This is a study of how a Klein Bottle can be modeled with vZome. This came about through a discussion on Discord.

Klein Bottle

Having constructed this model, it occurred to me that there could be a surface with the properties of the Klein bottle, but with multiple paths to traverse from the “exterior” to the “interior.” Below is an example of such a surface, with Tetrahedral symmetry. Hexagonal “end cap” panels are left open for viewing interior of “tunnels.”

Tetra-Klein Box

A more complicated model is this, with six “tunnel” directions. While it is in Icosahedral symmetry, the tunnels that are open from the center to the outside are in three-fold symmetry about only one axis, indicated with a single yellow strut. Red struts from the center are indicators of open tunnels. As with the above model, pentagonal “end caps” are left open for viewing.

Six Axis Klein Surface