It must be that we live in a higher dimensional world that is being projected down on this three-dimensional world

I was thinking a little about the real projective line and the real projective plane. This is how points on higher dimensional sphere get projected onto a lower dimensional manifold. So, all points in higher dimensional space are projected onto a lower dimensional surface. This seems at first to be abstract. However, I also noticed that spinors are members of the complex projective line. Since you can have higher dimensional spinors, it stands to reason that we are living in a complex projective plane. The reason for this is that spinors are the basic ingredient of quantum spin. Subatomic particles have spin. Furthermore, the only kind of surfaces that are possible are only three types, spheres, projective plane, and the torus. It must be that we live in a higher dimensional world that is being projected down on this three-dimensional world that has three kinds of projections. to see what I mean that there are only three kinds of surfaces that are compact and connected see https://www.youtube.com/watch?v=WIp63sPMNpc&list=PL7BFF10190F42006E&index=5 (In terms of higher dimensional spinors see the Poincare Sphere an the Bloch Sphere. The structure of the spinor is the same except you add a few terms to each spinor). This is not the same thing as string theory where the extra dimensions are wrapped up.