omino code blog

One may ask, “What is the volume of a Klein bottle?” It’s fun to say — and observe, of course — that since it has no apparent “inside” or “outside” that its volume must be zero. Or infinite, if you think everything’s “inside” it. Or finite, if you think everything isn’t infinite.

But then one may ask, “What is the area of a Möbius strip?” By analogy we may wish to say, Why, zero of course! But it’s also sensible to measure the area of the paper strip we used.

How can this be? Let us continue by analogy.

Here is our beloved Mr A Square trying to imagine a Möbius strip.

“How can this be?” he echoes. The original strip has an area, certainly… but the outline implied by this unimaginable “half-twist” intersects itself. It has no “inside” or “outside” really, does it? But of course in three dimensions we can see that the outline doesn’t really cross itself.

But how can we explain that to Mr A Square? This whole “half-twist” idea hinges (if you will) on one bizarre notion: We in spaceland can flip a flat object from left-right to right-left. In Dionys Burger’s Sphereland there’s a bit where a visiting spacelander graciously inverts several flat-dogs. This was quite a gift because, you see, right-handed dogs are quite common, but left-handed dogs are of a rare pedigree.

And in Roger Zelazny’s Doorways in the Sand there’s a similar bit where our hero discreetly puts himself through the alien Rhennius machine — on display at the local museum — and comes out inverted, down to the molecular level. You know, words appear backwards and sweet’n'low becomes fattening or some such. And then one of the bad guys goes through it twice and is turned inside out, and each organ and blood vessel is turned inside out, and each cell… and its not pretty and its a ripping good yarn, but we digress.

Mr A Square is of course familiar with keyed flexible flat-rods, which can be assembled into long chains, or looped.

What could he think of a mismanufactured piece, like this:

Like his mongrel and pedigreed dogs, it can be flipped… but we would flip only one end of it! Of course, the whole thing can’t fit in Flatland any more. That’s a Möbius strip. And what does this mean for us, and Klein bottles? Imagine a keyed flexible rod, like this.

How could we ever connect the ends? If we could move in four dimensions, it would be as easy as closing a Möbius strip.

And now we can ask — and answer! — the questions, “Does a Möbius strip have area?” and “Does a Klein bottle have volume?”

If we look at Mr A Square’s outline of a Möbius strip, we can decide that, actually, it’s a Möbius bottle! We can’t bring a Möbius bottle into Spaceland and fill it, because our three-dimensional liquids won’t stay between the lines.

A Möbius bottle has no area, but a Möbius strip does.

And the traditional Klein Bottle is an “outline” of the Klein rod.

A Klein rod has volume… but there’s no way to fill a Klein bottle. The four dimensional liquid will fall out the sides.

Portraits of Mr A Square reveal his internal biological structure including his central “germ”, brain, polyominous exterior skin, and his heretofore undocumented “aetherfeet”, which at last reveals how Flatlanders may “walk”. A similar physical property keeps buildings and trees in place.