CNRS-EHESS, Institut Jean Nicod
The Möbius strip is, for almost everybody, a piece of magic. If one is asked to predict the result of cutting it along its median line (one which runs parallel to its borders), one will probably infer that two strips are to be expected. (As a matter of fact, one gets one other Möbius strip, doubly twisted). If one is asked to predict the result of cutting our original strip along any other line which runs parallel to its borders, the expectations are less clear, but surely one will not easily predict that the outcome will be two strips, a and b, a being twisted once (like the original strip) and b being twisted twice; neither will it be immediately clear that b is twice as long as a.
The border of a Möbius strip is, even upon reflection, weird. One reasons from the principle that if there is a border, this normally implies that the two sides separated by the border are such that one cannot reach one of them from the other without crossing the border. Once you are on one side, it seems, the border bounds you to that side. But of course the Möbius strip is exactly what contravenes this principle.
Exposure to some topological science does not, by itself, correct one's propensity to misjudge some topological facts. Those who understand and master simple topological tasks, such as the detection of the topological equivalence between a sphere and a cube, or of a donut and a cup of coffee, are unwilling to accept that a cylindrical surface (such as the portion of a tube) and a disc with an internal portion removed (such as the old long-playing records) are topologically equivalent.
Double toruses are surprising too. The equivalence between the object represented in fig. 1
and that represented in fig. 2
is all but obvious even to trained mathematicians.
In spite of this an many other "bizarre" topological facts being widely known (their analysis often constitutes a standard exercise in topology textbooks) an explanation of why they appear bizarre and counterintuitive is still forthcoming. I am not claiming that there is one single such explanation, but I surmise that a constellation of facts points out the existence of an intuitive topology that is in the way and prevents us from appreciating correct topological equivalencies (or non-equivalencies). The case resembles that of intuitive physics (McCloskey 1983). Subjects' understanding of Newtonian physics is blocked by their implicit relying on an intuitive physics which in many cases is at odd with Newtonian physics. For instance, intuitive predictions of trajectories significantly depart from the correct prediction. Subjects required to draw the trajectory of a ball dropped from a vehicle in motion make the incorrect prediction that the ball will fall straight down, thereby failing to recognise that the ball will move forward and land ahead of the point where it was released. McCloskey has proposed a perceptual explanation of this incorrect prediction. The movement of the ball is perceptually analysed in the context of the frame of reference constituted by the moving vehicle, relative to which the ball, once dropped, moves indeed along the vertical direction. This description of the behaviour of the ball is then incorrectly extrapolated to the reference frame constituted by the ground because of the perceptual dominance of the local (moving vehicle) frame.
Not every intuitive physical principle seems to be explainable in purely perceptual terms, though (for instance, the principle that heavy bodies will fall faster than light bodies). Still, the existence of suchlike principles is, to some extent, accepted; from a methodological point of view we can for the time being accept that their explanation is piecemeal. In what follows I'll try to present some alternative approaches, mainly concentrating on the case of the pince-nez.
I'd like first of all to clear the ground from a general objection. Understanding of bizarre topological equivalencies, says the objection, is poor because of a complexity or length factor. The transformation that interlocks the two handles of the pince-nez, leading from fig. 1 to fig. 2, is "long" - no surprise the equivalence is all but evident (length could be measured here in terms of the number of intermediate steps, say phases of mental transformations of an object). The objection can be disarmed by considering some intuitively clear topological equivalence which requires nevertheless a "long" series of intermediate steps. The equivalence between a donut and a cup of coffee is available in spite of the length of the series of intermediate steps - the starting and ending point of the series could be quite far apart, the shapes of the donut and of the cup very bizarre.
Neither is the counter-intuitiveness of the equivalence between figures 1 and 2 to be assimilated to the counter-intuitiveness of very complex physical statements, such as those of quantum physics. A more appropriate analogue for the latter would be topological equivalencies in higher-dimensional spaces, which go beyond the limits of intuitive representation. On the other hand, the equivalence between figures 1 and 2 does not require mastering of superior mathematics, and the demonstration of the equivalence, albeit longish, is run in familiar three-dimensional space.
The perceptual account concentrates not on properties of the represented objects, but on properties of the representation itself, the perceived configuration. The account develops some suggestions from Gestalt theory. Gestalt theorists emphasise the figure-ground articulation of perceived configurations. Some portions of a perceived scene, due to their intrinsic wholeness, are perceived as figure, and are delineated against a background which is perceived as completing itself behind the figure. Now, the visual boundary that separates figure from ground is "oriented": it belongs to the figure and not to the ground. The configuration of fig. 1 has three such boundaries (one is "external" and two are "internal"), and so does that of figure 2. But in figure 2 the "same" boundaries appear to have been intertwined in a way that is incompatible with their having not been violated. In order to have one of the internal boundary go through the other, one boundary - or so it seems - must have given way, must have been opened and then closed. This motivates a judgement about the opening and closing of the object possessing the boundary, thereby implying a violation of the constraints on topological equivalencies (no cutting, no gluing). Obviously the judgement is mistaken, because the fate of the configurational boundary is to some extent independent from that of the object boundary. But this independence is overcome in the judging process.
