Perceptualand M o ~ oSkillr, r 1991, 73, 107-114. O Perceptual and Motor Slulls 1991
SOME THOUGHTS O N IMPOSSIBLE FIGURES
'B'"
JOSEP V. MOLINS Barcelona, Spain Summary.-Based on an artistic analysis of the construction of "impossible figures," their structure and the origin of the perceptual effects they induce are discussed, including the use of a basic indeterminate space of intersection, penetration of indeterminate solids, and symmetry. These and other points are illustrated by 17 progressively more complex figures constructed by the author.
I am a plastic artist and have recently been involved in the study, recreation, and manipulation of illusions and ambiguities of perspective Whde investigating the problems inherent in the two-dimensional representauon of space, based on the analysis of the basic structure of so-called "impossible figures," I have made certain discoveries that I consider deserve special attention by scientists interested in perception. I have consulted all the bibliographic sources to which I have access, finding only a few references to similar phenomena. The object of this account is simply to describe the effects observed, with the conviction that they are of interest, without preempting further in-depth research. My observations can be grouped into three areas: the first concerns intersection of impossible figures, the second impossible penetrations, and the third syrnmetry in such figures. Alpha Effect or the Space of Intersection in Impossible Figures Methodologically, Fig. 1 represents an impossible strip, which I present in its simplest form (it can be considered as the basic unit), to which most of the more complex figures may be reduced. I t allows the coherent unification of two inverse perspectives by means of a plane. Observation of Fig. 1 shows that the plane surface marked "a" has an ambivalent coherence, since it can be integrated just as easily into either of the two inverse perspectives. While carrying out my free compositions I became aware of strong sensations of depth, or floating, which led me to the observation that such sensations always appear when a new space with its own coordinates is 'This aper is not in the form of a standard scientific psychology paper; however, it re resents a great Bea~of thought and investigation about perceptual issues by a person highly slufed in the representation of illusions. Even though there may be technical problems in the language growing out of the individuahty of the author's ex erience, we believe that reading the paper and viewing the illustrations will be highly stimuLting. We have tried to leave the ori inal manuscript as intact as possible, allowing the reader to enjoy the integriry of rhe autaor's roduction .-Em. I' acknowledge the support and interest of the Museu de la Cihncia de La Fundacib Caixa de Pensions I de Barcelona.. In addition, I express my appreciation to the teacher who first !wakened my interest in this work, JosC Maria Yturralde. Address correspondence to J. V. Molins, C/.Torrent de I'OUa 37, 08012 Barcelona, Spain.
108
J. V. MOLINS
defined inside the portion of the figure marked "a," that is, with the appearance in "a" of any new volume or figure, independently of the inverse perspectives that define the impossible figure. Figs. 2 and 4 show the depth effect strongly, which appears to be more strongly defined the simpler the figure. Figs. 2 (Schroeder ladder) and 3 show how the depth effect can be extended to so-called reversible figures. Figs. 5 , 6, and 7 [the Penrose and Penrose triangle (1958)l extend the effect to the internal space of more complex figures. Fig. 8 deserves special mention: in this case, an illusion very similar to the preceding illusions takes place, but its analysis seems to be more complex. My interpretation is that here we have an impossible figure (the body entering inversely above the black plane) in which just one of its parts is directional, together with a neutral plane with which it operates in the same way as in the previous examples; that is, it gives directionality (coordinates) to the plane. Directionality, being neutral, plays the role of the ambivalent space "a." So that, as in Figs. 2 and 4, directionality inverts the figure-ground relationship, since the additional bodies change the black space, which was originally figure, into the ground (background). I t is interesting to note the analogy between this effect and the mechanism of abstract reasoning applied to the double negative where and So that in this case what is impossible or absurd in the combination of two inverses seems to become even more distinct than a 'normal' figure with the appearance of the space defined by the new body in 'la."
Impossible Penetrations Another kind of impossible figure is represented in Figs. 9 and 10 where apparently large volumes are contained inside narrow gaps. Considering the extreme simplicity, it is surprising that until now nobody appears to have noticed this illusory possibility offered by classical perspective. I consider that in the interpretation of these impossible penetrations, we should refer to a classical problem in the representation of space on a flat surface: the calculation of apparent size. One of the coherent solutions of the impossibdity of Fig. 9 would be to imagine the transversal slot being of huge size but seen by the viewer at an extremely oblique angle of vision. This revives the Euclidean idea rejected by Renaissance perspective, according to which apparent size is directly related to the angle of vision. Fig. 11 tends to corroborate this: here objects of very different sizes have very similar apparent projections on to the same plane but the apparent angle of vision is different.
THOUGHTS ON IMPOSSIBLE FIGURES
FIGS. 1, 2, 3, 4. Examples of impossible figures
J. V. MOLINS
FIGS.5 , 6 , 7, 8. Examples of impossible figures
THOUGHTS ON IMPOSSIBLE FIGURES
FIGS.9, 10, 11, 12.
Examples of impossible figures
J. V. MOLINS
15
FIGS.13, 14, 15. Examples of impossible figures
T H O U G H T S ON IMPOSSIBLE FIGURES
113
Also related is the effect shown in Fig. 12, wherein the tendencytowards grouping of the transversal bars is much stronger than the separation effect produced by the large cube apparently placed between them. Another speculative interpretation of the illusion aims at an approximation of a linguistic analogy. We can imagine that the body penetrating the slot is not solid, but a plane representation of a body (e.g., a piece of wood or paper on which a cube has been drawn). Then we would have a kind of logical metaproblem, the plane representation of a plane representation, which produces the illusionary effect of reality.
