We’re here in my office because today I would like to do some arts and craft. You see while everyone has been watching the Euro 2016 football competition games, because they want to see how team goes or to watch the sport, I’ve been watching it, because of the shape of the football. You see, in this football competition they changed the shape; this ball is actually based on a cube. So what shape is a football normally? Well obviously it’s roughly spherical, but when you have a closer look at a football, this is the classic white and black Adidas football, you can see to make it a sphere it needs to be assembled from smaller shapes. In this case, is made of 12 pentagons and 20 hexagons. Now, this shape is normally called a truncated icosahedron and to show you why, I’ve made a model where on the front it looks a lot like the standard issue football, but if you spin it around you can see it becomes an icosahedron. So an icosahedron is one of the platonic solids, it’s made of 20 identical equilateral triangles. I’ve got a dice over here somewhere, this one will do. That there is a icosahedron, and so the way you go from an icosahedron, to a truncated icosahedron, is you just slice off some of the corners, and you expose these pentagon shapes. I’ve actually made one of those sliced off bits, you can see if I put that back on there, there’s our original triangles that make up the icosahedron. We slice that off, it exposes a pentagon and you keep slicing you end up with the hexagons and pentagons that are so familiar. So there is your there is your stereotypical truncated icosahedron football, and I thought that is what all footballs had continued to be, until a few months ago I got on the tube in London, and on the platform I spotted an ad for something football related, and the ball, was not one of these. So as soon as I got to street level, I did something a little out of the ordinary; I found a sporting goods store, and I wandered around obviously blending in seamlessly. And sure enough I found some footballs, and even though, they look like they’re made of pentagons, and trapeziums, or trapeziods those aren’t actually separate panels; They’re just grooves within one of the panels. In fact, it’s only made from 12 pentagons. The current premier league football is a dodecahedron. Ok, dodecahedron – right, so another dice, this dice is a dodecahedron, and that’s the shape of the current premier league ball. So to make a model, I’ve actually cut out twelve pentagons. On each one, I’ve drawn lines for where the grooves are, and if I stick them all together, it looks, like this! So, here I’ve actually made the dodecahedral shaped premier league football, and it’s a lovely model, it’s exactly the same shape, but obviously it’s not very good for playing sports. I did actually buy one of the new premier league footballs, I’ve got it here. But what’s great is I didn’t buy one of the dodecahedral ones; I bought a cheating one, where they’ve just drawn the new pattern, onto the old panels. I wish I could have been in the football manufacturer’s meeting where they’re like, “oh, they’ve changed the premier league ball again, it’s so expensive to setup new panels and new manufacturing process,” and then one person from the corner pipes up and says “well actually, we can cheat; we can still use the same 12 pentagons and 20 hexagons, but on top of them we can line up and just print on the pattern of the new dodecahedral ball.” And sure enough, if you have close look, you can see this ball has the panels of the old ball, but the pattern of the new. I don’t know if whoever suggested this was a mathematician, but they were taking advantage of a fantastic maths principle called jewel shapes. So an icosahedron and a dodecahedron are actually jewels of each other. If you join together all the centers of the faces of a dodecahedron, link them up, you get an icosahedron, and vice versa; if you link up all the centers of the faces of an icosahedron, you get a dodecahedron. You can think of it as them being kind of underneath the skin they’re kind of the same shape; they’ve got the same kind of logical structure behind them, that just manifests different ways. So you can take the pattern from any dodecahedron and map it onto an old icosahedron. I love the fact that 3D geometry has been used by football manufacturers to save money. That’s why it’s actually taken me so long to make this video because when I noticed this with the new football, I went “oh that’s interesting and I like the symmetries and it’s lovely,” but at the end of the day it’s still a dodecahedral icosahedral thing, it’s not a dramatic change. Even later on when I was crossing a train station carpark in Bristol, and I saw this crazy monstrosity – what is that? That is the new champions league football, and so of course I went and bought one of those – here we are, and it’s pretty exciting; I love the funky stars, but again it’s the same story. If you link together the center of all the stars, you end up with an icosahedron, if you link together the center of all these six-sided shapes, you end up with a dodecahedron. So all of these are all just variations on the same family of shapes. It was seeing the Euro 2016 ball which finally prompted me to make this video. So what shape is this ball? Well, if you have a close look at the bits of material it’s made of, they’re all cross shaped, and there’re six of them. And that’s what I mean by cube shape. So I’m going to try and build a model out of cardboard, to get the exact shape of these crosses I’m actually going to roll the ball across a scanner so I can get the exact outline, and then I’m going to print those six panels out onto pieces of cardboard, of course, in a variety of different colors. [staccato printing] [printing accelerates] [scissors cutting paper] I have got to get a laser cutter. So each one of these now matches one of the panels on the ball, these are slightly smaller than the originals because I had to cut them out of a piece of card. But you can see on the ball, that they meet three to a point. Those are the corners of the cube – you can actually find all eight of them on the ball. So I can actually fold these to give me those original edges, and my cardboard model will be a lot more cube shaped. [folding paper] [peeling tape] Ok, I’ve now joined them together into the net of a cube, I’ve simply got to curl it all up, and stick it together. Ok, so that, is the complete cube made from those six pieces. So you can see, each corner has got three different pieces meeting, just like the football does there. And there’s the six different pieces making the six faces, just like the football. Obviously the football is not literally a cube, the pieces it’s made from are just based on the structure of a cube. This is what you would get if you took this and inflated it out to form a sphere. The Euro 2016 ball is fundamentally different to other footballs, because it has a cube structure to its panels, whereas other footballs, are either icosahedron or dodecahedron which is really the same thing. Although this should not come as such a surprise because even though this is the ubiquitous football that we all know and love now, it wasn’t always the case: this came around pretty much around the 1970 world cup was when this first became super famous. If you look at photographs from when England won the 1966 world cup – oh, that’s a little known fact that very few people in Enlgand bring up because frankly, it was half a century ago. But, if you look at those photographs, the ball they’re using is cube based; it’s got six different sections each made up of four panels. And so footballs used to be cube based – or rather, it used to be bladder based, but let’s ignore that. They used to be cube based, then they went icosahedral/dodecahedron on us, and now, they’ve gone back to cubes. The moral of the story is you can’t have footballs, without geometry. And I love that, I get so excited about different types of polyhedron, other people get very excited about football. But this is the intersection of the two; without the mathematics of these different 3D shapes, we wouldn’t have our modern football technology. A final word on balls: so obviously I have a growing collection of sport balls I have bought for geometric purposes. But I can’t keep doing that, because apparently this is getting ridiculous, and, “is that another thing for Youtube? I thought you had a real job.” So, I’ve had to borrow some, the Euro 2016 ball I actually borrowed off a friend of mine, or rather he stole it off his kids and then lent it to me. So, Adam, Jenna and Josie, thank you so much for letting your dad steal your football so I could use it. If anyone else out there has a ball they would like me to have a – If anyone else out there has a sports ball, specifically, they would like me to have a closer look at mathematically, then please do get in touch, I would love to see the geometry behind other sports balls, particularly across different sports.

