Geometry of Footballs and the Cube-shaped Ball
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- เผยแพร่เมื่อ 17 พ.ค. 2024
- Make your own cube football!
www.dropbox.com/s/idtkss3bped...
If you want the net for the hilarious "perimeter league" ball I made, here it is:
www.dropbox.com/s/561q7hiljt0...
This is the exact premier league ball that I bought:
www.amazon.co.uk/Nike-Pitch-P...
Here is the same ball on the US Amazon, but they don’t have that exact colour scheme:
www.amazon.com/dp/B00VRJSRJE/...
The icosahedral dice I show is a generic spin-down dice I happen to have on my desk. The dodecahedral dice is one of the Maths Gear "Go First" dice:
mathsgear.co.uk/products/go-f...
1966 football photo by Ben Sutherland, 2009.
Music by Howard Carter
Design by Simon Wright
MATT PARKER: Stand-up Mathematician
Website: standupmaths.com/
Maths book: makeanddo4D.com/
Nerdy maths toys: mathsgear.co.uk/ - บันเทิง
I'm an animator and have scanned thousands of pieces of papers, but I've never seen that technique of scanning a sphere. This is actually the most interesting thing I learnt from this information packed video.
+GuyWithAnAmazingHat As long as the contact point doesn't slip and you roll the ball straight, you get a good scan. Works a treat for cylinders.
+standupmaths Matt, you are really awesome :)
+standupmaths how often do you need to scan spheres/cylinders to discover this...?
Probably more often than you'd think.
That is the best thing I've learned this week.
Not related, but have you seen the 2010 World Cup ball? To me it seems to be a truncated tetrahedron (four triangle sides and four hexagon sides), although I have a hard time telling which grooves are actual stitches.
You bought footballs for geometrical purposes?
I hope you never change Matt xD
Well yeah. What else do you want to do with a football anyhow?
+Djorgal cut it up for the leather shapes?
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Girish Manjunath Indeed, and if you cut it correctly and reassemble it, you can Banarch-Tarsky it into two identical footballs.
thats banach.
I showed this video to one of my classes. The next day, a student came back with photos of the 2018-2019 ball, which is a tetrahedron!!
seems like it was a good lesson if it actally made one of the students expand on it
I believe the 2010 worldcup ball was a tetrahedron as well
Old football is identical to current volleyballs. Quite interesting that the volleyball design has remained the same, while the football has not.
That's exactly what I thought.
There is less money involved in volleyball than in football. They want an exclusive design for each major event. To prevent from illegal copies, they have to come with a different shape and not only a different pattern, so come the time of the event, the copy pirates will not have a pirated model ready and only official balls will be available for sale.
I was hoping he would go into possible reasons as to why they would change
actually they did change the volleyball design in 2008 now they used a weird 8 panel shape, the mikasa mva200
+Simon Schmidt Maybe that shows Volleyball is now becomming so diva-ish it thinks it has to design-for-design's-sake-change everything (like that Teen Titans' "Fütbol" sphere already mentioned ;) and those divas. Fütbol is SOOOOOOO big it needs constant redesign... volleyball is catching up. What oddities is Lawn Bowls gonna throw up or are they like the infamous 'lawn-shark'?... perfect, so unevolved for the last 160 million yrs?
"If anyone else out there has a ball they would like me to have a... if anyone else out there has a SPORTS ball" OMG I DIED
The people designing the champions league ball were like:
Someone: Isn't it annoying, that every time we design a new ball companies make cheap imitates by just mapping the visual pattern of our ball on a regular football?
Math-Guy: Yeah, let's just screw them over by using a geometry that will make their balls look even worse than they do anyway.
You clarified that you only want sports balls, but I am genuinely curious as to the geometry of the testicle. Apparently there are over 150 metres of tubing in each testicle, wrapped into some approximation of a spheroid. Aren't you also curious as to what kind of geometry the wrapping of that tubing forms?
