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. Want even more hexagons? Well go get them at Brilliant, today's episode sponsor. Brilliant's full of courses, quizzes and problems to sharpen your brain; to lift your game. Everything's interactive as you can see here, and designed with loving care. It's made to make you smarter but also put a smile on your face. Whether you're a beginner or a veteran; a school student or a lifelong learner Brilliant has more than enough stuff to keep you busy. Already a subscriber? Then why not give Brilliant as a gift? By the way, get 20% off a premium all-inclusive subscription by going to brilliant.org/Numberphile Details and a link can be found down in the video description.
