Someone emailed to ask what that strange image was in the Hegel post, and why I illustrated the post that way. The image was probably not the clearest I could have found (I was writing a bit under time pressure, and illustrations weren’t my highest priority… ;-P), but is meant to be a picture of a Klein bottle – a figure I’ve occasionally toyed with using in place of the ouroboros as the basis for the site logo…
The animated image in this post – which is from Konrad Polthier’s article “Imaging Maths: Inside the Klein Bottle” in +plus magazine – provides a somewhat clearer sense of what a Klein bottle is. I know several people who lurk here who could explain the concept of a Klein bottle more easily and clearly (and accurately!!!) than I can… Perhaps one of them will step forward and bail me out here… ;-P But let me embarrass myself a bit first, to give them something to correct.
The basic idea is that a Klein bottle, like a möbius strip, is non-orientable – a concept that I won’t outline here (among other things, because this concept is easier to see than to read about): the Polthier article provides a nice illustration. In our everyday three-dimensional space, non-orientable objects appear to have only one side. So, in terms of the animated image in this post, if you were walking along the path mapped by the möbius strip then, at any given point along your journey, it might appear that you are moving across an object that has another “side”. As you continue to move along the surface, however, you will eventually reach what earlier appeared to be that “other” side without having to cross through a surface or clamber over an edge.
While all of this is quite cool to try to visualise, and non-orientable images – particularly möbius strips, but also the occasional Klein bottle – seem to crop up quite regularly as illustrations in social theoretic discussions of immanence, the underlying mathematics has no real implications for the social theoretic discussions about there being no transcendent “outside” from which to view our social experience or history… Nevertheless, there’s a nice aesthetic, metaphoric resonance between the social theoretic and mathematical concepts, which does no harm as long as it’s recognised as such… I tend to like the Klein bottle as a metaphor due to its various strange properties, as described in the Polthier article:
The bottle is a one-sided surface – like the well-known Möbius band – but is even more fascinating, since it is closed and has no border and neither an enclosed interior nor exterior.
Picture a bottle with a hole in the bottom. Now extend the neck. Curve the neck back on itself, insert it through the side of the bottle without touching the surface (an act which is impossible in three-dimensional space), and extend the neck down inside the bottle until it joins the hole in the bottom. A true Klein bottle in four dimensions does not intersect itself where it crosses the side.
Unlike a drinking glass, this object has no “rim” where the surface stops abruptly. Unlike a balloon, a fly can go from the outside to the inside without passing through the surface (so there isn’t really an “outside” and “inside”).
So we have a closed but borderless surface with no inside or outside, which can be embedded only in a four-dimensional space – not a terrible metaphor for the object of an immanent historical theory… ;-P
If anyone is looking for some holiday procrastination opportunities (or do we not have to call it “procrastination”, since it’s the holidays?), Beyond the Third Dimension has some nice animations of Klein bottles, including some interactive ones, as does the Polthier article referenced above.
Anyone needing ideas for belated Christmas presents (or perhaps looking forward to Valentine’s Day…) might consider purchasing a three dimensional immersion of a Klein bottle from Acme Klein Bottles – a company which, I note, also offers “industrial and post-industrial consulting”, boasts about its “finite but unbounded warehouse”, and displays diverse mottos, including “where yesterday’s future is here today!”, “since 1995, imposing on the impossible!”, and – my personal favourite – “where there’s one side to every problem!”
Even if you don’t intend to buy, I’d still recommend browsing the Acme Klein Bottles website – the “Important Information for Idiots” section might be a good starting point (not to imply anything about my readership, mind you… ;-P). It’s also worth checking out Acme’s pioneering lifetime guarantee – something that I suspect you might be able to convince them to extend to you, even if you don’t purchase a Klein bottle.
[Updated to add: my son noticed the animation on my laptop, and came over to have a look. He asked what it was called, and then stared, fascinated, for around fifteen minutes. He finally turned to me, all concern and wrinkled brow, and anxiously asked: “There’s no end to the bottle?! Where’s the end of the bottle??”]
[Note: animated gif @2003 Konrad Polthier from +plus magazine “Imaging maths – Inside the Klein Bottle: Klein Bottle with Möbius Band” September 2003.]