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Dennis Veasley

100 thoughts on “Geometry of Footballs and the Cube-shaped Ball”

  1. Russia 18 ball is different, but it kinda looks cube-based too: you can see the "6 faces, 8 vertices joining 3 faces each" structure there, too.

  2. But because of the duality ({5,3} to {3,5}), you can make that truncated icosahedron shape by starting with a dodecahedron and slicing each of its corners down enough.
    Those cuts will need to go deeper than the ones where you start with an icosahedron, but it still can be done.

    PS: That 1966 World Cup ball looks (pattern-wise) just like a present-day volleyball.


  3. I think that the champions league hexagons are the outline of a cube seen from an adge and with the FOV increased a bit

    Try it out in Minecraft, I'm pretty damn sure about it

  4. +standupmaths So … the previous football (base) shape is the same as the current volleyball (base) shape? I noticed that the official shape changed shortly after Volleyball went Olympic, I'm wondering if that's any correlation.

  5. I'm going to tell you this: When you flatten the dodecahedron into a sphere, you can remove the outer lines and turn it into something like an icosahedron, because there are hexagons (made of trapezoids), except three hexagons meet at one point. In fact, you can convert one to the other using a virus model

  6. Have you looked at the geometry of an American football? Most of them are made from four panels (plus the laces, but they don't contribute to the overall geometry really).

  7. As much as I hate sports, I kinda wanna see the geometry of an American football. It’s such an odd shape. I’d love to learn how the balls are made and maybe a little bit of history behind it.

  8. So would you say that the Euro 2016 football is a snark at the mathematical saying "squaring a circle"?

    And can you make triangle based footballs?

  9. Wait… If an octahedron is 8 equilateral triangles, attached 4 to a vertex, and a cube is 6 squares, attached 3 to a vertex, does that mean they're both dual shapes of eachother?

  10. the 'black and white' is the second design — the first was similar to a volleyball look…the reason for the black and white design was to show the ball rotation/movement on black and white TVs

  11. As a followup though it is not strictly math. How do the different geometries effect the spring constants of the balls?

  12. Tennis balls have two panels, I should think it ought to make a coin – a 2D round thing… not at all sure yours will end up being round.

  13. I love that the Euro 2016 ball is also a good example of creating a repeating pattern by displacing edges – like MC Escher did with hexagons to make that classic Gecko pattern.

  14. The only reason I can think of for having so many dice is so you can roll each one say 1,000 times and check how evenly weighted they are based on the statistical distribution of numbers rolled vs what would be expected mathematically. I know, I've done it myself with far fewer dice and fewer rolls.

  15. "If anybody has any balls-…. if anybody has any SPORTS BALLS they would like me to take a close look at" 😂😂😂😂 big mood

  16. I like "Bola 8", an octahedron by Penalty.

  17. I suppose someone will have pointed this out to you by now, but in case no one has: "dice" is plural. The singular is "die".

  18. Matt: what shape is a football normally?
    Me: well, its got two pointy ends, som–
    * Matt pulls out the soccer ball for Americans *
    Me: Oh, that football; the one the rest of the world besides America uses… Right…

  19. The Jabulani soccer ball from the South Africa World cup has a different structure; it uses 8 panels

  20. Wow! I’ve never actually seen those kinds of balls before. If anyone knows buy links to the dodecahedral balls, could you tell me? I’d love one of those for collection.

  21. Do you think you could show us the sculpture in the back of the room, the pointy paper looking thing

  22. The 2006 world cup ball (adidas Teamgeist) is a interesting truncated octahedron – so playing on the cube/octahedron pairing.

  23. There are others balls based on 'pyramid' shape, with four identical curved pieces that joint like a curved faces pyramid, makeing a sphere.

  24. Wanting to make a wooden fruit bowl using triangles. But I am having a little problem with ankles as each triangle will be a equilateral triangle on its outer faceand I want the wall to be 30 millimetres thick and the overall bowl to have a radius of 200 mm.could somebody help me with my problem thank you

  25. Looking for a dodecahedron, doesn't reach for the megaminx puzzle on his desk, no, reasonably so he looks for a tiny dodecahedral die. Pro gamer move.

  26. A fun game to play if you're in America: call these things "metric footballs" and see who figures out what you're talking about

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