Why is it that when we view an image in a mirror parallel to the object's front, we are aware of a left-right reversal but not of a top-bottom reversal, though the mirror does not optically discriminate between the two axes? An elaborate exposition of ideas formerly proposed by Navon (1987) is presented. It notes that the real source of the perception of reversal is not the one suggested by naive cause attribution: Whereas the mirror does not discriminate between frontal axes, frontal encounters do. Mirror images thus appear to be reversed along a planar axis - the horizontal one, in our ecology - because they suggest a prototypical frontal encounter, yet deviate from it in a lawful manner. The deviation is due to the fact that, unlike in mirror viewing, in any frontal encounter, homologue intrinsic sides are opposite to each other only along one planar axis. The distinguished axis, namely that particular axis whose homologue poles ARE opposite to each other in a prototypical frontal encounter within a given ecology, would constitute there the mirror-invariant axis. A generalization to other coordinate systems, other encounters and other ecologies follows. Finally, other accounts of the issue are critically reviewed.
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AUTHORS' RATIONALE FOR SOLICITING MULTIPLE COMMENTARY: This target paper is neither about any classic controversy nor about some hot issue. It is rather about a commonplace phenomenon. Hopefully, however, it might have some moral about scientific explanations in general.
When we view an image in a mirror parallel to the object's front, we are aware of a left-right reversal but not of a top-bottom reversal, yet the mirror does not optically discriminate between the two axes. A number of attempts have been made to solve this puzzle, yet no consensus has been reached. Quite interestingly though, it is often claimed that the solution is obvious. As that approach is common to people holding different views on the issue despite the differences, one cannot help being puzzled as to why the disagreement tends to be sometimes dismissed as much ado about practically nothing.
Since the issue is characterized by diversity of views on a quite narrow scope, it might profit from a discussion held at one forum in close time proximity among a number of participants coming from various disciplines. Such an open discussion could hopefully lead to a sharper characterization of the controversy. But even if it does not converge at something, it might at least help to make the conceptual gaps more transparent. In any event, readers who will find the puzzle worth their while may also find it thought-provoking.
Because the organization of the paper is somewhat unusual, it might be helpful before reading to look at the table of contents presented below. For those readers who would have just limited patience for the subject, I suggest five optional exit points. Those who are interested just in the account of the mirror reversal of the observer's image in our ecology may skip sections III-VII; those who also wonder whether mirror reversal is restricted to our ecology may skip only sections IV-VII; those who are further interested in mirror reversal in a variety of cases may skip only sections V-VII; those who would also like to read about object reflection may skip only sections VI-VII; true completeness lovers may choose to read section VI as well. Should I add that reading the whole paper is another option?!
I. The issue II. A proposed view III. An extension to other encounters and other ecologies IV. But is the account universal? V. And how about object reflection? VI. An appendix for true completeness lovers VII. But isn't that an instance of a more general conceptual rule? VIII. And what have other people said? IX. And finally
1. Both mirror vision and spatial cognition are discussed in a number of delightful books and sober articles. Notwithstanding the ample common wisdom cumulated, there is still one topic related to both fields that outstands in its obstinate defiance of becoming uncontroversial - the old puzzle of left-right mirror reversal. The puzzle is intriguing not only in its own right. It can also serve as an illustration of how a grain of real puzzle can agglomerate into it a number of other, quite mundane issues. Accordingly, let us first delineate the scope of the real puzzle.
2. There is nothing puzzling about most mirror reflections. The optics of the mirror dictates a reversal along the axis perpendicular to the mirror's plane (see, e.g., Gardner, 1964, 1990), namely the one that coincides with the observer's midline body axis when she faces the mirror plane. Hence, it is evident how the mirror is able to produce different reflections for different spatial relationships it happens to be put at with respect to the mirrored object. For example, depending on how we approach the mirror, we can get an optical reversal of our back and chest, or of our right shoulder and left shoulder, or of our head and feet. Also, when the mirror plane is being set perpendicular to a flat surface, we can get anything drawn on that surface to be reflected in any desired manner depending on where exactly we place the mirror - next to the figure's top, figure's right etc. Thus, some cases of left-right mirror reversal have a simple optical explanation. So, where is the puzzle?
3. Left-right mirror reversal IS puzzling when it appears to occur in a frontal view, namely when we view in a mirror the image of that facet of an object which is about PARALLEL to the mirror plane (e.g., the front of our body when we stand in front of the mirror). We observe, once we notice it or once it is pointed to us, that the image does not look like the actual facet does but rather like a left-right reversed version of it (namely, what would be seen of it in a flat mirror placed at its right or at its left). Thus, the mirror appears to discriminate between the two planar axes of the mirrored facet. If we were as curious as were many people, including celebrated thinkers (for an illuminating review of the history of interest in the puzzle, see Gregory, 1997), we would then try to find the source of that selective reversal. The natural scapegoat, namely the mirror itself, would be sooner or later exonerated (but see Haig, 1993, and the following rebuttals by Morris, 1993, and Takano, 1998), once it is realized that it just faithfully reproduces the environmental directions of all the planar sides of a surface which is parallel to its plane (for example, the upper side - up, the lower side - down, the easternmost side - eastwards and the westernmost side - westwards, in the case that the mirrored surface is erect and aligned with the east-west environmental axis).
4. Thus, there appears to be a contradiction between two facts - one phenomenal, the other physical:
(F1) We are aware of a left-right reversal but not of a top-bottom reversal.
(F2) The mirror does not optically discriminate between the left-right axis and the top-bottom one.
5. Since it follows from (F2) that we cannot ascribe (F1) to the optics of the mirror, what can we ascribe it to? Quite a few scholars found that puzzling, and the intuitive solutions - elusive. On the other hand, some other scholars seem to think that the solution is obvious, hence the issue must be trivial. Ironically, since there is no undisputed solution to the mirror reversal puzzle, different people exhibit equal degrees of conviction, each in his own "obvious" solution.
