Klein bottle and Hugo in 3D

A few evenings ago I got to scratch my itch in an opportunity to watch the film Hugo in 3D for the first time, courtesy JA. I can still rate this movie high on my would-I-ever-watch-it-again list. And to mark the event JA presented me with a Klein bottle which their 3D printer produced whilst we watched the film.

All 60mm of my very own Klein bottle

Most people are familiar with the Klein bottle, but not so many can follow its mathematics. Sentences like "an immersion is a differentiable function between differentiable manifolds whose derivative is everywhere injective", which occurs in Wikipedia's definition of the Klein bottle, leave me feeling I am too old...  However some attributes are more down to earth, like:

A Klein bottle has only one surface, just as a Möbius strip has only one side. But you might say that any bottle has only one surface - if you start at a point on the inside surface you can travel smoothly to the outside via the neck. The catch is that, topologically, both shapes should have zero wall thickness, just as a line is defined to have no width, and, if so, to get from inside to outside you have to negotiate an edge at the rim, and an edge is what divides two intersecting surfaces. With a Klein bottle with zero wall thickness, however, the transition from "inside" to "outside" is smooth and continuous: there is no edge, there truly is only one surface.

The Klein bottle is to a Möbius strip as a cube is to a square.  Thus a Klein bottle is properly a 4D shape, so that my very own pictured above is in fact only a 3D representation in which it must intersect itself (where the top bends over and disappears into the body) and that intersection is clearly a cheat because it creates an edge, a discontinuity which a true Klein bottle doesn't have. In 4D that intersection would, apparently, not occur thus the surface would be continuous.

I still use a 2D printer in my office. JA used a 3D printer to make my bottle. Roll on the 4D printer!

Back to Hugo: need I repeat how it is the perfect story in which each colourful and larger-than-life character is introduced in turn at the start and exeant at the end having gained in some way? Apart from Hugo Cabret who in contrast is a normal, coming-of-age kid who we love for his persistence.