I hadn't though about it, but actually that could be quite interesting. I suspect it will be something to do with how much tubing can be fit into a given space, similar to en.wikipedia.org/wiki/Space-filling_curve
+LeoMRogers both questions (he original from +codebeard and yours) are just as interesting as disgusting xD
You want him to buy testicles ?
Testicles are sport balls, bed sports.
+Blue Bill Sports balls get kicked. Don't kick those balls please.
Can’t we just agree that the pyramid shaped football is the best?
It definetly looks like it hit the peak
No, I want a plane shaped ball. (Or whatever it is called what a tennis ball is) It's the best next to the point shaped ball (aka blatter).
Damn, now I made myself wonder if there is an equivalent to a line shape 🤔
@@theaxer3751 technically, tennis balls are dihedral in shape, as there are only two funkily-shaped pads of fabric on either side. If you were to make a flat approximation of its shape, it'd be akin to a coin.
It's really nice to see Matt handling balls so well and speaking so enthusiastically about them.
What's the mathemagical term for the shape of a nut?
Dr. Duck: gaming the slow way
Legume.
I'm funny.
Night Angel
Not in any botanical sense you're not.
(Just being a biology pedant, no hard feelings).
Can we call it the Parker Cube?
That's one example of a parker cube.
Omfg yes! xD
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I wanted you to mention the megaminx on your desk
I was hoping so much that that would be the example for the dodecahedron
Did you click on the video because of the megaminx?
Yes, but I would've watched it anyway bc I really like this channel
+BrodytheCuber ya it's pretty good
Yay! Megaminx!
I like how he specified the type of ball that he was talking about in the end because of the internet.
MY MIND IS BLOWN. I never knew that footballs could be so interesting.
Same. I thought they were boring spheres.
Did you see that ludicrous display last night?
What was Wenger thinking sending Walcott on that early?
The thing about Arsenal is they always try and walk it in.
Thing about Arsenal is they're always tryin' to walk it in.
That is true. See you later.
You're saying football things in a football voice! How do you know about football things!?
The fact that the historical anecdote of buckyball wasn't mentioned in this video, deeply hurts the chemist inside of me.
Make your own video you lazy fuck
I think about the shapes of footballs sometimes, too. I remember a few years ago at school I saw a football that was like a rhombic dodecahedron! (Except I live in america so the only "footballs" I saw were egg shaped, it was a "Soccer ball" that had a rhombic dodecahedron pattern)
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.
Matt, you're totally the kind of mathematician I want to become. :)
I have a question though: what has your mathematical career been like?
What do you do research on and so?
I can't believe that a football is just a spherical truncated gyro-elongated pentagonal bipyramid
If footballs weren't so expensive.
A nice arts and crafts project would be deflating them, unstiching them/cutting out the individual shapes.
And then gluing them to the respective polyhedron they are based on, so it has nice straight edges.
Aside from the cube, constructing those polyhedra out of wood, is probably not as trivial tho xD
So even under inflation pressure, it seems like the cube-based tesselation would be subtly less spherical than the dedecohetral one. Would make an interesting followup video to explore if this changes the way pro players can use them in a match.
The puzzles on his desk are:
Megaminx
Axis cube
Mirror Cube
Soduko Cube.
I have all but the Soduko one.
+Brandon Boyer It's not a sudoku cube. :]
Oh. Is it just numbers then? Or even dates?
I think I know what puzzle it is. If I am not mistaken, when solved the six faces show the triangle series, prime numbers, digits of pi, the Fibonacci series, digits of e, and a magic square. Although, I don't know what its called.
This is all I can find. gizmodo.com/ugh-who-invited-math-to-the-rubiks-cube-party-1473657020
Sounds like it but it's blue.
+standupmaths I'm am surprised you didnt use the megaminx to explain the dodecahedron after all that is its form lel
The white and black soccor model looks like a cartoon in the coolest way. Those edges are CRISP. I also want to say how much I love the models you make for your videos, they look so nice and really do help my understand that idea you're trying to explain.
a nerd in a sports shop to get nerdy about sports equipment...