6. Undaunted by the risk of sounding either wrong or obvious, quite a few authors nonetheless published attempts to solve this puzzle (for a review of those, see e.g., Gregory, 1997; Ittelson, Mowafy, & Magid, 1991; Morris, 1993; Navon, 1987; Takano, 1998). Some of those attempts differ only in nuances, some - more substantively. Part of the diversity results from different stances on the level of account being required - conceptualization, specification of conditions, or causal analysis. Another part is due to some variability in focus - the puzzle proper or the cognitive apparatus involved in the perception, or deduction, of reversal. Several motifs recur in most accounts: the linguistic convention - due to ambiguity of direction terms in natural language - to refer to a mirror distortion as left-right reversal (e.g., Bennet, 1970; Corballis, 1989; Corballis & Beale, 1976; Gardner, 1964, 1990), the symmetry of our body about its vertical axis (e.g., Block, 1974; Corballis, 2000; Corballis & Beale, 1976; Gardner, 1964, 1990; Ittelson, 1993; Ittelson et al, 1991; Morris, 1993), the high frequency of locomotion in which the vertical body axis stays erect (e.g., Pears, 1952), the mediation of operations of mental imagery or orientational frame rotation (e.g., Pears, 1952; Shepard & Hurwitz, 1984; Takano, 1998), the prior rotation of the observer or of the mirrored object (e.g., Gregory, 1987, 1997). The relevance of those motifs cannot be denied. The question is whether any of them is either necessary or sufficient. In my view, none is.
7. To reason that, I first present my own view of the puzzle, including what I regard as its adequate solution (originally presented in Navon, 1987), and then try to answer briefly several putative questions. The discussion of other accounts thus follows at the end.
8. To get a better intuition for the source of mirror reversal, it might be useful to consider the following thought experiment. Imagine that it was possible for you to generate a perfect copy of yourself whenever you wanted it, in no time at all, and put the copy at any position or posture you wanted.
9. Now suppose you have just copied yourself and made your copy to stand facing you, side-by-side with a full size flat mirror (see Figure 1). If your copy faced you in the typical manner people frontally encounter each other in our world, you would see two abutting images that are identical except for a left-right inversion.
Figure 1. An illustration for the thought experiment (a view from above)
10. That phenomenon results from two facts. The first is obviously the optical law mentioned above, namely that the mirror faithfully reproduces all sides of a surface which is parallel to its plane in the same environmental directions.
11. The second fact is somewhat less obvious. You decided to put your copy at a posture of a typical frontal encounter with yourself. That, in our world, entails cross-lateral inversion, namely that the object-centred RIGHT side of the encountered object is opposite to your own LEFT side and vice versa. Just the opposite of how your mirror image "encounters" you.
12. Of course, if you had rather decided to put your copy at a different posture (say, upside down but still with its face facing your front), the abutting images you would have seen would have been top-down reversed with each other rather than left-right reversed. However, such a decision would serve no good cause. If the objective was to judge how veracious the mirror is in telling you what you would see if you at the moment encountered yourself, the more reasonable option would be to pick the prototypical encounter.
13. In sum, the mirror reversal in that thought experiment is due to the combination of an optical law and a human decision. The decision, in turn, is based on typicality.
14. In what way is mirror reversal in a more familiar environment any different? You actually cannot have a copy of yourself right next to the mirror to enable a quick comparison, but you can harness either your imagery or your perceptual schemata to make up for that distressful handicap. You rather compare the mirror image with what you expect to see had you managed to frontally encounter yourself. Since you never had any chance to do just that (though watching yourself in the video comes pretty close), you base your expectancy on the prototypical frontal encounter in our ecology and some other, non-perceptual knowledge (e.g., I wear my wristwatch on my left wrist).
15. So far, simple enough - so it seems. However, it is not that simple. There is another premise underlying your expectancy: You judge that the mirror image should be compared with a typical frontal encounter. You cannot base that premise on postulating that the mirror itself is frontally encountered in the strict sense, since the mirror plane does not have sides of its own: However you rotate it in plane, you would still see the same thing. Neither can that premise be due to a perfect match with any schema of encounter, since there is no geometrical encounter in which all homologue planar sides are opposite to each other. Thus, you must derive that premise from your feeling that, based on what you see, the mirror image appears to be what you would see in a frontal encounter more than in any other encounter. A schema of the view afforded by a prototypical frontal encounter is activated and a correction is then noted: A typical frontal encounter indeed - so it appears, yet the left and right are inversely related with what the schema anticipates.
16. The next step is naive cause attribution. The inverse RELATION detected by our perceptual system or deduced by our cognitive processes is construed as a reversal OPERATION imputed to the mirror. That attribution is, however, wrong. The mirror does not optically reverse anything but the axis perpendicular to it. The perception of mirror reversal on the plane parallel to its surface is due to the interface of an optical law and a conceptual habit. But what is the habit due to? It must be associated with the knowledge - embodied in our perceptual system - that:
T1. In our ecology, cross-lateral inversion along the horizontal axis is the rule in prototypical frontal encounters.
Thus, strictly speaking left-right mirror reversal is just a naive non-sequitur. It is, however, based on absolutely valid premises. Objectively, the spatial relationship between an object and its mirror image differs from the spatial relationship between two objects in a prototypical frontal encounter. Like any other difference, there is no sense in "blaming" it on either of the compared entities. One might wonder, though, which of them requires further explanation, and then note with some amusement that it is not the optical law: The mirror, for obvious reasons, does not discriminate between frontal axes. Ironically, prototypical frontal encounters do. Hence, if there is still anything yet to be elucidated, it is the cross-lateral inversion mentioned in T1.
17. Yet, T1 is just a particular instance of a more general geometric fact:
T2. Some cross-lateral inversion is the rule in ALL frontal encounters in ALL conceivable ecologies.
In other words, it is impossible to have a frontal encounter - with any angle of encounter (namely, angular disparity between the corresponding axes of the encountered objects) - in which all homologue intrinsic (namely, object-centred) sides of the fronts of both objects are opposite to each other, as they are in a mirror image (Footnote 1). That is true not only for sides we have verbal labels for (namely, top, bottom, right and left). Generally, in any frontal encounter, points on the fronts of both objects having the same intrinsic polar coordinates are not opposite to each other, except for those located along one planar axis (out of the infinite number that exist in a polar coordinate system). The particular axis which is exceptional in the modal type of frontal encounter within a given ecology would naturally have some special status in the schema of frontal encounters of residents of that ecology. Let it be termed: the Distinguished axis. But even more generally:
T3. Some cross-lateral inversion is the rule in ALL possible encounters between ANY two homologue object facets in ALL conceivable ecologies.
In other words, it is impossible to have any encounter of homologue facets in which all homologue intrinsic sides of the facets being faced are opposite to each other, as they are in a mirror image. For example, the rule holds also for an encounter between the top of one object and the top of another one.
