I don’t know what that means.
But Asteroids space is curved, so that the top of the screen meets the bottom, and the left meets the right… isn’t it?
Maybe we’re using different meaning of “curved” and “flat.”
Whatever you mean by “flat”, if a flat piece of paper is “flat” then so is an Asteroids screen because they are indistinguishable on a local basis. Yes, the top of the screen meets the bottom, but that is a large-scale phenomenon (imagine you are an ant, then you couldn’t tell without walking all the way around). There are no seams or bends. Also keep in mind this is an Asteroids flat torus and not a bagel-shaped surface.
To a topologist, the local curvature doesn’t matter. A sphere and a cube are the same topology. A simple handle coffee cup and a donut (torus) are the same topology.
As to an arcade game with a toroidal geometry there are two primary mappings: The top and bottom edges wrap through the center of the donut and the sides go around the hole or vice versa. Once you introduce such a mapping then there is a distinction that can be made.
The terminology can be a bit muddled in terms of talking about a purely Mathematical model or something that is more of a Physics notion.
This might be helpful: https://www.youtube.com/watch?v=AwwIFcdUFrE
Quite true, but I want to consider not only the topology of the toroidal surface, but the additional Riemannian manifold structure where we can measure lengths, angles, etc., and in particular the curvature is defined at every point, and in the case of the arcade screen is zero, which is not the case if you twist it into a doughnut.
If you try to cram a flat torus into three-dimensional Euclidean space it ends up very wrinkled…
I’m a topologist, not a geometer, but yes, the torus can be given a metric in which the Gaussian curvature is everywhere 0. It’s just the metric inherited by quotienting the plane. See Wikipedia’s section on the Flat Torus.
You can’t embed a torus into a three-dimensional space without giving it some curvature. But who said anything about wanting to embed it into 3-dimensional space? The Universe is presumably not embedded into any space larger than itself, and it has whatever topology it has independent of any embedding or lack thereof.
Yeah, there’s that Math vs. Physics issue. Yeah, Math people like to be able to define flat toruses. Physics people know that’s crazy. And these are people who love to model particles in 11 (or is it 13?) dimensions.
A lot of Math is crazy and impractical. Until it isn’t.
We do?
Tori are not that crazy. Maybe Seifert–Weber space or something…
He’s actually right, though. Stuff that’s very far away does appear to accelerate away faster and faster (In an identical way to how it would appear if a mysterious, invisible energy was pushing on it very hard - hence the term “dark energy” which you may have heard).
It’s been known since Hubble (the human, not the satellite) that things further away are receding faster than things nearby. It’s only in the past couple of decades, though, that we’ve learned that they also have a positive acceleration, rather than a negative one like you would expect from gravity. That’s what the dark energy is all about.
Yes, but that doesn’t mean he understands the analogy correctly, which is what I was pointing out. A lot of people do get that analogy wrong, because they think of a balloon as a 3D object, but the analogy is only about the 2D surface of the balloon.
But doesn’t the balloon analogy still work whether your location is in the middle or toward the edge? Like continuously dividing by any number, the result is never 0? I know complete newb rambling, but still enjoy reading the responses.
One of the points of the balloon analogy is that there is no middle or edge, just as the real universe has no middle or edge. It’s a 2D analogy, so only the surface of the balloon counts in the analogy. The central part of the balloon is not part of the analogy because there’s no equivalent in the real universe.
Think of the world’s surface instead of a balloon. Put your finger on it anywhere. It looks like the centre of the surface right? Ok now move your finger somewhere else. Huh…looks like another centre. Basically everywhere is a centre.
Now draw grid lines on your globe. Put you finger on a “centre” and imagine the distance between grid lines expanding. You’re still in a centre but the overall distances between points is increasing.
Wow! Yes thank you for that practical analogy. That’s for a flat theory right? Is there a great anology like this for the spherical theory?
That is the spherical analogy (because the Earth is, after all, a sphere). But the flat analogy isn’t too different, and still gives the result that any point has equal claim to being the center.