I LOVE IT!!!
First time I realised there were weird footballs, was the world cup 2006 in Germany. And I think it's still one of the more intersting designs I've seen.
I agree! Every one else has seemed to have forgotten that awesome design.
Exactly my thoughts
^^ was gonna say
That one also had cube-octahedral symmetry, in principle. The thing is it used two different types of patches as "sides". Non official manufacturers also printed that design on top of standard truncated icosahedron footballs, because if you examine a trucated icosahedron closely, you can find three opposing pairs of hexagons, where the long 8-shaped patch of the official ball would fit.
Yes, wanted to say this as well. But I wonder, if there are any tetrahedral balls out there...
9:30 starts asking for viewer submission then realizes phrasing 4 secs in.
A few months obsession with modular origami has payed off and helped understand this video, which, in turn has suggested that learning more about polyhedra might help with the origami projects. If only I could have predicted this at scool.
I'm not only impressed you found someone in the UK that remembers that england won the world cup once in the past, but they remembered the year.
9:31 :D I certainly... :D 9:35 oh :(
When he said under his breathe "I've got to get a laser cutter" I thought he was going to say "I've got to get laid" 😂😂😂😂😂
Up to this date my favorite football is the 2006 Germany World Cup ball. Both the design and the feeling of that ball was master class.
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
Matt, it seems to me that that "cube-shaped" ball does not have cubic symmetry! It looks more like it has pyritohedral symmetry, as the three axes are all rotated differently.
just putting this here for further developments
exactly my thought
They are 6 sides assembled exactly like a cube, I've build digital versions of this layout and have studied them a lot, they're exactly symmetrical like Matt says.
i understood some of those words
it can have cubic and pyritohedral! as long as each "face" can be rotated onto the next face rather than reflected or arbitrarily placed, it fits both categories.
1:09 Magic the Gathering spindown counter :D
Came here to say that.
D&D Dice!!!!
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This one in particular is actually a mtg spindown counter and not a D&D dice (although, both can act as the other in a pinch). You can tell because the numbers are ordered.
I got yelled at for using a spin down as a d20 in d&d
You overlooked something interesting regarding the duality of the shapes, a classic adidas ball is just as much of a truncated dodecahedron as it is a truncated icosahedron, the only difference is that the truncated pieces intersect each other, this intersection explains the hexagons, when you truncate a dodecahedron, triangles appear, and when they intersect, you get cropped triangles, which are essentially hexagons.
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.
Fred
Link?
Is it extra nerdy of me to immediately recognise your d20 as a Magic: the Gathering spindown life counter rather than a fair and legal die?
I am also guilty of that.
The font for the numbers gave it right away hah
Yeah I noticed right away, and I hardly ever play
Bunny (∞) Yup I got that. Trying to remember what set that colour/texture was from
It's actually not a spindown since it's numbers are 'randomly' put onto it, so you can't just spin it down.
The Parker sphere
Tennis balls are pretty interesting, because the typical two-panel design, it could be argued, is essentially a flat structure in 3D (it only has one edge).
While the cube is a different base solid, it is closely related to the dodecahedron. Specifically, a dodecahedron can be formed as the compound of 5 cubes. You can see this easily if you join the non adjacent vertices with 5 different coloured lines. You can also "extract" a cube by truncating 6 non-adjacent edges of the dodecahedron, leaving a cube and 6 fastigium-like solids.
"I'm gonna get this dice". This die. Dice is plural.
willprogresivo no, dice can also be singular, check your facts.
damn u, i cant look at a football thinking its a sphere anymore
I love these channels.
Not in a thousand years would I have stopped to contemplate balls and think about them or listen to anybody analysing them. I mean not anywhere else.
But here it's natural... and it's interesting.