Now, since the mirror image suggests the view afforded by a prototypical frontal encounter, the deviation noted would be a reversal ABOUT the distinguished axis and ALONG the planar one orthogonal with it. In other words, the axis along which both sides are opposite to each other in the modal angle of frontal encounter within a given ecology would be perceived in that ecology as the mirror-invariant axis. By the same token, the axis orthogonal with it would be perceived there as the one being reversed by the mirror.
We might have difficulties in analytically thinking of this general, abstract formulation of the issue, because we often get confused by the way coordinate systems are confounded in our environment. To unconfound those, it might be advisable to take a trip to an unknown land. Let us call it Clockland.
18. Consider, for example, the eight-character "clocks" in Figure 2. Two clock faces can be interfaced so that only opposite characters along the vertical axis are identical with each other (see Figure 2a), or only those along the positive diagonal are identical with each other (see Figure 2b) etc., but that correspondence cannot be obtained for more than one planar axis at a time. Hence, if in Clockland clocks could see and if they sensed their own vertical axis as the one passing through the squares, yet typically they frontally encountered each other so that only opposite characters along the orthogonal axis - namely, the horizontal one - were identical with each other (see Figure 2c), then their prototypical notion of frontal encounter would be characterized by VERTICAL inversion, namely one's filled square would be opposite to the other's hollow square, and vice versa. When such a Clocklander sees himself in the mirror, he would ask himself: In what way does what I see now differ from what I would have seen had I frontally encountered myself? His answer must be: By reversal along the VERTICAL axis. Thus, in general the axis perceived to be reversed by the mirror is the same axis that is cross-laterally inverted in a prototypical encounter (Footnote 2). In our ecology, that axis happens to be the horizontal one.
Figure 2. Three possible frontal encounters between clocks (a side view). [dashed lines are drawn to indicate homologue side-pairs].
19. Having said that, would it be very enlightening to further explore what gives rise to mirror reversal within our ecology? This is primarily a matter of intellectual taste. Notwithstanding, let me mention one reason why I think the worth of such an exploration would be limited. Gravitation surely constrains human locomotion in a way that must considerably affect the distribution of angles of encounter. Human anatomy clearly adds further constraints. However, for arguing that those are crucial for the explanation of mirror reversal, one would have to imagine how it would be like in an ecology that differed from ours in aspects like gravitation, locomotion etc.
20. Imagine, for example, an ecology with no locomotion at all. When A and B want to meet, they send to each other real-time video displays. The angle of view is of course arbitrary, but whatever the modal angle is, it would give rise to a certain discrepancy between a mirror image and whatever one expects to see in "real-life", namely in a video meeting. Would that be left-right reversal? Not necessarily. It depends, of course, on what the modal angle is.
21. Now, how much does this help us to understand the puzzle of mirror reversal? In my view, not much. The crux of the puzzle is that there is a reversal of a planar axis in the first place. The fact that the reversal just so happens to be located along the horizontal axis is interesting yet not inherent. What IS inherent is that a mirror image suggests a prototypical encounter, yet deviates from it in a lawful manner. It must do so, because unlike in a mirror reflection, some cross-lateral inversion is the rule in real frontal encounters. The rest is subsidiary.
22. Does the perception of mirror reversal depend on prior movement of the mirror or of the mirrored object? Not at all. When one stands in front of a mirror, she is aware of a left-right reversal regardless of whether she has gotten there by having rotated about her vertical body axis or by having crawled up from the floor.
23. Does the perception of mirror reversal at least depend on some information about prior movement? When we view mirror images and notice that they reverse left and right, we seldom note anything about prior movements of us or of another mirrored object with respect to the mirror. Furthermore, we do not need to note. What we do note and certainly do need to note is just how objects would have to look when frontally encountered.
24. Do we in turn need to IMAGINE rotation to find out how objects would have to look when frontally encountered? At least in the case of an upright familiar object, the deviation is glaring, so that no mediating operation would be necessary. In the case of the observer herself, there is actually nothing to mentally rotate, since the observer cannot see her own front but in the mirror (and her memory of her real, non-reflected front is typically poor). In the case of an unfamiliar object, since the observer cannot directly see its facet which faces the mirror, there is nothing for her to rotate unless the appearance of the facet is deduced from its mirror image. But to do that she has to have some mental model of mirror perception. This circularity brings up a more general question: What function such an operation is expected to serve anyhow? Mental rotation is used to generate the image of a stimulus as would be seen from a different perspective (e.g., Cohen & Kubovy, 1993; Cooper & Shepard, 1973; Hinton, 1979; Shepard & Hurwitz, 1984). However, in mirror viewing the stimulus is the mirror image, and what it mirrors (namely, the facet facing the mirror) is hidden from the observer. Hence, the outcome of any mental rotation would have to be compared with memory. Why then bother to rotate? Trusting memory from the outset seems like a more reasonable option. In any event, since reversal detection which does not require mental rotation is not impossible, the onus of proof is on those who surmise the necessity of mental rotation.
25. Does the perception of reversal depend on having bilateral symmetry of the observer (or of any other mirrored object)? Not quite. A congenitally asymmetrical cripple would most probably have the same intuition about mirror reversal as we do. If anything, he would notice the reversal most readily, just as we notice more easily that an R is reflected than that an N (let alone A) is. Consider the Clocklanders in any panel of Figure 2. They do not have any problem in identifying the aligned axis, as well as in noticing the reversal along all other frontal axes, since they are totally asymmetrical. Had they been asymmetrical just ALONG one axis and symmetrical ABOUT it, like the hypothetical creatures in Figure 3, they would have had some difficulty in noticing reversal - at least when the axis being aligned in the frontal encounter was the asymmetrical one (see Figure 3a). To be sure, the inversion exists, but it is camouflaged by symmetry. To detect it, some defect in symmetry (like a single local scratch on the clock's face) would have to be introduced. In any event, the identity of the frontal axis which the mirror is blamed for reversing depends on the modal angle of frontal encounters, regardless of front symmetry.
Figure 3. Three possible frontal encounters between laterally symmetrical creatures (a side view). Dashed lines are drawn to indicate homologue side-pairs.
