As you can see in the picture, on this page you can download a 3D model of the Klein surface, mostly known by the name ‘Klein bottle’.

The Klein bottle is a geometric shape (two-dimensional manifold) that does not confine any section of space, i.e., it is not possible to determine which point is outside and which is inside, so that its surface has only one side (one-sided surface). The Mobius strip is a geometric shape that has similar features to the Klein bottle, except that the Mobius strip is a surface that has boundary while the Klein bottle has no boundary at all. For example, unlike a drinking glass, the Klein bottle has no “rim” where the surface stops abruptly, and unlike a balloon, it is possible to pass from the outside to the inside only by moving on the surface (without passing through it).

 

Klein bottle 3d surface
3D CAD software used for modeling

K3DSurf (MathMod software)

Available NURBS and BREP file formats

Surface 3D models: .igs (Standardized 3D graphic exchange file format) and .3dm (Rhinoceros software native file format)

Available polygonal mesh file formats

.obj and .blend (Blender 3D graphics software native file format)

Mesh - Quads Only; Vertices: 2880; Edges: 5760; Faces: 2880; 3D model without texture and materials

- Mesh has faces with directions different from the mesh as a whole. Although this does not necessarily mean that the mesh is bad, it can cause problems if you're doing mesh boolean operations with it.
- Mesh has pairs of faces that intersect each other. Although this does not necessarily mean that the mesh is bad, it can cause problems if you're doing mesh boolean operations with it.

 

This 3D model contains only the surface of the Klein bottle, i.e., it has no thickness. In recent years, although of no practical value, the Klein bottle has become very popular. One could say that its value is only that it is one of the few things that originates in mathematics and that is of interest to ordinary people (similar to mobius strip). A large number of people use such 3D models to create math-art 3D models that are printed on 3D printers and published on the Internet. For the purposes of online marketing, glassware workshops have competed to make as creative or complex Klein bottles as possible. You can even find pendants, children's hats and Klein bottle in the form of a Klein bottle.

If you are keen on this kind of creativity, feel free to download this 3D model of the Klein bottle and try to do something creative with it.

 

Download 3D model (polygonal mesh)