Having a laser cutter at your disposal is amazing, let me tell you. I joined my high school robotics team as an incoming freshman and now have pretty much unlimited access to a 3d printer, a laser cutter, and pretty much any tool i want. It's amazing. I printed some rouleaux tetrahedrons in halves and am working on figuring out support structures so i can make it all in one
So... does that make make a tennis ball a two sided shape?
I'm pretty sure tennis balls have only two panels right? Them what are they based on?Ps. Keep up the great work!!
Still a cube. Just the equivalent of two 1x3 rectangles forming after being folded into thirds.
each 'petal' of the tennis ball is made from three-sided piece. then if you fold those and connect them on borders you will get a cube (three pcs+three pcs=6 sides - a cube). if it is inflated the shape will be a sphere
You could just as easily make the same argument for any regular polyhedron. They can all be split into two identical halves joined together.
Ian Abbott No you can't (sort of). They always resemble a specific one. The tetrahedron is also close, but the hexahedron is closer.
In any case, you'd need to place some imaginary vertices and some imaginary edges for it to resemble any specific polyhedron, including the cube.
Matt: Now, this shape is normally called-
Me: BUCKMINSTERFULLERENE!
Matt: -a truncated icosahedron.
U can't have football without mathematics.
How amazing !
I wouldn't believe a video about football can be that interesting.
Amen to that
Hooray the British have gotten to you. You said Football most other Aussies say Soccer
I know right. It's infuriating
Matt is Australian ??? My whole world is tearing apart !
He doesn't have to talk about Aussie football anymore, so it was only a matter of time.
...although I doubt he cared about Aussie football to begin with
Yep he's Australian, but lives in the UK so his accent is a blend of the two.
Finally someone has managed to get me interested in something related to football. In general, I find it completely unisteresting, even though I live in one of the most football-enthusiastic countries in the world and lots of people have tried to get me to like it.
the editing at the end to make it seem like you throw it straight at the camera was amazing
This is what footballs would be in Minecraft.
@@metachirality ???
"Local english man is tricked into printing swastikas for maths"
Parker N@z!
I've always been fascinated by footballs and how they design them to make interesting shapes and patterns. Great video Matt
Cubes and octahedrons are also related in this way, and a tetrahedron is related to itself, meaning you can get a tetrahedron by joining the centers of the face of another tetrahedron, and cubes become octahedrons and octahedrons become cubes. Also, octahedrons are my favorite Platonic solid!
What about the Teamgeist and Jabulani? What geometry are they based on? I think the Jabulani might be based on an octahedron. Also, tennis balls and baseballs seem to be based on cubes, but have just two panels that are equivalent to three sides each. Oh, and oblong football variants (rugby, American football, etc.) share a similarity with beach balls in their panel structure in being polar.
Took a closer look; I think the Teamgeist is based on a truncated octahedron.
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Looks like Jabulani is based on a truncated tetrahedron.
+Proceed Anyway Oh, nice catch. That just led me to learning that a fully-truncated (rectified) tetrahedron is an octahedron (as well as learning the term rectification). Now we can learn about polyhedral duals and rectification from footballs!
Tennis balls don't actually have "panels"; the line is just drawn on top of a spherical rubber (typically built as two hemispheres glued together).
I want to know about Golf Balls
You should watch the Mythbusters show, where they showed how dimples can be applied to cars for good effect
If you look at a basketball, there are 3 great circles at 90 deg intersections. Choose one as the equator, and the other two are circles of longitude intersecting each other at the poles. Add two more circles of longitude equally spaced from the first two. Now the problem-- at the poles, there's lots of seams and little points. This will make the ball uneven. So the last two circles of longitude are modified: 1) Trim these circles at the poles to form semicircles. 2) Before the semicircles reach the poles, at about 75 degrees of latitude, curve the tops as follows-- the 45 deg line curves to the right, the 135 deg line curves to the left. Do the same to the 225 deg and 315 deg lines, respectively. Do this top and bottom, always holding the ball with North facing up. The curved lines should meet each other at 90 deg angles to the line they are crossing.