26. Would the above account fail to explain mirror reversal in the case of an observer who is not upright? Not at all, as long as the mirror plane is parallel to the observer's front. Although most often frontal encounters occur in the upright position, some frontal encounters (not of the kind one is liable to forget) occur when both parties are perpendicular to the line of gravity. Anyhow, the ecological frequency of the current body orientation is immaterial. For example, even when one hangs from the ceiling in front of a mirror, he does not feel that his mirror image is vertically reversed with respect to how he expects to see himself at the moment. True, the image is seen to be inverted with respect to gravity, but so is the observer. Since his mirrored head is opposite to his real head, not to his real feet, just as it is in a frontal encounter taking place between two erect persons, he has no reason to blame the mirror for reversing up and down. Thus, mirror reversal is defined with respect to intrinsic coordinates (namely, object-centred in the case of mirrored objects, egocentric in the case of the observer's reflection). The fact that just one of the intrinsic axes - the one labeled vertical - correlates highly with some environmental axis - the axis of gravity, whereas the others do not, does not in itself determine the axis being reversed. It might just indirectly contribute to that, by affecting the distribution of encounter angles.
27. Would the account fail in the case of a mirror image viewed in an atypical mirror (see Gardner, 1964, pp. 4-6)? No, the account would be still valid, mutatis mutandis. Consider the reflection in a silver spoon. When you find there an upside down image, you still note the relationship between what the image is (due to the optics of a small concave mirror surface) and what you expect to see in a frontal encounter. If you compared it with an atypical frontal encounter in which tops face bottoms, you would still note that left and right are reversed. If you rather compared it with the prototypical frontal encounter, you would rather note that the vertical axis is inverted. It is my hunch that most observers would choose the latter comparison. However, that empirical point is of secondary importance. What is of primary importance is rather that regardless of the particular comparison chosen, that choice dictates the way in which we claim the mirror "distorts" the image. And the choice probably follows from the specific schema that is being activated.
28. As an analogy, think of a case in which an observer who is familiar with the Roman alphabet but not with the Greek one sees for the first time the letter Sigma. Whether she interprets it as a W rotated 90 deg clockwise or as an M rotated 90 deg counterclockwise depends on which mental schema, the W schema or the M schema, is more primed at the time. Whichever it is, the perceived "distortion" follows from the relationship between the sensory image and the expectancy generated by the schema being activated.
29. And if that analogy seems too remote, consider the case of our shadow projected on a wall we frontally face by a source of light behind our back. We do not normally feel that our shadow reverses left and right. If asked about it, we will probably be at a loss. A plausible reason is that our shadow does not contain enough clues to suggest a frontal encounter to our perceptual system. For what we see, it might as well be a front-to-back encounter. When there is no overriding schema, there is no clear sense of distortion.
30. Finally, can the account be applied to mirror images of other observer postures? Suppose, for example, that you stand on a flat mirror. If you bend your head a little, you will see yourself upside down. That is the optical consequence of your vertical body axis being in that case the one which is perpendicular to the mirror plane (see paragraph 2). However, if you now focus on the facet of your body which is parallel to the mirror plane, namely your heels, you will notice that they look left-right reversed in the mirror. An even more interesting case is when the facet of your body which is parallel to the mirror plane is your right (or left) side. Not surprisingly, in that case the axis which is perpendicular to the mirror, hence optically reversed, is aligned with the transverse axis of your body. That reversal is not puzzling, but in that case as well there appears to be a reversal of the sort that IS puzzling. Not surprisingly either, it occurs along the horizontal axis of the facet parallel to the mirror, namely of your right side (as explained in Footnote 2). In other words, in that case a non-optical reversal appears to reverse the front and back of your body. You would see it, if you turn just your head towards the mirror. Thus, left-right reversal of the facet facing the mirror plane does not necessarily appear to reverse the left and right of the observer's body (Footnote 3) This observation is quite compatible with the present account but hard to reconcile with a bilateral symmetry account (see paragraph 25).
31. Is the explanation fundamentally different for an object other than the observer? Not quite. To realize that, imagine that your instant copy-and-send capacity assumed in the above thought experiment (see paragraph 8) applied also to objects you have with you. Further imagine that you wore a T-shirt with the letter R printed on it, held a cup in your right hand and hugged a one-eyed teddy bear with your left hand. The mirror image would still be left-right reversed with respect to the copy you had sent to abut it - a copy that aside of yourself includes the R, the cup, the teddy bear and all other objects you had with you. The same would be true, had your mental imagery or perceptual schemata substituted for the copy-and-send capacity in case you lacked one.
32. The account can be phrased similarly for all objects, including our own body: When the front of an object is placed in front of the mirror, its environmental sides are reproduced accurately, hence the side of its mirror image that is opposite to the observer's left corresponds to the side which is actually her left. That would not be true of a real frontal encounter of the kind suggested by the mirror image. As stated in T1 and explained in paragraph 17, once labels are assigned to the intrinsic sides of the fronts of two dorsiventral objects in any consistent manner, arbitrary or non-arbitrary as the case may be, some cross-lateral inversion is entailed. In our ecology, the inversion is along the horizontal dimension.
33. Note that the teddy bear is different in a way from both the cup and the R. In the case of an object that does not have a well-specified front (e.g., a cup) or of a two-dimensional stimulus (e.g., the letter R), we often denote their sides in egocentric terms (see, e.g., Levelt, 1984). However, regardless of the verbal labels we apply to the sides of the stimulus, our perceptual system notes that the side which normally is opposite to our left is seen by the observer of its mirror image to be opposite to her right. In other words, what the perceptual schema of a particular object notes as its side that is opposite to our left during a frontal encounter need not correspond to any verbal label. Thus, however the sides of a stimulus are being assigned - in an egocentric manner or an object-centred one - the sides of its mirror image might violate the expectancies derived from the activated schema, provided that there is some noticed asymmetry. True, one cannot very well talk of right and left before a conceptual distinction between different coordinate systems - egocentric and object-centred - has matured or developed. However, even an observer who does not have in memory any well-defined concepts of right and left - either because she is preverbal or because her spatial skills are not well-learned - might presumably feel something odd about mirror images, though she obviously would have a hard time verbalizing that feeling.
34. The account stated in paragraph 32 applies regardless of how sides are being assigned, even when the definition of a front is not unique. In the case of a flat figure such as an R, the front is probably taken to be the facet of the object on which the R is mounted (e.g., a certain face of a 3X5 card). In the case of an object whose front is not specified a priori (e.g., a cup, a match, a peephole view of our index finger pointing rightwards), the front may be taken ad hoc to be whichever facet is at the time parallel to the mirror (Footnote 4). Yet in that case, mirror reversal seems to be a tough inference rather than an immediate perception. Violation of expectancies must be readily available only when derived from an existing schema.
