[Ono-Sendai]

The belt trick is about smoothly mapping paths through SO(3) to each other, not about if any point in SO(3) is reachable from any other point. I think you are confusing the notion of 'simple-connected' with just 'connected'.

[nyrikki]

No, the belt trick is about the fundamental group of SO(n) where n>2 is Z_2and the fact that a 4pi rotation is homotopic to the identity.

It is often introduced as RP^3 being isomorphic to SO(3) and as RP^3 is not simply being connected thus SO(3) cannot be.

They don’t explain why until students have the ability to deal with 4D 3-spheres being projected on 4D hyperplanes.

You can build an intuition with a 2-sphere being projected through the origin onto a 2d plane. This will demonstrate how the antipodes are not uniquely identified.

If you go from SU(2), which is the double cover of SO(3) you can use basic algebra to reduce it to a^2+b^2+c^2+d^2=1 where a,b, and c are your free variables and figure it out.

It works well without that complexity at lower levels because people are actually rotating the clopen set, which is only allowed to be the entire body or the empty set.

The 4pi symmetry applies to the topological properties of neurons and the plate trick in the real world because you are translating a portion or a property of the body and not the entire body.

Same thing on how they don’t introduce Tori topology when using Euler angles.

Pick up basically any topology or Lie algebra book targeted at first year grad students and the topic will be covered.

Bredon’s Topology and Geometry is one I personally like.

[nyrikki]

I should add that “connectedness” is a local property that has to be everywhere.

The antipode pairs don’t have unique identities they are identified together, but IIRC they are never close enough to break that local condition of connectivity.

You could use the unit circle and a line at x=2, connecting points of that line through the origin and see how that identification of two points on the unit circle happens with that one line.

That may help but it wouldn’t explain the issues with so(3) that results in a 720 deg rotation being equivalent to a 0 deg rotation.

It would explain the point at infinity concepts as the poles are indescribable with a line from X=2 through the origin.