So what's the geometry of a basketball? It has an equator, and, ignoring the fudging at the poles, eight spherical triangles that meet at the poles. I propose that it's a Bipyramid based on an octagonal equator. en.wikipedia.org/wiki/Bipyramid This has geometric similarities to an Octahedron where just the equator is divided to degree-2.
Golf balls are interesting. Their texture helps them go farther than smooth balls
Have you seen the "Teamgeist" ball of 2006 world cup?
+Agustin Blanco (AugustB) A few people have sent me photos now. It's odd.
Isn't it an octahedron with sliced off edges just like the icosahedron?
Why take the tiny tiny dice when you had a megaminx on your left?
The "cheat versions" of Teamgeist look like traditional truncated icosahedrons, suggesting it's just a variation of that concept.
It is a cube-based ball made by a schizophrenic designer, I think
I'm a bit disappointed that Matt has little interest in at looking at my non-sports balls...
Might have just helped a challenge I have with a project. I need to generate in Unreal Engine a sphere using polygons. What I was lacking is the math behind it, how to tell the code where to generate each pentagon/hexagon. Not only that, but also have it flexible in size which means I got to add more pentagons/hexagons where needed - to the 10,000's if needed.
This video gave me an idea on how to do it.
On the Nike ball, if you somehow erased the lines dividing the trapezoids that are connected by their long edge, you get the familiar pentagons and hexagons shape.
Your unsolved rubiks cube variants are triggering me
What about tennisballs, aren't they basically coins?
@@metachirality What are they related to?
@@metachirality I don't speak Greek
Fundamentally the football geometry hasn't changed. Icosahedron is cubic:
On each face of a cube draw one line segment. It has to be parallel to sides, centred in the face, approximately 3/5 long. For each pair of neighbouring faces, on line has to be parallel to the edge they share and the other one perpendicular.
The ends of these line segments are vertices of an icosahedron. For each pair of neighbouring faces you take the nearer end of the segment perpendicular to the edge and connect it to each of ends of the segment parallel to the edge.
Just did some calculation, the exact length of those line segments has to be inverted golden ratio.
I think the reason for the new design is that you have 1 type of piece not 2 as with the truncated icosahedron, and only 6 pieces. These pieces also tile nicely into a plane unlike a pentagon, so in total its way easier to manufacture.
What's with the numbers on that D12?
Kyle Marie thats not a d12
@@supercool1312
At 2:37. Yes, it is.
At least, it is a die with 12 sides. I can't explain the numbering.
What about baseballs? Aren't they made up of only two parts?
Tennis ball
Still a cube, but three of the sides are joined.
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Yes, if you take their number of panels alone into account they map to spherical dihedrons (a regular shape that can only exist in 3D if the sides are allowed to bow out). Though, as others have pointed out, they can be mapped to a cube by taking each panel as a set of 3 linked squares or to a tetrahedron if you take each panel as a set of 2 linked triangles. Or any two halves of a regular polyhedron.
Your office looks like a hybrid between a toy store and a storage room in a role-playing game shop ...
You can get footballs made from six double pentagons. So based on the dodecahedron but with a link back to the cube. The seamless edges of the double pentagons are at the faces of an imaginary cube, in a similar orientation to the panels from the 1966 cube derived ball.
Oh. A spin-down d20. Play some MTG?
MrBrainFog I was thinking about the same thing.
Why does your 12-sided die have a 47 on it?
1959Edsel thats a d20 is it not?
Seeing you create the cube out of those squiggly crosses was so satisfying!
your office is full of enough dice for the entire UK Parliament to play D&D
Golfballs! Different brands/models have different shaped dimples (callaway vs titleist) as well as different numbers (usually around 400 I think). Some even have dimples inside of dimples
To make the list of platonic solids used to make sports balls more complete, rugby ball is loosely based on octahedron.
i have a non-official euro 2016 ball that is made up of a panel pattern that i've never seen before. It's two different kinds of octagonal panels, one being a regular octagon and the other being the outer border of the shape made by putting two pentagons together. Really interesting design. I'm pretty sure that the overall shape is a truncated cube.