35. Nonetheless, there is some difference between reflection of the observer and other reflections. The observer never sees himself upside down in a flat mirror which is parallel to his front. On the other hand, he might occasionally see an upside down object (e.g., another person) in such a mirror. That occurs in the case that the object facing the mirror plane is also upside down with respect to the observer. Note, however, that in that case as well, we observe the same phenomenon: The mirror image is left-right reversed with what the observer would have seen had the object faced him in the same way it faced the mirror. And the reason is the same one stated in paragraph 32, considering that the kind of frontal encounter suggested by the mirror image is in that case an encounter with an upside down object.
36. To realize that indeed the account of object reflection follows from the same basic difference between real frontal encounters and "encounters" with a mirror image (discussed in paragraphs 16 and 17), let us make another trip to Clockland. It is tempting to assume that when a Clocklander encounters another Clocklander, he naturally tends to attribute to it an analogous coordinate system: Since I have a filled square at my top - he would think - the other guy must take his top to be where his filled square is; since I have a filled circle at my right, he must take his right to be where his filled circle is, etc. However, a Clocklander (or his system of ingrained schemata) does not have to know anything, nor to conjecture anything, about the intrinsic coordinate system of any other Clocklander he might encounter, or of any other encountered creature for that matter, to be able to perceive it. A purely egocentric approach strategy would suffice. When the Clocklander encounters a different creature (see Figure 4), he could just assign as its front the facet of it that has been most frequently encountered, and his idea of the appearance of its front would be derived from the modal angle of that encounter: What is most often opposite to my top - he would say - is the creature's saturated red spot; what is most often opposite to my right is its saturated green spot, etc.
Figure 4. A prototypical frontal encounter between a Clocklander and some other creature (a side view).
37. Now, what would the mirror image of that creature's front be in the eyes of the Clocklander? In other words, what would the Clocklander see when both his front and the creature's front were parallel to the mirror plane? Clearly, regardless of whichever the creature's front is and whichever is its typical appearance, an encounter with its mirror image would be different. But different in what respect?
38. Here is where geometry has nothing more to say. Geometry has an answer only as long as the state is definite. This is not the case here, since the number of angles that the front of the creature can subtend with any arbitrarily chosen axis of the observer's (say, the vertical axis) is infinite. Hence, some conceptual decision has to be made here: What would "mirror image" be taken to mean? To impose some parsimony on the observer's conception of his perceptual world, he would presumably choose the angle which is modal in his encounters with other CLOCKLANDERS. For example, if it was the one illustrated in Figure 2c, namely 180 deg, hence he felt that his own mirror image was top-bottom reversed, then by fiat the mirror image of any other creature as well would be DEFINED as the one reversing top and bottom as seen from the observer's egocentric perspective. In other words, the observer would say: Since I know that what is opposite to my top in my mirror image is actually the bottom of my front as seen by any other Clocklander, so it must be with any other creature, hence its mirror image is what I see when it faces the mirror with its top side down and bottom side up.
39. However, encounters with other persons in OUR ecology are typically like the one illustrated in Figure 2a, hence we define the mirror image of other creatures (or objects) as the one that reverses right and left from our perspective.
40. Completeness lovers may frown on how in the last section we deduced the consequence in spite of bypassing T1 and T2 (see paragraphs 16 and 17). It would appear, then, that mirror reversal of objects is after all conceptually different from mirror reversal of our own portrait. However, if we did make a commitment to a specific definition of frontal encounters between other creatures - arbitrary as it may be - we would be able to see how it implicates mirror reversal in the same way that it did for encounters between Clocklanders.
41. The Clocklander might assign to the creature's front creature-centred sides. But how would he do that? For example, to which side of the creature from Figure 4 would he assign the saturated red spot that typically is opposite to his own top? Since the creature cannot inform the Clocklander of its favorite choice of sides, creature-centred sides have to be assigned arbitrarily. However, the most parsimonious way to do that is to draw the analogy to the prototypical encounter with other Clocklanders. For example, the Clocklander from Figure 2c would say: Since in my normal encounters with other Clocklanders, what is opposite to my top is the other Clocklander's bottom side, let me postulate that that is also the case with the other creature, namely that the saturated red spot is at its bottom side.
42. From then on, the story proceeds just as it does in paragraph 18. How would the Clocklander conceive of two creatures of the same type that frontally encounter each other? By the same token that T2 is true for Clocklanders, an analogous assertion can be made by a Clocklander on other creatures:
T4. When other creatures frontally encounter each other, points on their fronts having the same intrinsic polar coordinates are not opposite to each other, except for those located along one planar axis.
Without any evidence to the contrary, the Clocklander from Figure 2c would naturally postulate that the distinguished axis is the vertical one. Thus, he would expect two frontally encountered creatures of the type illustrated in the colored part of Figure 4 to most often have one's red spot opposite to the other's pink one. That, however, would not be the case with an "encounter" of a creature with its mirror image. Hence, the Clocklander would naturally blame the mirror for reversing the creature's image along the same axis that it does for his own image. QED.
43. Thus, as a rule mirror images appear to be reversed along a planar axis - the horizontal one, in our ecology - because they suggest a prototypical encounter, yet deviate from it in a lawful manner.
44. The fact that mirror images are perceived to be left-right reversed might seem suspiciously similar to another phenomenon - our tendency to say that members of any pair of enantiomorphs (namely, pair of objects that look alike except for an inversion of some frontal axis; e.g., your pair of shoes, Kant's pair of gloves, etc.) are related by left-right inversion. The truth is, of course, that which axis is inverted depends on how the members are positioned with respect to each other. When a left shoe is put back-to-back with a right shoe, they are rather seen to be vertically inverted. So, why do people conceive of enantiomorphs as left-right reversed? Probably because they represent them in memory in their canonic or prototypical spatial layout (Footnote 5). Shoes, for instance, usually abut each other laterally. Hence, when asked about shoes outside of a specific context, one retrieves the prototypical layout and answers accordingly.
