Triple-nested Klein bottle

Here's glassblower Alan Bennett's astounding triple-nested Klein bottle, a beautiful thing:

A single surface model made by Alan Bennett in Bedford, United Kingdom. It consists of three Klein bottles set inside each other to produce, when cut, three pairs of single-twist Mobius strips. A Klein bottle has no edges, no outside or inside and cannot be properly constructed in three dimensions.

Klein bottle, 1995. (via Neatorama)

(Image: Science Museum/Science & Society Picture Library)



  1. “…cannot be properly constructed in three dimensions.”
    Well, wtf is that in the picture then?

    1. The three-dimensional shadow of a four-dimensional object.
      Klein bottles are defined as non-self-intersecting.  In three dimensions that’s impossible, but in four dimensions it can work.  So all the glass objects advertized as Klein bottles are actually 3 dimensional embeddings of four-dimensional objects.  A necessarily imperfect simile, in other words.

      You can see the same sort of embedding in general relativity embedding diagrams where “down” would require some kind of meta-gravity, and in visualizations of tesseracts (4D cubes)

      1. I like your answer much more than @boingboing-430cc6899313afb79f799dc72fb58493:disqus ‘s. That answer is right, but very unsexy. :-)

    2. Cliff Stoll (above) can construct them in his better moments, but you can’t have one because UPS insists on shipping them in 3 dimensions.

  2. “cannot be properly constructed in three dimensions.”

    Looks pretty proper to me.

    1. Theoretically,  with a Klein bottle the “tube” passes through the side of the bottle without there being an intersection in the surface. The tube passes from one side to the other in a fourth dimension. Not possible in three dimensions, so this is a representation only. Not a true Klein bottle.

  3. A proper klein bottle has mobius-strip like properties along two axes, the pseudoklein bottles that can be constructed in three dimensions only have them along one.

    1. You are thinking of the projective plane, not the Klein bottle. winkybb is correct: the problem is self intersection in 3-dimensions, which can only be avoided in 4-dimensions or higher.

      1. A real Klein bottle wouldn’t contain the smoke, heh.  Since it’s a 3D object that contains no volume, there’d be no logical places to put the bowl and the mouthpiece, since any smoke generated could flow out of the Klein bottle without having to pass through an orifice.

          1. As long as we keep incrementing spatial dimensions and leave time 1d. I can’t figure out 1d time myself, and I doubt anyone can think in 2d time in any meaningful way. It’s both unintuitive and there are exactly 0 examples in real life that map onto it in any way shape or form. I’m just glad I can imagine a couple of static 4D objects using time as an extra spatial dimension.

      2. My first thought was, “How would I clean that? I wonder if kosher salt and white vinegar would do the trick?”

        It is a stunning piece of art though.

        1. You went straight to cleaning? I’m still trying to figure out how to pack the damn thing.

  4. Cliff Stohl of (Cuckoo’s Egg fame) makes and sells a variety of klein bottles and other odd glassware at

    dig the retro (lack of) web design. that’s straight up html, that is.

    1.   Thank you, Lectroid … yes, I’ve made Klein bottles for about 15 years; much fun!  I’ve made an externally coupled, triple Klein bottle, as well as a kleinbottle-wine-bottle.  And yes, m’website is pretty much handcoded, slowly built up from 1997 or so.  Someday, I’ll recast it with modern tools…
        Best part of my microbusiness of mathematical manifolds is meeting interesting people; if you’re visiting Berkeley, give a shout…

  5. “It consists of three Klein bottles set inside each other to produce,
    when cut, three pairs of single-twist Mobius strips. A Klein bottle has
    no edges, no outside or inside and cannot be properly constructed in
    three dimensions” Does no one see the problem with setting inside each other things which have no insides?

  6. That is very cool, but it is three separate Klein bottles overlapping in the same general space. I think it would have been cooler if it was a progression of kleing bottles from the largest to the smallest, so that instead of three separate surfaces the entire thing would have been a single surface.

  7. Interesting point, Lavadera! 
    When I made a triple-klein bottle, I interconnected three “ordinary” klein bottles (welded in glass, of course).  Math folk say that these are homeomorphic to each other … that is, any odd number of linked klein bottles have the same characteristics as a single klein bottle.   I just took the attached photo to give an idea of this other immersion of the triple Klein bottle.  My son, Danny, and I have also built a hole-through-a-hole-in-a-hole; this is homeomorphic to a triple-torus.Warm cheers to all! -Cliff

  8. I can see a Klein Bottle having no inside, but… no outside?  If a Klein Bottle actually did exist, where are we in relation to it?   Or by “no inside and no outside” do you just mean there’s one side that encompasses both so you can’t really say which you are in any given time?

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