Representations of surfaces by physical models

When a surface is represented by a physical model in some opaque material (for instance paper) and a point is marked in pencil on that model, it is only visible from one of the sides of the model (notice that near each point on the sheet there is no difficulty in identifying both sides). It is then possible to mark a point with a different coloured pencil on the other side at the same position of the model. The question of knowing whether what has been marked is considered on the mathematical object being represented (the surface) as a set of two different points is a legitimate one:

(Place the mouse near the figure or move the figure with it.)

Notice that the question we are asking is not necessarily associated to the existence of thickness of the paper - which in a physical model will always exist -  and has a sense even if we can abstract from that thickness.

Before we consider this question, let us imagine two other types of models:

  1. the material is opaque, but the representation of a point is made by a tiny hole made with a sharp needle; the hole thus visible from both sides suggests in this case that what is being represented does not consist of two different points;

  2. the material is transparent and extremely absorbent and the mark is made with a tiny drop of ink; it is immediately visible from both sides and the the conclusion is the same: what is being represented is not seen as two distinct points.

So, which is the good model? What is "correct"? For mathematicians, in the surface "at that position" is there only one or are there two distinct points?

From this point of view, it is really the last two models which more faithfully express the notion of surface: in each "position" only one point is considered on the surface being represented, not two. However in other contexts mathematicians are sometimes led to consider what they call a double covering of the surface and there to consider what can been described as "two points for each position"; the formally precise description is beyond the level at which the text is being written at the moment, but an intuitive idea can be obtained imagining the surface covered "on both sides" (near each point) by a layer of ink (supposed without thickness). That covering of ink can be connected, formed out of just one piece as is the case of the Möbius strip in usual space, which cannot be painted "continuously" with two colours; or it can be formed out of two separate pieces, two sheets of different colours which "do not touch each other", as is the case of the cylinder in usual space.