45. Sometimes, however, a typicality account may not be good enough. Consider an ink stamp. We conceive of it as left-right reversed. But is that necessarily so? It depends on how we view it. From a certain angle it indeed looks left-right reversed. But turn the stamp 180 deg, and it will be seen to be top-bottom reversed. Of course, we might consider the latter to be a 180 deg rotation of the former. On the other hand, we could with equal justification choose to consider the former to be a 180 deg rotation of the latter. What is then the reason for our preference to interpret the ink stamp as left-right reversed? That can hardly be explained by the frequency of our encounters with ink stamps. We seldom encounter them anyhow. Thus, in this case we must be using our general notion of frontal encounters to impart directions on direction-wise ambiguous enantiomorphs. That notion dictates that the vertical axis of the front of the stamp must be the axis which is not reversed. Hence, left-right reversal is not perceived but rather imposed by fiat (Navon, 1987).
46. How relevant is this neo-Kantian notion to the mirror puzzle? Not much. The instances above illustrate how prone people are to attribute left-right inversion to images or concepts retrieved from memory. However, the mirror puzzle does not concern what we SAY about mirrors in general, but rather how we PERCEIVE specific mirror images. Mirror images are real stimuli, hence when the object is set next to a mirror plane, no retrieval of other mirror images is necessary. Whereas the mirror image is SEEN to reverse the object along its axis which is perpendicular to the mirror plane, whichever that axis might be, the mirror image of the object facet which is parallel to the mirror plane just LOOKS left-right reversed. While the former is due just to an optical operation, the latter appears puzzling because the mirror does not optically discriminate between axes. On the other hand, the phenomenon need not be ascribed to a conceptual or linguistic habit, since it is a simple consequence of an implicit comparison between the input image and the memory of a frontal encounter with the object.
47. It remains to examine in what way the view presented here differs from other ones. That should emerge from the following critical review. I made it brief, since the authors may elect to present their own views in commentaries, probably better than I can do it. The interested reader is referred to a fairly recent extensive review by Takano (1998).
48. The case of mirror reversal is one of those cases in which the phenomenon constrains its explanation to a considerable extent. Most of its students realize that the fault is to be sought outside of the mirror. They typically point to some difference between the right-left axis and the top-bottom one (Footnote 6). They still disagree, however, on what exactly that difference is. The main motifs recurring in most accounts are listed in paragraph 6. In the next five paragraphs I address those motifs as put by several early authors (following arguments initially presented in Navon, 1987). The subsequent discussion expands on three more recent accounts.
49. Gardner (1964, 1990) claimed that it is convenient for us, because we are bilaterally symmetric creatures, to describe the mirror world as if it reversed left and right. It is unclear how bilateral symmetry serves to establish the cognition of left-right reversal. If anything, it must impede the detection of reversal: An asymmetric creature would presumably have discovered the reversal more easily (see paragraph 25). In any event, symmetry is certainly not necessary for the reversal to be perceived.
50. Even more disturbing is the argument that left-right reversal is just a linguistic convenience (Bennett, 1970), or "a convention in the use of words" (Gardner, 1964, p. 24). Note that by using two or three mirrors perpendicular to each other, or by using curved mirrors, one can obtain reflections that are seen to reverse both axes, no axis at all, or just the vertical axis (see illustrations in Gardner, 1964, pp. 4-6). Hence, the sort of reversal perceived evidently depends on the actual image. The choice of words to describe the reversal is not at all arbitrary, once specific intrinsic directions have been designated by those words.
51. Pears (1952) argued that our conviction that the mirror reverses left and right is contingent both on the fact that we are symmetrical only bilaterally, and on the fact that we turn about the vertical body axis. He based this conclusion on the claim that the reversal can be determined by a test that he calls "getting into the portrait": If you imagined yourself going behind the mirror and facing its "back", you would find that although your body fits a portrait superimposed on the outline of your mirror image, left and right are nonetheless reversed. In Pears' view, this is so because you imagine yourself moving about your vertical body axis, and because you are bilaterally symmetrical. The bilateral symmetry is critical in Pears' account merely for the fit with the portrait, which is used just as a metaphor. Hence, that symmetry seems immaterial for the reversal per se. The notion of rotation through the third dimension is more relevant, since to give rise to left-right reversal, the rotation must be about the vertical axis. However, an explanation that resorts to such an operation is appealing only as long as that operation is natural from the point of view of our system of locomotion. It is clear, though, that our perception of left-right reversal is as compelling when we lie down in front of the mirror as when we stand up (see paragraph 26). Furthermore, whereas the motion of people relative to each other indeed most often preserves the parallel relationship between their vertical axes, this is not generally true of our dynamic relationships with many objects. Still, the mirror images of those objects appear also to reverse sides. Thus, our intuition about mirror reversal cannot be adequately attributed to the greater frequency of movements about the vertical axis. Finally, a psychological model for the source of the mirror intuition that would be based on Pears' argument would assume that to infer the reversal we imagine rotation, namely we mimic it by some mental process. However, noting the deviation of the image from what the schema of frontal encounters prescribes will suffice. Hence, invoking a mediating mental operation of rotation transgresses the rule of parsimony. Our everyday movement about our vertical body axis is clearly related with our notion of frontal encounters, but that is a far cry from positing a mental operation of imaginary rotation in depth to account for our intuition of mirror reversal (cf. paragraph 24).
52. Block (1974) pointed out that the question of mirror reversal is equivocal, and that different authors may have construed it differently. Then Block offered the explanation that whereas up and down are defined extrinsically (say, with respect to gravity), right and left are defined intrinsically. To prove the point, he suggested a thought experiment involving a person who looks at a mirror positioned at the center of the earth through a long hole drilled across the earth. Block claimed that the person's image would be up-down reversed. However, this is not how the image would APPEAR to that person. The vertical axis whose directions are perceived to be preserved by the mirror is NOT defined by gravitation, as evidenced by the indifference of the apparent reversal to the posture of the observer (see paragraph 26). Block then argued that even an intrinsically defined top-down axis was qualitatively different from the left-right axis, since the former could relate to anatomical body features. However, would the intuition of a congenitally asymmetrical cripple about mirror reversal be any different from ours? Would it be different in a world in which all people are crippled in exactly the same manner? It probably would not, because the cripple's mirror image would be left-right reversed with respect to what she expects on the basis of encounters with other people (cf. paragraph 25).
53. Thomas (1980) attributed the left-right reversal in our conception of mirror images of objects to the tacit assumption that the mirror is placed at the object's side rather than above or below it. This explanation does not, however, solve the puzzle of why a mirror that actually faces the observer appears to reverse her own left and right.
