Impossible shape with no distinct "inside" or "outside" - in Art, Fashion and Architecture
A geometric enigma, a convoluted mind-bender dropped upon us from the wonderful extra-dimensional realm of topology, the Klein Bottle is perhaps even more popular with artists and architects than the ubiquitous Moebius strip. In fact, the Klein Bottle is what happens when you merge two Moebius Strips together: the resulting shape will still have only one side - with its inside and outside merging into one!
Such a paradoxical shape is clearly not possible within our three-dimensional reality and requires a fourth dimensional jump at some point to make it all come together. Also because true Klein bottles do not have discernible "inside" or "outside", they have ZERO VOLUME. As a result, these objects can only be simulated as an "impossible art" in our world, or only modeled with a "fake" 3-D intersection, instead of a true extra-dimensional joint. The more such intersections you add, the more it would look like some sort of a Spaghetti Monster:
The first Klein Bottle was described by the German mathematician Felix Klein in 1882; here are his lecture notes (left image below). On the right is the famous Escher's Moebius Strip - a structure from which Klein Bottle can be formed by topological extension:
This would be perhaps the first object that we'd print on a 3D printer
The whole gallery of 3D models by Torolf Sauerman (also known as "jotero") is worth looking at (and drooling over the possibilities of 3D printing it for your office) - click here. He also runs a YouTube channel full of topological animations.
The California-based Acme Klein Bottle Company sports quite a catalog of different glass and plastic bottles. They describe their products as "the finest closed, non-orientable, boundary-free manifolds sold anywhere in our three spatial dimensions":
Speaking about more sophisticated topological glass objects, you should check out glass Klein bottle collection on exhibit in The Science Museum in London - for example, this "triple bottle" variety (made by Alan Bennett in Bedford, United Kingdom, in 1995):
These are the "two single surface glass vessels made by Alan Bennett in 1995. Part one is a double loop Klein bottle which when cut gives a pair of three-twist Moebius strips. Part two is a triple loop Klein bottle which when cut gives a pair of five-twist Mobeius strips."
Do you want your Klein object DOUBLE, or TRIPLE?
Produced by the same "Acme Klein Bottle Company", the double Klein bottle looks much like an hourglass - it was featured on the cover of the book, "Endless Universe" by Paul Steinhardt and Neil Turok. Both double and triple bottles are externally linked in this case, but it is also possible to link them internally.
In the middle you can see another Acme innovation - the spiral top vessel (more info). They also call it the "Spiral top Kleindensor" which are commonly used as condensors in 4-dimensional stills. On the right is the Acme Klein wine bottle, which might present a few problems in actual usage:
"The Wine Bottle Klein Bottle is difficult to fill with wine, because of vapor-lock. As you pour water (or wine) into it, there's no place for the air to go. So the wine is trying to go down while the air is trying to go up the spout. Result is slow filling. Pouring wine out is equally frustrating. Not only are these difficult to fill and empty, but cleaning them is a real challenge. Since there's little air circulation within the Klein Bottle, moisture doesn't evaporate. Worse, you can't reach in with a towel. So you'll need to dry the interior surface using alcohol. I've had good luck with a pair of small magnets wrapped in cotton cloth."
This wonderful structure is made from 11 faces of brightly colored and illuminated glass, which certainly creates the illusion of complexity in its reflections; a sort of the stained glass 3D puzzle. it was created by glass sculptor Istvan from Hungary:
Klein Bottles in Architecture: A Klein Bottle House
Remember a great 1940s science fiction story by Robert A. Heinlein "And He Built a Crooked House"? The bizarre topological house featured there was a tesseract, but a Klein bottle would lead to similar fantastical adventures if one were to get stranded inside of it by some freak accident:
The Klein Bottle House was built by architects McBride Charles Ryan just outside Melbourne, Australia, has all the trappings of a science fictional dwelling and mathematical Garden of Eden for those who like their brains and sense of orientation challenged - but it also looks surprisingly liveable and warm (more info).
The Klein Bottle Playgrounds: Just don't end up with arms and legs tied in a topological knot!
How do you build physically impossible structure, especially turning it into a playground? Well, you could use the so-called "figure 8 immersion" of Klein Bottle: American artist, designer and architect Vito Acconci has been commissioned to create such an intriguing shape as part of a playground in the Miami Design District in 2014
I wonder if you open a wine bottle with a physically impossible bottle opener (made by Bathsheba Grossman), would that lead to some miracle while drinking the wine? Perhaps the wine would never run out? I can only dream.
Similarly, impossible topological coffee cups should lead to a never-ending supply of coffee inside (siphoned from some helpless extra-dimension, straight into our coffee-starved reality). The cups on the right were made by Cunicode, a design agency in Barcelona, which came up with a unique cup for each day (a new coffee cup design every 24 hours); see lots of them at this link.
Fold it!... and don't let it fall into an extra-dimensional wormhole -
A single sheet of paper with no cuts or tape can be folded into this interlocking model of Klein Bottle: the crease patterns are available here. Dr. Robert Lang is responsible for this puzzling origami masterpiece:
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