All is well until the poor bug finds out it's on a Klein Bottle, or worse yet RP2.
Is there any way to distinguish between orientable or nonorientable surfaces when you're simply walking on it? If the bug was also 2d, I might have had some ideas hut I'm totally blanking for 3d.
Are you familiar with the construction of a torus by taking a square and "gluing the edges together"? It's effectively the game Astroids, where going off one edge of the screen wraps around to the opposite edge.
If, instead, you twist the edges 180 degrees before gluing, we then get the Klein Bottle. I.e. when you warp from, say, the left edge to the right edge, you also get flipped vertically.
I'm guessing you're already familiar with that, but by a similar construction, we can start with a (hyper-)cube and glue faces together to get higher dimensional analogues.
To my knoweldge the easiest visualization of a non-orientable 3D manifold is just by thinking of a cube that warps around to the opposite face. However, in addition, it also reflects you into your mirror image. This is essentially a Klein bottle in one higher dimension.
It's then not too hard to start grappling with the 4D case and beyond!
I took that idea to its extreme and released "Geodesic Asteroids" on iOS (free) a couple of years ago. You can play on the 2D map or an embedded torus. There is also an old blog post I wrote at the time: http://nbodyphysics.com/blog/2015/03/06/asteroids-on-a-torus...
Maybe it would be interesting to have somewhat of a switch between the 2D map or the torus map, in that while playing 2D you can enter a challenge state where the map switches.
For a Mobiüs strip, the bug would find the world had been mirrored along the width of the strip if it made a round trip and came back to the same point. Other bugs that stayed behind would claim that the traveller itself had become mirrored.
(If you do the experiment with paper, the bug would be on the other side of the paper from where it started, but I don't think this concept of "side" exists when the strip is described as a manifold. The bug lives in the strip, it's not walking on top of it. It helps to imagine the strip to be transparent.)
For a Klein bottle the same would happen, but there the bug could also find the world mirrored along a different direction, depending on how it travelled.
> Is there any way to distinguish between orientable or nonorientable surfaces when you're simply walking on it?
Draw a circle on the surface and orient it. If the surface is non-orientable then there's a way to take a long walk, return to the circle, and find that the direction of the orientation has changed.
If the surface is non-orientable then I can draw a very small circle, put an orientation on it, then there's a path I can take that brings me back to find the orientation has changed.
Let's be more explicit.
Take a non-self-intersecting embedding of RP^2 in R^4, a point P in our RP^2, and a sufficiently small epsilon e. Then take three points on the circle of size e centred on P, and think of them as going in order, thus defining an orientation. Now take an appropriate walk around RP^2 and return to the circle. For an appropriate walk I will now find that the order of the points on my circle has reversed.
That's a more precise way of saying what I originally intended, and in that context your comment doesn't make sense to me. Can you expand on it?
I think when they say "circle" they're talking about your path, not your orientation reference. That's probably wrong terminology, but it makes some sense if you're thinking about how the path has to be a closed loop.
Yes, I took "circle" to mean any closed path since the difference is immaterial in generic manifold. You wouldn't normally be talking about "orienting a circle" in the manner the original poster did, but I see now what they mean. I took it to mean "choose an orientation along a closed path" which would be impossible to do for some (but not all) closed paths in a non-orientable surface.
A closed path is S^1, which is certainly orientable, no matter what it's embedded into. You just draw an arrow on the line.
ColinWright's point is that drawing such an arrow on a tiny circle is equivalent to drawing the letter R on the 2D manifold. It gives a local orientation to the 2D surface. (If you wish you may think of this as an arrow into some 3D embedding space, but you don't have to.)
If the 2D space is orientable, then when you take a copy of this little circle (or letter R) and go for a long walk, when you get home your copy will always match the original. That's all that orientable means. In the standard usage, it's a property of the 2D manifold, not of the particular walks you take. I think this is the point of confusion here.
Yes, you can always choose an orientation of the path but not necessarily of the manifold along the path. I understand what he was saying now and misunderstood since it is not the usual way one talks about specifying an orientation at a point and in fact had imagined he was specifically specifying a "big" circle, like a nontrivial loop on the torus. You'd more typically talk about choosing a basis of the tangent space at a point - presumably ColinWright wanted to avoid involving extra definitions like tangent space and so chose a visual definition that wasn't quite as precise. I don't think there's any further confusion.
Cool, I think we're on the same page. I know what you mean by "orientation of the manifold along the path" but this isn't precisely the standard usage. (My vague memory is that you can define an orientation without needing a tangent space, but I can't think of an example, so could be wrong here.)
My background is in riemannian geometry so I never had to worry about lack of a tangent space. Certainly you can define orientability of some topological spaces that aren't even manifolds. I'd forgotten but you can even define a local orientation at a point for a general topological manifold in terms of it's top-dimensional homology. I think that's the most general situation it makes sense in, you need a well defined dimension to consider this.
Is there any way to distinguish between orientable or nonorientable surfaces when you're simply walking on it? If the bug was also 2d, I might have had some ideas hut I'm totally blanking for 3d.