To construct the classic Klein bottle, we merged the two horizontal blue edges to form a tube whose cross section is a circle. Alternatively, we could glue together these two blue edges to form a ‘double tube’ whose cross section is a figure-8. In the figure below, the bottom edge rolls out of the page while the top edge rolls into the page, and the two blue edges merge along the red line segment.
To turn this double-tube into a figure-8 Klein bottle, rotate it around an off-center vertical axis while applying a half-twist. The half-twist is needed to align the two green edges correctly. You can view this half-twist by varying the section setting in the following program.
The figure-8 Klein bottle is made from glueing two Möbius bands together along each band’s edge. You can view one of the Möbius bands by varying the opacity of the other band. The two Möbius bands intersect in a circle which corresponds to the red line segment in the previous figure.