- #1

Jimster41

- 783

- 82

I'm trying to get comfortable with the idea of a flat torus topology that is also an everywhere a smooth manifold like the video game screen where you got off the screen to the right and pop out on the left (because as I understand it this topology could be a model of space) I can't get how this kind of surface can be everywhere smooth (differentiable?).

If I assign any coordinate system (I can think of) to the grid and start marching from the left edge of the screen toward the right (or vice versus) at some point isn't that counting going to have to be discontinuous. How can a step from 10 to 1 be considered continuous in an ordinal set form 1 to 10. Or a sequence that goes 8,9,10,-9,-8 (turns around, or starts counting down - if that's the coordinate system).

I don't understand exactly what constraint such a topology would place on the geometry of space, or how the topology and geometry specifically have to relate. But why it's considered a candidate topology has been bugging me... forever. Every since I heard this chestnut of how counter-intuitive the universe is.

I can imagine the answer is obvious, but it totally eludes me.

It occurs to me that a radial angle goes from 360deg to 0deg. But that just adds to my confusion - as to why a circle is everywhere smooth. Or why the tangent between those two points isn't busted, compared to all the other tangents.

If I assign any coordinate system (I can think of) to the grid and start marching from the left edge of the screen toward the right (or vice versus) at some point isn't that counting going to have to be discontinuous. How can a step from 10 to 1 be considered continuous in an ordinal set form 1 to 10. Or a sequence that goes 8,9,10,-9,-8 (turns around, or starts counting down - if that's the coordinate system).

I don't understand exactly what constraint such a topology would place on the geometry of space, or how the topology and geometry specifically have to relate. But why it's considered a candidate topology has been bugging me... forever. Every since I heard this chestnut of how counter-intuitive the universe is.

I can imagine the answer is obvious, but it totally eludes me.

It occurs to me that a radial angle goes from 360deg to 0deg. But that just adds to my confusion - as to why a circle is everywhere smooth. Or why the tangent between those two points isn't busted, compared to all the other tangents.

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