Klein bagel (4-D non-intersecting parameterization z-coordinate)
Description
In mathematics, the Klein bottle is an example of a non-orientable surface, informally, it is a surface (a two-dimensional manifold) in which notions of left and right cannot be consistently defined. A Klein bottle has no boundary. The “figure 8” immersion (Klein bagel) of the Klein bottle has a particularly simple parametrization. It is that of a “figure-8” torus with a 180 degree “Möbius” twist inserted. In this immersion, the self-intersection circle (when v = 0, π) is a geometric circle in the xy-plane. In four dimensions this surface can be made non-intersecting modeled after that of the flat torus.
Related formulasVariables
| z | Z-coordinate (dimensionless) |
| P | Constant that determine aspect ratio (dimensionless) |
| θ | The angle that determines the rotational angle of the figure-8 as well and the position around the z-w plane (dimensionless) |
| ϵ | Small constant related to the dependent "bump" to the fourth w axis at the intersection point (dimensionless) |
| v | The angle that determines the position around the figure-8 as well as the position in the x-y plane (dimensionless) |