What does the sixth dimensional equivalent of a Mobius strip look like /sci/?
I need this to complete my loop.
>>7890192
Like a six dimensional mobius strip
>>7890214
Yeah yeah, I get the whole "paradox queen" thing and everything, but something about taking six of my words and jumbling them about seems mighty similar to circular reasoning to me.
(Technically five words and a subset of the fourth, really.)
Imagine 4 4D connected Klein bottles.
>>7890192
>expecting somebody to tell you what something in six dimensions "looks like"
how do you think you're going to visualize something three dimensions higher than your visual faculties
not to mention that your question doesn't even make any sense
"equivalent" of a Mobius strip in what sense? What property is supposed to carry over from three dimensions to six?
>>7890230
I guess a more specific question would be HOW? Which ways are they supposed to be connected? What does "connected" mean in this context? I'm not sure how else to phrase my question.
>>7890192
Pretty much a non-orientable 6-manifold desu
All you can visualize is a 3D cross-section, which really wont help
Try abstractly constructing it
>>7890224
>Yeah yeah, I get the whole "paradox queen" thing and everything, but something about taking six of my words and jumbling them about seems mighty similar to circular reasoning to me.
>(Technically five words and a subset of the fourth, really.)
Autism test poz
>>7890287
>Try abstractly constructing it
That's what I'm trying to do. Renaming/rephrasing the question 20 times doesn't help me. I need the math, not the visualization. You can't help me with the visualization, so help me with the math. That's what I meant to ask.
Come on /sci/. I know at least one of you has this kind of formal mathematical knowledge in you.
I can't be the ONLY one that contemplates higher dimensional structures.
>>7891471
try miller indices, not gonna say more because I want to do it myself, also you are kind of annoying but you don't seem stupid
>>7891725
>also you are kind of annoying but you don't seem stupid
Thanks! I think that's the best compliment I've ever received.
>>7891744
I don't know either, that's kind of the problem. I don't know which kinds of properties emerge in hyperdimensional space, but we can go from Mobius strip to Klein bottle.... Those are both 2D manifolds, aren't they? Is a Mobius strip just one subset/"section" of a Klein bottle? Is this a topology question?
Anon, you need to narrow down your search. What you are asking is too vague.
The mobius strip and the klein bottle are both constructed as quotient spaces of the unit square, that is, the cartesian product of two unit intervals.
If you were interested in doing something similar in six dimesnsions, you could start fucking around with quotient spaces of [0,1] ^6 generated by gluing opposite sides together.
One of the distinct characteristics of the mobius strip is that sides are glued with a twist. That is, 0 is identified with 1 and visa versa in the equivalence relation that generates the quotient space.
>>7892583
Yes, a Mobius band is a "part" of a Klein bottle (more accurately, you get a Klein bottle by gluing two Mobius bands, kind of like how you can get a torus by gluing two cylinders along their edges).
I don't know how to think about this. You can define the Mobius band as the only nontrivial line bundle on S^1. Maybe if you look at vector bundles with 3-dimensional fiber over S^3, you can find the analog you are looking for (but I don't know how many there are in that case, there might not be a unique nontrivial one).
>>7892707
Thank you, that's what I need. Why is it called a quotient space though?
>>7892739
>You can define the Mobius band as the only nontrivial line bundle on S^1.
>nontrivial line bundle on S^1
I, uhh, notation?
>there might not be a unique nontrivial one
>unique nontrivial one
Unique... Nontrivial...
Why do I care about nontrivial? What is trivial in this case?
>>7892739
>only nontrivial line bundle on S^1
Is that supposed to be S^2? Would a torus be a trivial line bundle... "on" S^2? Or... S^1?
>>7892912
Well since the Mobius band is a nontrivial line bundle over S^1, you might think that the 6-dimensional analog of the Mobius band would be a nontrivial vector bundle with 3-dimensional fibers over S^3. Now, if you do not know what any of these terms mean, it might be a bit complicated. Maybe check the wikipedia page.
The point of a vector bundle over a topological space X is to formalize the idea of a vector space "varying continuously" over the points of X (you can think of a mobius strip as a "continuous" family of lines "over" a circle).
>>7892957
My brain hurts. Let me process this.
>>7890192
Who's that semen demon?
>>7892957
I don't even know which Wikipedia page to look at. You sound like a shill. If you can show me a world where S^1 is a standardized notation for vectors in a topological space, I'll believe you. Note that you can't interfere with any Mochizuki-like process on the path to exposing me to such a world.
>>7893002
Look at the dimensions. It's a wallpaper. Might not be any anime behind it.
>>7893556
Just reverse image search and the answer shall be given.
>>7892707
Wow, okay. I can see what you're saying now. I'm not sure if I want to ... I don't... What would the cubic equivalent of a Klein bottle be? Are Mobius strips and Klein bottles [0,1]^2 quotient spaces?
>>7893575
Would a cubic Klein bottle be a [0,1,i]^2 quotient?
