Every spherical football is a branched cover of the standard one.
August 17, 2006 7:31 AM Subscribe
Bending a soccer ball - mathematically. Found via Ivars Peterson's short exposition on Braungardt and Kotschick's The Classification of Football Patterns [pdf, technical].
How is the soccer ball "torus" folded into two concentric spheres without tearing its surface? From topology class, I thought that was "against the rules" — like ripping a knot to unwrap it, and calling the two equivalent. The animation seems to fold the one side of the "ball" across the other.
posted by Blazecock Pileon at 7:39 AM on August 17, 2006
posted by Blazecock Pileon at 7:39 AM on August 17, 2006
I'm with Blazecock; I don't get this. Isn't this the whole Poincare (sp.) Conjecture?
posted by inigo2 at 7:41 AM on August 17, 2006
posted by inigo2 at 7:41 AM on August 17, 2006
Blazecock: They probably pop part through the fourth dimension, like the intersection in a Klein bottle.
posted by edd at 8:03 AM on August 17, 2006
posted by edd at 8:03 AM on August 17, 2006
(although there must be some more to it as a torus is clearly one surface and two concentric spheres are two... so I don't quite get what they're doing. The surface-crossings should be legitimate though)
posted by edd at 8:06 AM on August 17, 2006
posted by edd at 8:06 AM on August 17, 2006
Edd: The page says the torus is embedded in ℜ3.
posted by Blazecock Pileon at 8:08 AM on August 17, 2006
posted by Blazecock Pileon at 8:08 AM on August 17, 2006
Will you philomaths please stop kicking my ass? Does this soccerball stuff help you keep your underwear from falling down?
posted by gorgor_balabala at 8:10 AM on August 17, 2006
posted by gorgor_balabala at 8:10 AM on August 17, 2006
Yeah - the elastic in your boxers is in the form of a torus embedded in ℜ3.
posted by edd at 8:12 AM on August 17, 2006
posted by edd at 8:12 AM on August 17, 2006
How is the soccer ball "torus" folded into two concentric spheres...
Some of the most important mappings to study in topology "break the rules" in the sense that they glue or tear things. Covering maps, for instance - which is the sort of thing we're looking at in the article - are fundamentally important and they're interesting because they glue an object together in interesting ways to make a new (inequivalent to the original) object. Also, homotopy - which is also in play here - is a sort of "continuous deformation" which may seem to "break the rules" of topology, but is nonetheless an essential part of the subject. It's a weaker relationship, but it's one that works very well with algebra, more so than the rigid relationship of homeomorphism.
In the torus-to-two-soccer balls transformation the final two spheres aren't disjoint; they share a few points of contact. That makes all the difference in the world to the homotopy type of the result. The torus and the end result are not homeomorphic (perfect, bijective, bicontinuous equivalence) as you can see from the fact that the end result isn't manifold at the joining points. But just imagine taking a torus, and allowing two of the meridian circles to shrink (as if we were tightening two belts around the torus). What you end up with in that case is clearly (right?) two spheres joined at two points, although they aren't concentric. It's a continuous process, but doesn't end up defining a homeomorphism.
Isn't this the whole Poincare Conjecture?
The Poincare conjecture doesn't have a lot to with this directly, since the question is about the combinatorics of patterns drawn on the surface, whereas Poincare's problem is about recognition of the sphere among other objects, based solely on algebraic information. (And the Poincare fact is not a difficult fact at all when the objects are 2-dimensional surfaces sitting in regular 3-d space as we have here). But the algebraic topology of the objects certainly plays a role in determining how many different patterns are possible - roughly, the trivial algebraic structure of the sphere makes their classification of patterns result (everything is just a branched cover of one pattern) possible, whereas they point out that topologically more complicated surfaces do not have that property.
posted by Wolfdog at 8:12 AM on August 17, 2006 [1 favorite]
Some of the most important mappings to study in topology "break the rules" in the sense that they glue or tear things. Covering maps, for instance - which is the sort of thing we're looking at in the article - are fundamentally important and they're interesting because they glue an object together in interesting ways to make a new (inequivalent to the original) object. Also, homotopy - which is also in play here - is a sort of "continuous deformation" which may seem to "break the rules" of topology, but is nonetheless an essential part of the subject. It's a weaker relationship, but it's one that works very well with algebra, more so than the rigid relationship of homeomorphism.
In the torus-to-two-soccer balls transformation the final two spheres aren't disjoint; they share a few points of contact. That makes all the difference in the world to the homotopy type of the result. The torus and the end result are not homeomorphic (perfect, bijective, bicontinuous equivalence) as you can see from the fact that the end result isn't manifold at the joining points. But just imagine taking a torus, and allowing two of the meridian circles to shrink (as if we were tightening two belts around the torus). What you end up with in that case is clearly (right?) two spheres joined at two points, although they aren't concentric. It's a continuous process, but doesn't end up defining a homeomorphism.
Isn't this the whole Poincare Conjecture?
The Poincare conjecture doesn't have a lot to with this directly, since the question is about the combinatorics of patterns drawn on the surface, whereas Poincare's problem is about recognition of the sphere among other objects, based solely on algebraic information. (And the Poincare fact is not a difficult fact at all when the objects are 2-dimensional surfaces sitting in regular 3-d space as we have here). But the algebraic topology of the objects certainly plays a role in determining how many different patterns are possible - roughly, the trivial algebraic structure of the sphere makes their classification of patterns result (everything is just a branched cover of one pattern) possible, whereas they point out that topologically more complicated surfaces do not have that property.
posted by Wolfdog at 8:12 AM on August 17, 2006 [1 favorite]
If you'll read it closer, you'll notice that the 'concentric' balls are joined at four points.
on preview: yes.
posted by sonofsamiam at 8:13 AM on August 17, 2006
on preview: yes.
posted by sonofsamiam at 8:13 AM on August 17, 2006
My roommate, a math major, tried to explain this to me last year when he took a topology class. I still don't get it.
posted by whoshotwho at 8:39 AM on August 17, 2006
posted by whoshotwho at 8:39 AM on August 17, 2006
Thanks, Wolfdog.
posted by Blazecock Pileon at 8:43 AM on August 17, 2006
posted by Blazecock Pileon at 8:43 AM on August 17, 2006
I KNEW IT!!!
posted by gorgor_balabala at 8:46 AM on August 17, 2006
posted by gorgor_balabala at 8:46 AM on August 17, 2006
To be honest, I found the technical challenges in making the animations more interesting to think about than the original mathematical problem about classifying patterns, though it is a pretty neat result. I'll probably get my pure mathematics card revoked for admitting that.
posted by Wolfdog at 8:48 AM on August 17, 2006
posted by Wolfdog at 8:48 AM on August 17, 2006
Why has it taken so long for someone to come up with the obvious Beckham reference? I mean, come on, just because the guy has resigned the captaincy and was left out of the squad, it doesn't mean he has vanished completely, does it?
posted by Sk4n at 9:25 AM on August 17, 2006
posted by Sk4n at 9:25 AM on August 17, 2006
So Beckham does all those calculations before he kicks it? I'm impressed.
posted by Smedleyman at 5:17 PM on August 17, 2006
posted by Smedleyman at 5:17 PM on August 17, 2006
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Kotschick shows that it's possible to create new soccer balls by using a mathematical construction called a branched covering.
from Peterson's article could be stated more strongly: the gist of the article is that branched coverings of the standard pattern are the only way to create new soccer ball patterns, if we use only pentagons and hexagons with pentagons and hexagons occurring alternately around each hexagon.
posted by Wolfdog at 7:31 AM on August 17, 2006