Topological Marvel: The Klein Bottle in Art

Link - article by Avi Abrams

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:

(images credit: Torolf Sauerman, Bathsheba, Satgnu)

On the left you see the intricate mathematical art by Anatoly Fomenko from Russia (Klein Bottle and Torus combination); on the right is a steam Klein Bottle cart "impossible concept" by Roger Shepard:

(images credit: Anatoly Fomenko, Roger Shepard)

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:

(images via 1, 2)

And here is how the bottle can be formed (note: this requires an extra-dimensional jump at the "intersection", so it's only a simplified visualization):

Taking a clue from the Moebius strip image with ants crawling all over it (see above), this animation shows the path that these ants would follow crawling on the Klein bottle:

(image via)

The Klein Bagel: do you want cream cheese with that?

Also called "the figure 8 immersion of the Klein bottle" (read about immersions here), this bagel-like impossible shape has an even more impressive, impossible cross section:

(images via)

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.

(images credit: Torolf Sauerman)

Californian 3D artist Erik Anderson rendered this remarkably life-like image of chained Klein bottles:

(image credit: Erik Anderson)

Nested Topological Glass Vessels

Here are a couple of rather more simplified, hand-blown glass emulations (the wine "glass" on the right you can even purchase here) -

(images credit: Clifford Stoll and Grand-Illusions, Encyclopædia Britannica)

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":

(image credit: Science Museum, London)

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):

(image credit: Science Museum, London)

Here is "a single surface, five-layered sphere: extension of the three Klein bottles above to infinity which when cut gives a pair of single-twist Moebius strips":

(image credit: Science Museum, London)

Adding more and more rounds of "handles" that theoretically fold upon themselves, results in this intriguing glass sculpture:

(image credit: http://www.sciencemuseum.org.uk/images/I065/10328081.aspx)

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.

(images via)

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."

... or, perhaps, even TRIANGULAR?

Triangular, triple Klein bottle form:

(image via)

Would you prefer your Klein manifold sliced? (more info):

(images credit: Science Museum, London, Klein Bottle)

Stained Glass Impossibility

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:

(image credit: Istvan)

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:

(photo by John Gollings, via)

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

(image credit: Vito Acconci)

On the right image above you can see another mind-boggling park installation by Vito Acconci, perhaps even more sophisticated and original.

(image via)

Here is a Krabbelknoten , a crawl-through playground structure installed at the ‘Math Adventure Land’ in Dresden in 2011 - more info

(images via)

Sophisticated mathematical knots make up great knitting projects!

Check out these single-sided Klein Bottle Hats (created by Kleinbottle and Majolo):

(images via)

Klein Bottle Bottle Opener and Coffee Cups

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.

(images via 1, 2)

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:

(image via)

Hide the M&M candy inside a topological machine, and kids would never get to forbidden sweets

As we said before, Klein bottles are supposed to have zero volume, yet you can apparently fill them with M&M candy? Well, this standard Acme bottom-mouth Klein Bottle does the trick:

(images via 1, 2)

Right image above shows the BIGGEST Klein bottle in existence: 1.1 meter tall, 50 cm diameter (the size of a 5 year old child) - made by the same Acme Klein Bottle Company.

Here is the Klein Bottle - SQUARED!

if you thought that simple Klein Bottle was something challenging to wrap your brains around, try this topological maze, created solely to befuddle our spatial 3-dimensional senses:

(image credit: Sea Moon)

And now, to cool your brain after all these impossible topological exercises, here is a picture a NORMAL BOTTLE, with normal "inside" and "outside", for a change! -

(image credit: Bree Walk)

Article by Avi Abrams, Dark Roasted Blend.


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Anonymous Anonymous said...

The klein wine bottle may run out, however it very difficult to get the wine in and out the inside....

Blogger zelph said...

Challenged my brain and sense of orientation to the max.Thank you! :-)


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