions. And the three solutions are these triangles,
the, the funny side triangles here. If you have x equals 2, you get your 2 right-angled triangles. If x equals 3
you get no new
solutions. And,
if you are looking at bigger numbers,
you'll find it's impossible.
If you look at these, this equation here.
On the left hand side x is the smallest number; x is smaller than y, it's smaller than z. Left hand side,
which is this, will be smaller
than 4 lots of
z plus z plus z. Z is the biggest number out of those three, which is 12z. And on the right hand side,
xyz, it's going to be bigger than: x is bigger than 4, so we can say it's bigger than 4,
y is bigger than 4 as well,
so it's bigger than 4 times 4 times - let's have z here - because that will be 16z
And we're - to solve it we're looking for a number smaller than 12z,
bigger than 16z, at the same time - which is impossible. Which means you can stop looking. End of proof.
So there's only a few things you have to check and you find all the possibilities.
So we found 5 triangles where the perimeter is the same as the area.
Which kind of, yeah, cute, right. And look - and you can see that the kind of like perfect numbers
Here's another triangle I wanted to show you.
It's best draw it out. This has
33, 34 and
65. And for that it's a perimeter of, if I'm all correct,
132. And an area, well we have a formula for that now, but I'll tell you the area:
264. So this is a triangle where the area is double the perimeter.
Well I can tell you there are 18 of those.
And I can tell you: if you want your area to be some multiple of the perimeter,
it's always a finite number. So, it's always these lovely little finite lists.
So if you want to be equal to the perimeter: we have our list of 5
If we want it to be double the perimeter,
we now have this list of 18, and same for triple, and timesing by 4 - and so on.
So I've got some news.
Even though all these
triangles have been studied for thousands years, we're still learning new things about them. And we've discovered some new
Hero triangles.
So I've printed them out here - (Brady: New heroes!)
Yeah, yeah, they're kind of interesting
You can see the point where my printer ran out of ink. Some of these triangles come in pairs,
I'll do - before I talk about these special ones - I'll do an example of a pair of triangles.
(Brady: Your triangles are getting worse, look at those two! They're terrible!)
They're great triangles! These really valid; lovely drawn triangles: 21,
20 and 29 and we've got a
17, a 28 here, and a 25. And, if you look at the perimeters and areas, because that's what we're doing, the perimeter of the first one
is 70, the area is 210.
Oh, so that would be on our list of triple,
the area's triple the perimeter. This one, perimeter of 70, area of
210. Like sometimes when we're studying numbers, numbers come in pairs
sometimes. These triangles have come in pairs because they have the same perimeter, same area - different triangles -
but they, they have this in common.
So, there's infinitely many of these pairs if you want to go looking for them, but this one is
special. And this pair is special and unique
because it's the only pair that has a right angle triangle and an isosceles triangle,
with the same perimeter, same area. And this was proven to be the only pair in
2018. So really recently - even though this stuff goes back thousands of years,
this has only just been shown to be this unique pair of triangles.
You can find these triangles without too much difficulty, but proving that they're the only
possible pair like this, it was quite difficult. So, I said there were no equilateral Hero triangles, it's just impossible. Erm, so
you've got
nice equilateral triangle, all got the same side, I'm afraid your area is always going to have some sort of roots,
it's always going to be some irrational number - horrible, but there are some near misses.
And so there are a few examples of this but I like this one in particular: 13, 14, 15.
So they're just either side of 14, if you take 14 as the middle number,
one below, one above and that - not an equilateral triangle.
Perimeter 42, and an area of
ooh,
84. Which also means it's one of our double
area on the list. Just misses being an equilateral. Also, there are examples of this that are bigger,
where you just, it's just the three consecutive numbers, and the bigger it is, the more it looks like an equilateral triangle as well.
We've got some superhero triangle merchandise like t-shirts and cups and things just because - well because it's good fun.
It's a good talking point, and also it's a way for you to support Numberphile.
Check out the links on screen and down in the description. Thanks for watching. We'll be back again soon
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