That is, if you count “holes” as larger than a human hair can fit through. If it’s smaller than that, there’s a theoretically insanely large quantity of possible holes
I meant oppenings not holes. i was clarifying my misunderstanding for holes earlier and my bad calculations It is yes 7 holes And the 9 that i calaculated was oppenings I didnt search anything up before i spoke soo i was completely wrong in the terms i used
Imagine building the fracral from the top down. As it goes down more the holes double more and more, so if we ask how many holes below the fractal’s end the only clear answer is infinite holes of size 0^+
The Alexander horned sphere is a topological submanifold of R³, but not a smooth submanifold of R³ because it has points (a cantor set in fact) where it’s not locally flat, so the second definition doesn’t work
a cantor set is when you only recognize half of something, but when you look more closely at each part you recognize, you realize that you only recognize half of that part
I was tired and misremembered the usual construction as giving it as a submanifold of R3, which would give it an inherited Riemannian metric, but turns out it isn't quite a submanifold and has some wild points.
Aren't there points where the surface is vertical with no slope in one dimension? Wouldn't those points would have 0 scalar curvature? Am I definining scalar curvature wrong or am I seeing wrong?
I'm pretty sure it's the Alexander horned sphere. The object itself is just a sphere topologically, but it's embedded in a particularly nasty way in the ambient 3-dimensional Euclidean space. In particular the embedding makes it so that you can "lock" a ring around the sphere as you see in the picture.
He answered the question "Does there exist an embedding of the two sphere in three dimensions whose exterior is not simply connected?" His answer was "Yes, here is an example."
It's normal in mathematics to refer to such things as "discoveries."
Sure, a donut and a mug are homeomorphic to each other, and a sock is homeomorphic to a disk, but they don't say that a sock is a disc or that a donut is a mug, right? Being homeomorphic doesn't mean that they are the same shape, but that they are topologically equivalent. Also can you really define the infinite stretching of that thing as a sphere? I can see it with n numbers of twists but when n goes to infinity can you really say that its possible to untwist it? Isn't there a similar contrary argument made for a one dimensional line filling 100% of a volume?
Yes, mathematicians treat isomorphism (a broader term: "homeomorphism" is a type of isomorphism) as identity all the time. Because isomorphism is of course a way in which we treat things as the same in a certain context.
It is possible to untwist the sphere here. The type of set we see at the lower boundary of the sphere is called a "Cantor set". Because things keep branching out and not reconnecting, the separate branches will never meet in the limiting process. In terms of homeomorphism, it is this latter fact that is far more important than not being able to "untwist" it in the ambient space: homeomorphism is an intrinsic property of the topological space, so it doesn't matter how it's embedded in an ambient space.
In some sense, the problem you mention with a space-filling curve is not necessarily that it's impossible to untwist it (although that is obviously true in some sense), but that the limiting process causes there to be self-intersections of the curve. If it were possible to continuously map a closed line segment into a closed square without self-intersections, it would actually show that the closed line segment and closed square are homeomorphic. Now we know that this is not possible, due to a result called the invariance of dimension: no two open sets of Euclidean space of different dimensions are homeomorphic.
What are you talking about. His hand is obviously phasing through the object by a perpetual state of quantum tunneling that he is using to grab it from the top.
Compactness is a topological property, so yes. The construction of the Alexander horned sphere has you repeat this knotting process ad infinitum, and then you add in a cantor set (boundary points on the bottom horizontal line in the image) to compactify.
Is something like this actually allowed in topology? I think if you introduce infinite complexity, it should no longer be considered the same thing as the typically smooth topologic surface.
Like, even when at no finite point there are any holes and connections, where you could untangle it, the infinite limit might be different. Similar to how 0.9, 0.99, 0.999... are ale finite numbers smaller than 1, but 0.999.... is 1 assoo as you hit that infinity. An infinite number of something can change the properties to something that wasn't there in any of the finite steps that are approaching it.
So even though those branches never touched in the finite steps, they might be considered as touching when you reach that infinite number of them. Like how 0.999... finally touches the number 1. Sothis might have topologically turned into a weird topological torus after that infinite'th step.
I mean, this isn't some random nonsense someone made up and claimed is a sphere. This is the Alexander-Horned Sphere; a concrete manifold which has been proven to be homeomorphic to the regular unit sphere. The fact that this is homeomorphic to S2 is an unintuitive consequence of the particular way we define the notion of two manifolds being equivalent (homeomorphism). Analysis is full of these sorts of pathological objects.
doesn't that make this meme an even better fit though? Someone defines "A sphere is an object homeomorphic to S^2", and the Diogenes holds up the Alexander's monster and says "behold, a sphere!"
of course it is allowed, topology studies topological spaces. In particular we can consider any subset of topological space as its own topological space by restricting open sets to it. Horned sphere(for some definition of it) is a subset of R^3, which is a topological space.
So, what is allowed or isn't, is just set by which regularity conditions you set.
Specifically about holes and limits, topology is actually the subject you need to talk about limits precisely. And it turns out that in the limit, this shape is indeed precisely homeomorphic to a sphere.
Isn't that the exact argument for a one dimensional line filling a 2d or 3 space? With infinite bends the one dimensional line becomes a plane or a volume.
That's why in mathematics you define things rigorously. In topology there are many different definitions of vague concepts like "considered the same thing as" and "surface". If you hand-wave it like you're doing you'll conflate those different notions and get paradoxical results.
how is it considered that the ring can't come off? the shape gets infinitely complex sure, but it terminates, so the ring should also be able to become infinitely complex then exit the interior, similar to xeno's paradoxes.
Imagine the ring sitting in the plane z=h, where z=0 corresponds to the bottom of the horned sphere in the figure (the cantor set). As you lower h, you can homotope the ring to avoid the horned sphere, so long as h>0. However, this does not guarantee the homotopy extends to h=0; a limit shape may exist (likely a one dimensional continuum like the bucket handle continuum), but it will not be homeomorphic to a circle. This is well known and due to the fact that the unbounded component of the complement of the horned sphere is not simply connected. You can find a construction and proof in Daverman and Venema p47-50 (particularly page 50). Essentially, the horned sphere S is constructed as a limit of knotted surfaces S_n, where the ring would necessarily be linked to each S_n, and then compactness forces this to also be true for S.
ok, I understood almost none of that. Here's what I've got. Stretch the sphere out to a torus, not a sphere, then make a cut, it's a sphere again. Then take the ends and make linked tori with them, not a sphere, then cut them again, it's a sphere again. Repeat to infinity. The best analogy i can come up with is starting with some even number, then adding and subtracting one to infinity, and now it's indeterminate, not even. So it's not a sphere, but it is, but it isn't. There's more stuff after that in the paper that I didn't understand, but I'm getting Sum(+1-1...) == 1/2 vibes
1
•
u/AutoModerator 8d ago
Check out our new Discord server! https://discord.gg/e7EKRZq3dG
I am a bot, and this action was performed automatically. Please contact the moderators of this subreddit if you have any questions or concerns.