54. Takano (1998) proposed a multi-process model that applies different processes to different reflection states. Each process compares a rotated representational frame for the mirror image with a perceptual frame for whatever is being mirrored. Takano distinguished between three processes corresponding to the following three cases: (i) the mirror image of the observer's front, (ii) the mirror image of an object front, (iii) the mirror image of the observer (or any object it carries) with her "shoulder against the mirror".
55. Takano noted correctly that the third type of left-right reversal is due to the optics of the mirror, since in that case the horizontal body axis is aligned with the axis perpendicular to the mirror plane (see paragraph 2). Of course, the object could be positioned so that the axis perpendicular to the mirror plane would be aligned with any of the object's axes. In this way we could "reverse" the left and right sides of an object in much the same way that we could "reverse" its top and bottom (see paragraph 26). There are numerous ways to do that. Would it be desirable to have for each of them a separate process? Probably not, considering the parsimonious alternative - that of noting (for any object posture) the alignment with the axis perpendicular to the mirror plane. But more important, that case falls outside the scope of the puzzle. As recalled (and see also paragraph 30), the issue that is really puzzling is: Why do we blame the mirror for reversing the left and right of the object's facet which is PARALLEL to the mirror plane?
56. As for the other distinction made by Takano, namely the one between the observer's reflection and object reflection, it was concluded in sections V and VI above that it may be gratuitous. True, there are some differences between reflection of the observer's front and reflection of any other object (see paragraph 33). Those differences lead to different consequences - the main one being the egocentric orientation invariance of the observer's mirror image compared to the myriad of orientations that object reflection can take. However, the difference in consequences does not require multiplicity of mediating processes. Consider the scene described in paragraph 30. It is composed of components that might be each related to a different frame of reference (for a discussion of multiple-frame scenes, see Navon, 1992). However, that seems of little relevance to the perception of mirror reversal. It would take some substantive evidence to be convinced that the reversals of the observer, the R, the cup and the teddy bear are not produced by one process. Furthermore, even if there were separate mediating processes, both would be compatible with the same basic logic stated in paragraph 32. It seems that one puzzle requires no more than one solution.
57. Gregory (1987, 1997) attributed mirror reversal to rotation of the observer or of the object about their vertical axis that typically takes place before the mirror image is viewed. As argued above (in paragraph 22; also see Ittelson, 1993; Takano, 1998), prior rotation is neither necessary for left-right mirror reversal nor common to most cases of mirror viewing. Furthermore, to paraphrase Takano (1998), the observer may turn forever about her vertical axis and still fail to find the view of her own face as seen by an outside observer (then presumably buy herself a mirror to make up for that distressful fact). Finally, Gregory sends us to Flatland, Edwin Abbott's imaginary land of 2-d creatures having a 2-d conception of space, to see how critical is rotation through the third dimension for mirror reversal. As Gregory correctly points out, there are no mirror images in Flatland, since even if there was a mirror there, no object would ever be mirrored in it. Sure enough, if there were no mirror images, there would neither be mirror reversals nor a mirror puzzle. Does that suggest that rotation in depth is the key to the mirror puzzle? Not a bit more than the absence of color constancy in a land of congenitally blind inhabitants would suggest that the possession of the sense of vision "explains" color constancy. In much the same way that the existence of vision is just the precondition for having any color phenomenon at all, so the existence of depth is the precondition for having any mirror phenomenon at all. Yet the existence of either depth or rotation in depth clearly does not tell us why the mirror appears to reverse anything, let alone why that reversal occurs along the right-left axis.
58. The fault in this kind of logic could be better appreciated, if we consider another peculiar environment: Pipeland. Pipelanders are 3-d creatures like you and me that, to their dismay, are all stacked on the top of each other in a very narrow pipe. Think now of a Pipelander that looks at her reflection in a mirror posted on the inner surface of the pipe. Since in Pipeland there cannot be any frontal encounters aside of mirror reflections, she would not sense anything unusual, and would certainly have no reason to claim that the mirror reversed left and right. That suggests that mirror reversal depends on having an idea of what a non-mirror frontal encounter is and what it would be like. That the ability to move about one's vertical axis is not critical is evident, once it is realized that nothing would change by letting Pipelanders move in unison (namely, as a whole pipe). That neither is that ability critical for having frontal encounters in the first place becomes clear, once it is realized that if Pipelanders lived in two quiescent, transparent pipes, they would develop a sense of frontal encounter that depends on the modal angle spanned by the two pipes. Most important for our concern, the Pipeland fable would be devoid of any moral, if it was assumed that Pipelanders filled the pipe so compactly that any mirror reflection was disabled. That could not be seriously taken as a support for the claim that mirror reversal is explained by frontal encounters, since obviously no mirror reversal could exist when no mirror images do.
59. Notwithstanding, the Flatland story is pertinent in some way. Note that aside of being deprived of mirror images, Flatland is deprived of frontal encounters as well. Flatlanders can "meet" each other only in the same limited sense that residents of different opera boxes do, or better yet, that animated figures on the same computer display do, namely by occupying locations that neighbor each other on some common planar axis in their flat habitat. Thus, if you ever needed a reminder that having depth as a dimension is a prerequisite for all events which take place in depth - e.g., frontal encounters, mirror reflections - here it is.
60. Finally, Tabata & Okuda (2000) recently claimed that the mirror is seen to reverse left and right, because the other two orthogonal axes are already determined by external asymmetry. In their words: "the top-bottom and front-back axes of the car are evident from its shape. We first align these directions (at least in our imagination) with those of our bodies, and then find the left and right of the car. The left and right of a mirror image are determined similarly by treating the image like a real object" (p. 171). One can only wonder what would be required to modify this account, if the mirrored object was symmetrical about its horizontal axis rather than about its vertical axis (as would be, for example, a futuristic aerial vehicle with a round front having just a small peephole on its left). One might also wonder why this account - that ascribes to the observer mental operations such as "align" or "treat" - is claimed by its originators to explain mirror reversal at line level "without recourse to psychological processes". Furthermore, the account neglects what actually is a cardinal question: Why is the mirror image taken by the mind of the observer to be a frontal view of a real object? The shadow example (see paragraph 27) suggests that such a surmise is not at all taken for granted, hence that the mind must do something to suggest the frontal view interpretation. But more important, the account postulates that the left side of the mirror from the observer's perspective must be the right side of the mirror from the image's perspective. That clearly appears to be a truism in our ecology, yet the video-meeting example (see paragraph 20) suggests that actually the truth in that apparent truism depends on our concept of frontal encounters: If the modal angle of a frontal encounter had been 180 deg, observers would have rather been guided by a diametrically opposed premise.
