The idea of a Klein bottle intrigues me, so my co-author Steve Cole and I incorporated the idea into The Ancestor Cell as the “bottle universe” that had first appeared in previous books. Some reviewers grumbled that the bottle was never a Klein bottle, but when one rereads Interference I don’t believe that anything in it makes that impossible, or even implausible, as a subsequent development. And “it was never intended to be a Klein bottle” is irrelevant. The Doctor Who books build and develop within a shared universe.
In the fictional world of the novel, Steve and I proposed that the extrapolation of a stoppered Klein bottle into a three-dimensional rendering could create an enclosed space, and that such a three-dimensionally-rendered container could be “filled” in the very process of its conversion into that rendering from a higher dimension – i.e. from its non-orientable (and theoretical) fourth-dimensional rendering. Simple, eh?
Acme make a Klein stein (buy one for yourself at http://www.kleinbottle.com/ if you wish). It plays similar games with the idea. One could consider this a three-dimensional rendering of a four-dimensional object, in which to exist in a three-dimensional space it has to make the physical concession that its surfaces intersect, and so the mug doesn’t leak – and you can put a lid on it, like a stopper in a bottle, so that your beer can’t leak out at all. It’s not four dimensional at all, of course, but (horrors!) they call it a Klein bottle. And yet the trading standards people aren’t asking them to recall all units because they patently are not closed nonorientable surfaces with Euler characteristic zero!
Now extrapolate that a “real” Klein bottle might have been part of the “methodology” for enclosing a universe in the first place – and if there’s a science for how one does get an entire universe into a conventional bottle, then it’s one that my own research failed to throw up – so let’s presume that a “methodology” may be postulated. One could conduct the “capture” in a fourth or higher dimension and then “snapshot” it down to the three-dimensional rendering in which the physics of that lower dimension “traps” the contents. (I’d show you how to do this, but I’ve left my notebook in a higher dimension.)
An analogy for this might be the (reverse) rendering of a two-dimensional artefact into a three-dimensional artefact. I can trap a column of two-dimensional ants in an endless route march by enticing them onto a two-dimensional strip of paper, and then when they’re all aboard I twist and join the ends into a Möbius strip. Now they cannot get off, because these two-dimensional creatures can’t go over the “edge” and can only march endlessly along the single plane.
The problem is that this confines only two-dimensional creatures. The analogy for The Ancestor Cell‘s “Klein bottle” is that it cannot confine four-dimensional creatures; the bottle “leaks”. And in the narrative of The Ancestor Cell, that leakage is caused when the Time Lords cast it into the Vortex – which, the novel implies, is a catastrophe along the lines of casting it into the fourth dimension where the three-dimensional snapshot rendering no longer applies.
The novel doesn’t go into such detail, of course; it’s an action adventure novel, not a PhD thesis. But for what it’s worth, that’s the thinking behind calling it a “Klein bottle”. We extrapolated imaginatively in speculative fiction without feeling hidebound by the general machinery of algebraic and differential topology.
Could we have chosen to call it something else? Yes, but we thought it was more fun to pick a name that the general reader would recognise from “popular science” (rather than because it was something a Maths postgrad student would quibble about). I imagine most folk would think of this animation as the familiar two-dimensional rendering of the three-dimensional animation of a Klein bottle. There is also a “figure eight” Klein bottle (animated here) which is rather less visually appealing for the purposes of The Ancestor Cell.
My current favourite image of a Klein bottle is this one, a Lego version! I was going to ask my kids to make one for me, but there’d be no end of complaining. (Geddit?!)
A mathematician called Klein
thought the Möbius strip was divine.
He declared: “If you glue
the edges of two
you can make a strange bottle like mine.”
Now, here’s an experiment you can do for yourself at home. My analogy is “stoppering a bottle” not “creating an intersection” or “severing a contiguous surface”. In this sense, a stopper touches the surface, it does not break it. I think I’ve explained the fictional logic for rendering a Klein bottle in three dimensions above. And the “Klein bottles in a three-dimensional environment” (like those links above) can, indeed, be stoppered.
If you take a pair of scissors to a Möbius strip and cut it, you may get a piece of paper (long or otherwise) with a twist in it – because you’ve cut across from “side to side” and severed the strip; and subsequently, if you wish, you can deform it without making any further intersections by simply untwisting it and laying it flat (i.e. reorienting it within the third dimension). But a different single cut may instead result in another single-loop strip. Try this yourself: try cutting a Möbius strip right down the middle parallel to the edge.
Now do the same thing again… and again… you have now made three cuts, and you still have something more than just “a piece of paper with a twist in it” – and what’s more, you cannot reorient it in the third dimension to get a single strip of paper lying in one plane, unless you make a further intersection.
A mathematician confided
that a Möbius strip is one-sided.
And you get quite a laugh
when you cut one in half,
for it stays in one piece when divided.
There are multiple other variants of the first cut, by the way, each of which depends on where the cut starts and ends and almost all of which just create a slit in the strip. Now cut along the whole length of a similar strip that has two twists in it to start with (i.e. it’s not a Möbius strip) and see what you get.
Now, analogously, imagine taking a pair of scissors to a Klein bottle (theoretically speaking, and in four dimensions – for Doctor Who fictional purposes, you may prefer to use Noel Coward’s pair from Mad Dogs and Englishmen). You may get a Möbius strip or something entirely different; it depends on the nature of the imaginary intersection, and in which dimension(s).
If you split a Möbius strip you get another single joined-up loop… but if you theoretically join a Möbius strip edge to edge, you get a Klein bottle. What’s going on there, eh? Putting a stopper in a Klein bottle rendered in three dimensions is not the same thing as cutting a Klein Bottle or cutting a Möbius strip. If you put a stopper in a “Klein bottle rendered in three dimensions” you get an enclosed space. To “stopper” a Klein bottle rendered in four dimensions, you’d need more than just a three-dimensional “stopper”. And this is the basis of one plot point in The Ancestor Cell.
Note also that to create the Klein bottle you need a fourth dimension. As mathematicians have noted, this doesn’t mean it has to be “the fourth dimension” (i.e. time) which is the game we play in The Ancestor Cell. We didn’t go into great detail in the novel, because we thought that would be… well… a bit dull.
Three jolly sailors from Bladon-on-Tyne
sailed off to sea in a bottle by Klein.
As all of the sea was inside of the hull
they found the whole voyage exceedingly dull.