Labeling a Klein Bottle

ACME brand Klein bottles (via)

Felix Klein’s non-orientable, one-sided surface was not originally imagined as a container, but was labeled as such because of a German pun:

The Klein bottle was first described in 1882 by the German mathematician Felix Klein. It may have been originally named the Kleinsche Fläche (“Klein surface”) and that this was incorrectly interpreted as Kleinsche Flasche (“Klein bottle”), which ultimately led to the adoption of this term in the German language as well.

Wikipedia

At lease one source asserts that Klein’s surface was, for a time, called the Klein jar, but I could not confirm this.

When did they change its name from “Klein Jar” to “Klein Bottle”? Look in any projective geometry book published before, say, 1960 and you will see the above “bottle” referred to as a “Klein Jar.”

Whatever you call it, a Klein surface can serve as a container, albeit a fairly impractical one. If we accept that it’s a container, then what sort of label does it get?

If you draw the letter “R” on a clear label, then slide that label around the outside of a sphere, when you return it to the same place, the letter looks exactly the same. So a sphere is orientable. On a Klein Bottle, you can slide that label around so that the letter reads backwards. To do this, you’ll have to slide the label all the way inside the Klein Bottle (you’ll need a long pipecleaner). When it’s on the other side of the glass from where it started, the label will read as the mirror image. That’s nonorientable.

ACME Kline Bottle FAQ

This idea of the label sliding on its one-sided surface all the way into the inside of the Klein bottle and then being backwards, is a recurring theme.


Labeled Klein bottle (via Matematita); Poster by IDeAS

The image on the left demonstrates the backwards inner label. The Klein bottle on the right is decorated with an abstract symmetrical design which would look the same whether it was on the inside or the outside. (A good idea for Klein bottle branding: ambigrams.)

While ACME does not, for the most part, label their bottles, they do sell a flask with their logo…

… and that logo employs backwards & forwards type on a Mobius strip to highlight the product’s non-orientability.

(The “Klein stein” and filling a Method/Klein bottle, after the fold…)

 

On the far left of the photo at the top of this post is the “Klein stein” — a patented drinking mug with Klein bottle features.

The video below demonstrates the challenge of filling Klein bottle with a Euclidean liquid, but also shows us another good re-use idea for old Method bottles

Filling a Klein bottle from Zach Backas on Vimeo.

Copiado completamente de BOX VOX

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