61. In sum, much of the diversity between accounts is due to difference in foci: They stress, and sometimes overstress, different aspects. In other words, the debate over the source of mirror reversal does not seem to be a rivalry between mutually exclusive conceptual alternatives. It rather resembles more an argument on how to bake a good apple pie: All the accounts contain some appropriate ingredients, but some are less complete or less parsimonious than others.
62. Yet to judge completeness and parsimony we first have to determine what is the crux of the puzzle. Once it is realized that there IS a real difference between the mirror image and any possible frontal encounter, and that the difference is not radially symmetric but really DOES discriminate between orthogonal frontal axes, the puzzle starts looking less puzzling, since the apparent contradiction between its two limbs (see paragraph 4) is resolved.
63. True, there are some questions left to be answered. The major one is why the discrimination is against the horizontal axis rather the vertical one. The answer seems to be evident: In our ecology the horizontal axis is the one typically inverted in frontal encounters. Why is that? This is a further question, the answer to which is relevant neither to psychology nor to physics. More important, it is gratuitous for removing the special perplexity induced by the puzzle, since it points to a possible cause after the apparent contradiction has been resolved. However, since this has been the most controversial issue in the debate over mirror reversal, it is worth realizing that one cannot expect to answer it just by pointing to evident facts about our ecology, such as our bilateral symmetry, typical locomotion, linguistic habits, etc. Since in our ecology all of these are confounded, one cannot help resorting to thought experiments to gain a better perspective on the role of those factors. What might appear like intellectual gymnastics is, thus, an inevitable ingredient of the apple pie. Another major question is why a mirror image suggests a frontal encounter. The simple answer is: Because by most sensory clues this appears to be a virtual frontal encounter. Which sensory clues? That seems interesting but subsidiary. Finally, it remains to be explained how the discrepancy is noticed and how one identifies it as a left-right reversal. Various process models may be proposed and possibly examined. But again, while validation of any of those models would gratify our scientific inquisitiveness, it is not required to resolve any apparent contradiction.
64. The methodical demons in our minds might not be laid at rest until they place this account in a single taxonomic box - psychological, geometrical, or physical. Unfortunately, our cognitive need to classify everything cannot be fully satiated here. Is there a sense in asking whether the account of the perception of an upper-case sigma as a rotated M is psychological, geometrical or physical? Both that account and the account of the mirror left-right reversal must take into consideration a mind trying to make sense of sensory data emanating from physical objects that are spatially related to each other as well as to the eyes in ways that are described by geometry.
65. Yet there is a bit more. The least that can be said is that the observer plays a critical role in this account. Well, it also does in a number of other theories that are not considered psychological at all, the special theory of relativity for one. However, unlike the abstract observer conceived by Einstein, here the observer is quite real. Furthermore, she does more than just observing. She must use her perceptual knowledge to infer a relation. For example, when she infers - consciously or unconsciously - that the image of her front in the mirror is left-right reversed, she does not do that by identifying a spatial relationship between two sensed objects. She has to invoke a mental image or an anticipation to compare the mirror image with. And what is invoked is not determined by physics. It rather depends on what the mirror image is presumed to be or is taken to simulate, namely a prototypical encounter. That clearly requires some information processing. So, psychology cannot be banished from the scene after all.
I am indebted to useful comments made by Marian Annett, Michael Corballis, Stevan Harnad, Marcus Munafo', Roger Watt and two anonymous referees, as well as to the technical help of my assistant Adam Shavit. I also owe some debt to Yohtaro Takano, whose paper published in 1998 spurred me dusting off my interest in the subject, which resulted in this target article.
. One might be tempted to "explain" that geometric fact by appeal to the set of 3-d transformations that map a prototypical front-to-back encounter (namely, one in which all the homologue intrinsic axes of both encountered objects are aligned in exactly the same way with the environmental coordinates) to any of the possible frontal encounters. That, however, would be gratuitous, since the particular transformations do not matter. In frontal encounters, by definition the front-back axis of one of the objects is reversed with respect to the front-back axis of the other one. That in itself entails T2. The reason is that any two object-centred coordinate systems, CS1 and CS2, can be made to coincide (namely, be positioned in space so that all corresponding axes of both systems are aligned and their poles point to the same environmental directions), but there is no way in which just one axis of one of them (say, the front-back axis of CS1) is inverted yet coincidence with respect to all other axes is preserved.
. Note that it follows from T3 that the analysis is not restricted to the observer's front but applies to other facets as well, provided that they happen to be positioned parallel to the mirror plane. For example, suppose the "clock face" in Figure 2a was considered by Clocklanders to constitute the appearance of their right facets (namely, what would be seen by an observer who looks at them from the right). A Clocklander can see his own right side in the mirror by turning rightwards just his pivot mainspring knob which houses his vision apparatus. He might then ask himself: In what way does what I see now differ from what I would have seen had I been able to actually view my right? His answer must be: By reversal of left and right, since now I see the filled circle at the left of my visual field, whereas another Clocklander viewing my right would see the filled circle at the right of his visual field.
. This appears quite confusing. It seems that the source of the confusion is that when focusing on the appearance of a certain facet, we tend to assign to it coordinates of its own that are not necessarily compatible with the overall object-centred coordinates. Imagine, for example, that you face the back of another person. If you attended to its appearance (e.g., the drawing on the back of a T shirt the person is wearing), you would probably interpret its right side as the one that is opposite to your left. However, that is exactly where the LEFT side of the person's body is. Clearly, what mirrors appear to reverse is the horizontal axis of the facet facing the mirror (see paragraph 16), not sides determined with respect to overall object-centred coordinates (when such exist).
. As stated in T3 (see paragraph 17) and illustrated in Footnote 2, cross-lateral inversion is bound to obtain regardless of which object facet happens to be parallel to the mirror.
. Note that a prototypical layout is defined as a spatial relationship between two related objects as seen by the observer, whereas a prototypical frontal encounter is used here predominantly to denote a relationship between the observer and another object. What the observer's schema represents is the typical view of the front of that object.
. Yet, some were led astray by the confound between different coordinate systems. For example, relating to the optical reversal along the axis perpendicular to the mirror plane as reversal of front and back might make one gloss over the fact that left-right reversal is not puzzling at all in the case that the body axis perpendicular to the mirror plane is the transverse axis rather the midline axis. Further misunderstandings of that sort are discussed in paragraphs 26 and 30.
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