Klein bottles and One-sided walls: Are we inside or outside ?

Walls separate spaces. But which side is outside and which side is inside? This is a non-trivial question. And In some cases, walls can be in fact one-sided. Let's explore together some interesting cases:

Warm-up case: The infamous balcony and directionality.

My bedroom view a few weeks ago.

Aside from the window/door duality (we leave that for another post), the balcony space seems to neither belong to the outside nor to the inside. To avoid this debacle, instead of talking in terms of spaces, let's talk about directions. What direction does the balcony point to? obviously outwards you might say. But it could be as well pointing inwards. It is up to you to choose one of either direction. Once you have chosen your convention, say all balconies point outwards, then you can define all the space in that direction as the outside.

A First Case: The courtyard and normality.

The courtyard of a mathematics department I visited.

Is it outdoors or indoors? Let's try to apply what we have learned in the warm-up case. We try to define a direction, but now it's not clear where the courtyard is pointing to. Is it point to the sky? is it pointing to itself? The trick, in this case, is to choose a reference surface. The most straightforward surface, in this case, is the joint surface of the floor, walls, and pillars. Now, we are able to pick a unique direction with respect to the surface, the right-angle direction to the surface. Also, known as the normal direction. Let's say, we pick one and we call it the outward direction. Then all the space outwards from the surface at right-angles is the outside. Hence, the courtyard is outside.

A challenging case: The Klein bottle and one-sided walls.

| An illustration of the Klein bottle. Source: Wikipedia commons. |

What do you think of this case? Let's apply what we learned so far. We first choose a surface, say the glass of the bottle. Then, we try to define a normal direction to the surface. But here we run into trouble, there does not exists one unique normal direction to this surface. Thus, we are unable to define the outside or equivalently the inside space of a Klein bottle. It is called a non-orientable surface. It is an example of a truly one-sided wall.