>>7893532
Not him, I'm >>7892707
Basically a line bundle is a way of assigning a 1-dimensional vector space to every point of a topological manifold. Kind of like how you can generate the tangent line to a curve at any point along said curve. The tangent line in that case is a 1-d vector subspace of the ambient manifold that the curve is embedded in (generally R^2 or R^3).
The mobius band can actually be defined as the only nontrivial line bundle on S^1 (the circle).
What I was talking about earlier in my post here >>7892707 was essentially generating 6-d mobius like manifolds as quotient spaces of [0,1]^6.
Basically, this would amount to a taking a 6-d cube and gluing certain sides together, being sure to give one of the sides a twist before gluing.
The ordinary mobius strip takes two (opposing of course) of the four sides in [0,1]^2, gives one a twist and then glues. The klein bottle glues two opposite sides in a standard fashion and then glues the remaining two with a twist in one of the sides.
If you messed around long enough with quotient spaces of [0,1]^6, you might get what you are interested in.
>>7893617
Oh, I think I get quotient spaces now. It's basically the formalism that solves the (Q,*) group problem, right? You need 0 as the additive identity and 1 as the multiplicative index to construct multiplication, and from there you have to take 0 out of Q to get a group. Or a field? I always get the two mixed up.
But your S^1 notation basically states that the Mobuis strip can be formalized in one dimension, meaning that the Klein bottle and its doublytwisted counterpart are strictly 2D notions (can't be constructed in 1D formalisms). Unless the doubled form of a Klein bottle is topologically equivalent to a torus? So this IS a topology question after all?
I mean, if I take a 3-cube and glue-twist it pi/2, I'll get the same "one surface" property as a Mobius strip. Glue-twisting at a pi ratio will make surfaces separated by one edge, but my understanding of topology is that the "edge" doesn't count and it's still topologically equivalent to a torus. In any case you've helped me a great deal and given me much to study and research.
>>7893645
>But your S^1 notation basically states that the Mobuis strip can be formalized in one dimension, meaning that the Klein bottle and its doublytwisted counterpart are strictly 2D notions
No this is not the case.
The "S" in S^1 stands for sphere. The circle is a 1-sphere, the standard sphere you are I are used to is the 2-sphere and it keeps on going from there.
As for your confusion on quotient spaces. Basically, a quotient space is a way of taking a certain topological space and connecting certain parts together. That is as simple as I can break it down.
Rigorously this amounts to constructing an equivalence relation on the space, and setting the parts you want to glue together as equal under this equivalence relation. In the quotient space, those points are now considered the same. The quotient space is then the set of equivalence classes under this equivalence relation.
Also for your last comment. If you took a standard 3 cube ie [0,1]^3 and glued one of the faces to the one on the opposite side. One construction would look much like a torus and the other like a klein bottle (depending on the orientation of the gluing). The huge difference is that the result would be 3 dimensional where the klein bottle and the torus are 2 dimensional manifolds, ie the torus is merely the surface of the standard doughnut shape one thinks of.
>>7893532
I was thinking about the wikipedia page on vector bundles, or Mobius band. Basically any wikipedia page on the subject will be talking about this
>>7893790
I guess my only other question now is if manifolds are a concept in topology of if they're defined with some other formalism.
>>7893913
In a sense, yes. Manifolds are defined as topological spaces satisfying a number of properties (every point has a neighbourhood that is homeomorphic to R^n through a "coordinate chart" in such a way that going from a set of coordinates to another is a smooth mapping).
It is not a topological notion per se (it is additional structure on top of the topology) but the fact that a topological space admits a manifold structure gives information on what it looks like. It is the whole point of the field of differential topology (Morse theory, cobordism, etc): What topological information can one gain on a space given the fact that it has a manifold structure.
I'm quite curious now. What was your original interest in understanding the 6-d mobius like structure?
>>7894664
It has to do with paradox logic and seven-fold arithmetic. Mobius strips are just the simplest geometric expression of a logical paradox.
>>7893938
Thank you. I had the sense that manifolds were just a hyperdimensional notion and topology was the study of mapping between arbitrary coordinate spaces. Effectively boundary manipulation AFAICT.
Thanks /sci/! I'll try to ask better question in the future.
>>7890192
A mobius strip in our reality is:
In 3 dimensions
Is a 2D surface
Has a single 1D boundary.
In 6D your mobius strip is
A 5D surface
has a single 4D boundary.
>>7896119
>Has a single 1D boundary.
That also describes a Klein bottle.
Okay, does anyone mind if I start calling it an inverted torus?
>>7896144
Wrong dude, the klein bottle is a manifold without boundary. You might want to at least visit the wiki pages on these objects before saying things that reveal your ignorance :[
Roughly how many minutes are between the big bang and the theoretical heat death of the universe, and how many minutes along this course are we?
I am not especially smart and all of the articles I've looked into give very confusing information
What are some new topics in operator theory ? All I find is chinese circlejerk.
