In/out equivalent to left/right "chirality"

Question

Apologies if this is off-topic, but we're having a problem over on English Language with this question, and I thought you guys might be able to help.

Basically it's a matter of topology. We know the word chirality, for distinguishing between two things that are either identical in all respects, or differ only in their left/righthandedness.

Is it meaningful in mathematics/theoretical physics to distinguish between "inside-outness" and "outside-outness" in the same way? If so, is there a word analogous to chirality to convey that distinction?

I ask here because I know there is debate about the "shape" of the space-time continuum (which I don't fully understand), so it seems at least possible to me that some hypothetical frameworks which are mathematically describable might actually be verbalised using a word such as we seek.

Answer 1

At first I misread the aim of your question, but I'd rather not waste what I had so I'll use my initial write-up as a segue. This isn't the most mathematically careful of an exposition but I hope it's useful conceptually.

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Of course topology has notion of interior and exterior (much like home designers). Problem is, the topologist's notion of them usually refers to merely belonging to a set of points or not (exception: the Jordan curve theorem) rather than how the outside is divided, so it is actually more of the differential geometer's notion we want. If a hypersurface living inside an external space, like a 2D surface living in three-dimensional Euclidean space, is nice enough (e.g. connected, closed), it will divide all points not specifically on the surface into an inside or outside. In more topological terms we can say that the complement (set of all points not on the surface) is partitioned into two connected components - in our Euclidean example, one component has infinite measure (the outside) and one finite (the inside). Although this can be applied to some manifolds that are embedded or immersed in larger ambient spaces - the ambient space providing the substance for the "inside" and "outside" - there are two important things to keep in mind about this idea:

• Some manifolds, with or without boundary, are non-orientable (e.g. Klein bottle) which means that if they are immersed, you can take a point on the hypersurface and an outward pointing normal vector (i.e. a "direction" out from the point at a right angle to the surface), then transport the point and the associated perpendicular vector around the surface back to where it was but with the normal pointing the other way, like an ant crawling on a Möbius strip and ending up ostensibly on the other "side." Here there is no inside or outside.
• Manifolds - and I should have made this a point earlier, but these are the abstractions mathematicians use for discussing curved and non-Euclidean spaces - are supposed to be viewed intrinsically. While differential geometers make a distinction between local phenomena (roughly, behavior on an infinitessimal scale, e.g. curvature) and global phenomena (features of the space taken as a whole, e.g. its genus, or roughly how many "holes" or "handles" it has), there is also something to be said about the distinction between viewing a manifold intrinsically versus "from the outside." Most people (in my experience) conceive of space as Euclidean and curved spaces as (essentially) hypersurfaces embedded in Euclidean space, but the ambient space, to mathematicians, is completely superfluous and is considered separate or additional structure. Hell, non-Riemannian manifolds can't even be embedded in Euclidean spaces because of the incompatible metric signatures.

Since manifolds are intrinsic creatures, being apathetic about anything outside of themselves, when they are immersed, their points do not naturally have a preference for inside or outside and so it doesn't technically make sense to say a space is inside-out. However, if we arbitrarily equip it with such a preference (that is, if it is an immersed oreintable manifold of codimension 1 and we give it a normal vector either pointing always outside or always inside) then we can justifiably speak about whether any continuous deformation we care to consider either preserves or reverses the normals (I think this can be made more rigorous with the Gauss map or something).

In this way, we see that "inside-out" is always relative to some arbitrarily specified initial data that is extraneous to the object itself (or perhaps relative to an outside invariant, like the person a shirt is on, where the tags and stitchlines could intuitively represent inward normals, or the impressed stripes on a tube sock relative to a foot). Unfortunately, I'm not aware if there's standard terminology for this initial data or relative position - like "orientation" or "evertivity" - so I don't think there's a true answer to your question at this time.

Yet such a thing has already been studied in the particular case of a sphere (here is a twenty minute video explanation with visual demonstrations, and here is the Wikipedia article on Smale's paradox). The action of turning the sphere inside out is called sphere eversion. I'm not aware if eversion has been broadened in study to objects other than the sphere, but the mathematical groundwork needed to parse and derive further facts if we wanted to looks like it is already set in modern topology and differential geometry.

Answer 2

Chirality presupposes some sort of motion or transformation (of the objects together with the surrounding space) to superimpose the left and right handed objects upon each other. It can be either a mirror-reflection that in one step exchanges every point with its chiral counterpart, or a smooth physical motion (requiring one extra dimension, such as placing the 2-d surface of one hand against the 2-d surface of the other by moving it through three dimensional space).

Inside and outside are perfectly sensible concepts mathematically but it is not that common (in particular there is nothing as generic as "inside" and "outside" themselves that would tend to apply when those words do) to have a transformation that exchanges the two. That is true even if one is allowed to distort distances by stretching and twisting the space unevenly. There are operations like inversion in elementary geometry that, loosely speaking, exchange inside and outside of a circle, but these really are switching the in/outside of a punctured circle, or an annulus, which in this context is an after-the-fact complication of the original geometric figure to make the transformation "work" (*), and not a manifestation of any fundamental interchangeability of the two sides.

The transformational picture makes some sense for local orientation. A curve in the plane, a surface in space, and generally an n-dimensional surface in (n+1)-dimensional space, will locally (that is, near any point of the surface, or in small regions) divide the space into a "left" and a "right" side relative to the surface. If there is a global division of the space into parts inside/outside the surface, this will coincide with a notion of local left/right orientation for the surface (one direction will point IN and the other OUT, near any point). If you place this picture into a space one dimension higher, such as drawing a circle in the plane and placing the plane into a 3-dimensional surrounding space, then the local arrows that point "left" and "right" in the plane at each point of the circle, can be rotated in the additional dimension until they are exchanged. There is another type of transformation between left and right in the case of non-orientable surfaces like the Moebius strip. There, you can switch left and right by following particular types of loops through any given point, like an ant walking the length of the Moebius strip, and finding that when it returns the notion of left and right (or up and down, from ant's point of view) have been reversed.

One difference between these transformations and the case of chiral hands, gloves and socks in 3-dimensional space is that for the familiar physical objects, the space and the objects are directly moved by the transformation. In the case of local orientations and higher dimensional placements of the object, the placements and orientations are a sort of extra structure added to the object at each point, and the transformations are moving the structure around while keeping the object in place.

notes:

• There was a much upvoted earlier discussion here, of topological inside-out transformations and how to formalize them: Why can you turn clothing right-side-out?

• (*) by making the transformation "work" I mean that the inside and outside generally are topologically different, so no transformation of the space can make them truly equivalent without some modification of the figure. For example, the outside of a circle or a sphere looks (topologically, i.e. with stretching allowed) like the inside with a solid piece removed. After removing a similar piece from the inside component it becomes topologically the same as the outside, creating the potential to use an inversion or some sort of continuous flow of the space through one higher dimension, to interexchange the two regions. Such a transformation cannot be a distance-preserving operation like a mirror reflection, so even if you allow this sort of modification of the original figure, there is no perfect analogy with the intuitive motivation of an orientation-reversing, but distance-preserving, correspondence between left and right handed gloves, socks, shoes or sugar molecules.