In an alternative accont, the analysis of fig. 1 and 2 does not focus on the configuration, but on the object themselves. The pince-nez are decomposed into primitive components or geons (geometric ions, following Biederman's (1987, 1990) Recognition By Components guidelines, possibly emended along a suggestion of Casati and Varzi 1995 to the effect that toruses are allowed among the geons) as follows:
The suggestion is that one now concentrates on the two donuts only. As it is clear that there is no equivalence between the left and the right configurations in fig. 4,
and there is a feeling that the middle cylinder of fig. 3 couldn't make any difference, one is led to judge that the difference between figure 1 and 2 reduces to the difference between the left and the right configuration in figure 4. As these configurations are not equivalent (you do have to cut and glue in order to get any of them from the other), one infers to the non-equivalence of figures 1 and 2. The mistake is in the unauthorised neglect of the bridge between the two donuts, over which the interlocking is performed.
One quick explanation for the counter-intuitiveness of equivalencies such as that of fig. 1 is that the metrical understanding of the situation dominates over the topological constraints. Rigidity (the relative preservation of metrical properties between parts of an object) can in some cases act antagonistically relative to topological properties. For instance, a scattered object whose parts move as one can be considered as more unitary that a connected object whose parts move in great independence, such as a quantity of water (Casati and Varzi, forthcoming).
Thus, the open pince-nez and the interlocked pince-nez are construed as metrically rather similar to one another, and any intervention to achieve the topological equivalence without altering the metric as potentially disruptive of the topology.
There is a subtle synergy between the Gestalt, the non-holistic and the metric account.
The Gestalt account is reinforced by the metric account, for the figural boundaries can be selected by some metric factors such as similarity of size or closeness of parts of the object which act as Gestalt factors. This in turn entails a non-holistic view of the object, and a focusing on parts taken in isolation.
On the other hand, the correct assessment of the role of our sense for metric equivalencies is not even prima facie clear. It might just be a side factor. Metrics should dominate also in the case of the transformation of a donut into a cup of coffee, but there it does not impede the recognition of the equivalence.
A bit more promising seems the attempt at explaining the failure in recognising the pince-nez equivalence in terms of features of the complement of the object. This account (Casati and Varzi 1994), combines some of the insights of the gestalt account and of ordinary topology. It withdraws the attention from the holed object, and suggests instead to concentrate on the geometric (including topological) properties of the void that intrudes the object. For instance, the pince-nez shaped object of fig. 1 has two such intrusions, circular shaped. Holes, thus conceived, are bestowed the dignity of quasi-physical objects, although they have negligible physical properties of their own. As they are cognitively accepted as full-fledged entities, let us call them reified holes.
Reified holes have a number of interesting features of their own. They offer a way of cognitively simplifying the topological analysis of an object. They can be easily perceived, counted and computed. We have in the norm an easy grip on their shapes, sizes and volumes; easier, in most cases, than the grasp we have of the topological complexity of the surface of the objects hosting them.
But they are in the way when one tries to appreciate this very topological complexity. Given their thing-like character, they have boundaries. These boundaries correspond to the boundaries of the Gestalt account above, but only geometrically. Their orientation is now two sided: they bound partly the object and partly the hole. The situation resembles perceptually what we have when two ordinary objects come into contact - the boundary dividing them is shared by them - and differs significantly from the previous Gestalt construal of fig. 1, in which the solid object only - not the hole - is the proprietor of the boundary.
Now, the boundaries of the reified holes are undoubtedly being violated in the passage from the configuration of fig. 1 to that of fig. 2 - for the same reasons as are violated the boundaries of the rings in the passage from the left to the right configuration of fig. 4. In this the reified hole account aligns with the non-holistic account, for the boundary of the hole and that of the object coincide; only, there is this new character, the boundary of the hole, that exists on its own next to the boundary of the object. It is also clear why the "bridge" that was overlooked in the non-holistic construal is irrelevant: it does not enter the representation of the two holes and of their relations.
An explanation of the incorrect analysis of the topological equivalencies including double toruses might make appeal to different factors, but a simple solution is available when one construes the cognitive representation of holed objects as making ineliminable reference to holes. Holes, under this interpretation, are empty pockets of space that, being taken seriously by the cognitive system, are credited with boundaries of their own and assigned a thing-like character.
To be sure, the fact that our cognitive systems take voids seriously is not the end of the story. It accounts for some of the most common mistakes, but not necessarily for the whole range of bad judgements. Other intuitive principles, such as the one about the borders of a Möbius strip stated at the beginning of the paper, can account for some other class of mistakes.
It is now possible to delineate a psychological research program: individuate as many simple and counterintuitive topological equivalencies (or nonequivalences) as possible; taxonomize them; look for possible intuitive topological principles that hinder the understanding of these equivalencies.
The pedagogical moral of this story is that the topology teacher should make as little reference as possible to intuitive models that can interfere with the correct appreciation of the topological situation to be explained. Some existing alternative models, such as oriented computer screens (in which the geometry of the movements of the mouse is constrained be the orientations of the borders), already exist and are used. Their advantage is that they do not suggest the creation of troublesome holes, which only emerge when the properties of surfaces are embedded in a three-dimensional model.
Biederman I., 1990, Higher-Level Vision, D. N. Osherson, S. M. Kosslyn, and J. M. Hollerbach (eds.), An Invitation to Cognitive Science. Volume 2: Visual Cognition and Action, Cambridge, MA, and London: MIT Press, pp. 1-36.
Casati, R. and Varzi, A.C., 1994, Holes and other Superficialities, Cambridge, Mass.: MIT Press.
Casati, R., and Varzi, A.C., 1995, "Basic Issues in Spatial Representation", Proceedings of the 2nd World Conference on the Fundamentals of Artificial Intelligence 1995, Paris: Angkor, 63-72.
Casati, R., and Varzi, A.C., 1999, Parts and Places: The Structures of Spatial Representation. Cambridge, Mass.: MIT Press.
McCloskey, M., "Intuitive Physics", Scientific American, 249, 114-122.