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David Cowles

Jul 21, 2022

Living in an orientable universe is no fun! Thank God we don’t live in such a universe!

*Editor's Note: Pssst, be sure to check out the challenge at the end of this blog.

Nothing could be more important, or more obvious, than this: all 3-dimensional objects have an inside and an outside! Imagine you’re caught in an unexpectedly powerful New England blizzard. Whether you’re inside or outside could be a matter of life or death. At minimum, it would make the difference between comfort and misery.

3-d surfaces divide spaces into inside and outside. I am inside, you are outside. Barring some kind of ‘tear in the fabric’ (e.g., a door), it is certain that we will never meet. (Is that a sigh of relief I’m hearing?) Space divided into inside and outside is what a ‘thing’ is, it’s what makes a thing a thing.

A house divides space into interior and exterior. So does a gift wrapped Christmas present. Otherwise, there would be no surprises on Christmas morning! Every kid would know exactly what she was getting as soon as the box was placed under the tree. But that’s not all. Without a well-defined inside/outside, there could be no such thing as a glass of wine…or a bottle of wine…or a cask…well, you get the picture.

You may be thinking that you could do without Christmas. You might even welcome a slower paced, less commercial holiday season. But could you live without wine? Didn’t think so! But wine is the least of your worries. Human beings have skin that divides their internal organs from the external environment…unicellular animals have membranes that do essentially the same thing.

Obviously, the relation we know as ‘inside/outside’ must be universal, right?…except it isn’t! Turns out that objects with a defined inside/outside are only a special class of 3-d objects. The phenomenon of inside/outside only occurs as a property of closed 2-d surfaces in 3-d spaces whose topology is “orientable."

How can there be an object without an inside or an outside? Simple, meet Mr. Mobius! But before you do, take a strip of paper, about two inches wide and 24 inches long. Connect the ends to make it into a band, like the rim of a paper hat. Notice that it has an inside surface and an outside surface. If you’re inside, well sorry, you’re inside forever. Topology has no parole board. And if you’re outside, well, worse yet, ‘you’re dead to me!’ In fact, you don’t even exist, never have, never will; you are quite literally ‘nothing.' Living in an orientable universe is no fun!

Thank God we don’t live in such a universe! Just disconnect the two ends of your 24” paper strip; then before you put them back together again, give one end of the band a 180° twist. Now reconnect the two ends. Voila! No more inside, no more outside. If I decide to go for a walk, I can cover the entire surface of the band, without ripping a whole the fabric. Like Alice in Through the Looking Glass, when I walk one way, I end up back right where I started.

Except I’m upside down! Wait, am I upside down or is the world upside down? Who cares, if I want to get right side up again, I just have to retrace my steps back to my starting point, right? Yes, that will do it, or I could just resume walking without changing directions. Either way, I end up back at ground zero and I’m right side up again. If you are interested in non-orientable space, check out Science and the Yellow Submarine or, for younger readers, Vacuum Monster in AT Magazine Issue #1.

Reader’s Challenge: write a short story about an adventure you might have in a non-orientable universe, i.e., a universe with no insides or outsides. Email your short story to We'll publish the winner in a future issue of ATM!


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