I vaguely recollect that a broken tennis ball from my childhood, made of two separate pieces, it was like two indented rounded rectangles of material. I'm curious to know which is more spherical, a ping pong ball or the rubber 'bouncy' ball. Also there are also the snow balls everyone throws during winter, but throwing those isn't very sporting. Thanks for another great video.
I'm pretty sure I'm not the only one who loves how the cube looks with these patterns on each face
Everything about the closing of this video was perfect.
Wow nice megaminx.
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.
Quite coincidentally after viewing this video, I was introduced to the Kelvin problem of packing and subsequently the Kelvin shape and the Weaire-Phelan structure. Any chances of seeing those shapes unpacked and a construction of them? If you want a sport tie-in (as you seem to be so fond of sport), the aquatic center for the Beijing Olympics seems to have been modeled on this!
+NSNick I actually visited the aquatic center in Beijing to see it! I should do a video about it. It's in my book so I've already done all the research.
I look forward to the video, whenever it may happen!
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Didn't know soccer balls had so much geometry!
The 1966 football has a symmetry like most volleyballs. Because each square is made up of three strips, the squares do not have four-fold symmetry, only two-fold symmetry. This is called pyritohedral symmetry, a type of tetrahedral symmetry.
You should look at the 2006, 2010, and 2014 World Cup balls if this is going to be a thing.
So Matt I'm happy to see you becoming a sports enthusiast. Have a go at tennis balls and baseballs please. My guess is that the base shape is a tetrahedron. Then put a circle around each vertex and connect two circles with a small piece to fill the gap. That gives you two peanut shaped pieces which are then stitched or molded together.
The way they projected the dodecahedron print onto the icosahedron is interesting. Mathematically it probably makes more sense, but design wise I would have thought it was neater and easier to just draw the internal pentagon shape onto the pentagon on the truncated icosahedron and transect the surrounding hexagons on parallel to the sides of the pentagon. Essentially adding a break on the dodecahedron that appears if you fully truncated the icosahedron. That would look a lot more like the newer ball shape in my mind, and now I think about it it's another mathematical equivalency.
The divots on a golf ball would be a good one, it seems they're truncated corners which are then pushed in.
Footballs I played with when I was a kid mostly had cubic symmetry, featuring 12 rectangles of one distortion and 6 of another. For a while in the 90s that was a more common shape in sports shops than designs with icosahedral symmetry.
Actually now I look, it seems this design originated in volleyball.
Cubes and dodecahedra are both special cases of the pyritohedron. The third special case is the rhombic dodecahedron, which has cubic/octahedral symmetry.
What's also interesting are balls based on "illegal polygons" like the U.S. football that's made from four "digons" with four edges and two vertices.
That cube looks really really really cool.
Wow. Great stuff, Matt. Would've never noticed any of this if not for you. Thanks!
I'd love to see you mathitize a tennis/baseball and/or a basketball. A two panel ball(s) and an 8 panel ball.
You can make a dodecahedron from a cube if you put a gable roof shape on top of each square and turn them so each triangular part of the roof shape blends into the trapezoid part of the next roof shape forming a pentagon. Also a cube truncated becomes an octohedron. I believe all the regular polyhedrons can be some how morphed into each other.
such dedication, would love to hear about other balls, the 2006 germany world cup ball was beautiful, and the jabulani's design was pretty weird as well
I don't know if you'll see this comment since this video is pretty old now, but I just wanted to say you should check out the Adidas +Teamgeist ball from the 2006 World Cup, it was the first competition football to break the traditional 32 panel design that had stood since the 1970's, although football manufacturer Mitre never actually stopped making cube based ball, the Premier League used Mitre balls in the 90's, and they were still using essentially the same shape as the 1960's balls!