Symmetry in Impossible Figures At first glance it may seem axiomatic that impossible figures, just like any other representation of a solid on a plane surface, give figures which are symmetrical with respect to an axis. However, the analysis of these symmetries makes it clear that it is conceptually correct to talk about two impossible triangles, and so on, as can be seen in Figs. 13, 14, and 15. I n Fig. 13 I return to the strip, the simplest impossible shape, to point out that the inverses fuse when we perform an imaginary rotation. I n symmetrical figures this imaginary rotation would be clockwise in the figure on the right of the symmetry axis and counterclockwise in the figure on the left. Perceptually, one can note that clockwise rotations in volumetric figures produce a movement of the figure into the paper, while counterclockwise turns move them outwards, i.e., towards the observer. I n Fig. 15 we see the movements which the clockwise and counterclockwise triangles of Fig. 14 would make were they not closed off and made into impossible figures: the clockwise triangle is continuously (and infinitely) rotating inwards, while the counterclockwise triangle does the reverse. We could perform the same operation on any sirmlar figure. I n closing, I would like to repeat that the intention of this paper is solely to describe the observations I have made, without wishing to preempt more rigorous subsequent work. A theory of impossible figures has proven to be a difficult undertaking and so far lacks a universally accepted Formulation. Richard Gregory (1968) semantically resolved the issue when he pointed out that any representation in a plane of a three-dimensional object is in fact an illusion. This statement is certainly true; however, the human effort involved in this representation comprises the whole history of perspective, and the millions of 'unreal' images to which we are exposed every day make it important to study representation itself, including all the illusions which it produces by its very nature. For this reason, I am convinced that a proper study will need to be interdisciplinary, since it relates to subjects such as geometry, and ultimately, even to the theory of knowledge and semeiotics. REFERENCES GREGORY, R. L. Visual illusions. Scienhfic American, 1968, 219, 66-76. PENROSE,L. S., & PENROSE,R. Impossible objects: a special type of illusion. Britirh Journal of Psychology, 1958, 58, 49-138.
Accepted July 2, 1391
J , V, hlOLINS
APPENDIX'
-Although u7e are not able to reproduce the striking colors of the original prints, we find the effects.-Eds. Author's more complex figures present highly interesting
SOME THOUGHTS O N IMPOSSIBLE FIGURES
'B'"
JOSEP V. MOLINS Barcelona, Spain Summary.-Based on an artistic analysis of the construction of "impossible figures," their structure and the origin of the perceptual effects they induce are discussed, including the use of a basic indeterminate space of intersection, penetration of indeterminate solids, and symmetry. These and other points are illustrated by 17 progressively more complex figures constructed by the author.
I am a plastic artist and have recently been involved in the study, recreation, and manipulation of illusions and ambiguities of perspective Whde investigating the problems inherent in the two-dimensional representauon of space, based on the analysis of the basic structure of so-called "impossible figures," I have made certain discoveries that I consider deserve special attention by scientists interested in perception. I have consulted all the bibliographic sources to which I have access, finding only a few references to similar phenomena. The object of this account is simply to describe the effects observed, with the conviction that they are of interest, without preempting further in-depth research. My observations can be grouped into three areas: the first concerns intersection of impossible figures, the second impossible penetrations, and the third syrnmetry in such figures. Alpha Effect or the Space of Intersection in Impossible Figures Methodologically, Fig. 1 represents an impossible strip, which I present in its simplest form (it can be considered as the basic unit), to which most of the more complex figures may be reduced. I t allows the coherent unification of two inverse perspectives by means of a plane. Observation of Fig. 1 shows that the plane surface marked "a" has an ambivalent coherence, since it can be integrated just as easily into either of the two inverse perspectives. While carrying out my free compositions I became aware of strong sensations of depth, or floating, which led me to the observation that such sensations always appear when a new space with its own coordinates is 'This aper is not in the form of a standard scientific psychology paper; however, it re resents a great Bea~of thought and investigation about perceptual issues by a person highly slufed in the representation of illusions. Even though there may be technical problems in the language growing out of the individuahty of the author's ex erience, we believe that reading the paper and viewing the illustrations will be highly stimuLting. We have tried to leave the ori inal manuscript as intact as possible, allowing the reader to enjoy the integriry of rhe autaor's roduction .-Em. I' acknowledge the support and interest of the Museu de la Cihncia de La Fundacib Caixa de Pensions I de Barcelona.. In addition, I express my appreciation to the teacher who first !wakened my interest in this work, JosC Maria Yturralde. Address correspondence to J. V. Molins, C/.Torrent de I'OUa 37, 08012 Barcelona, Spain.
108
J. V. MOLINS
defined inside the portion of the figure marked "a," that is, with the appearance in "a" of any new volume or figure, independently of the inverse perspectives that define the impossible figure. Figs. 2 and 4 show the depth effect strongly, which appears to be more strongly defined the simpler the figure. Figs. 2 (Schroeder ladder) and 3 show how the depth effect can be extended to so-called reversible figures. Figs. 5 , 6, and 7 [the Penrose and Penrose triangle (1958)l extend the effect to the internal space of more complex figures. Fig. 8 deserves special mention: in this case, an illusion very similar to the preceding illusions takes place, but its analysis seems to be more complex. My interpretation is that here we have an impossible figure (the body entering inversely above the black plane) in which just one of its parts is directional, together with a neutral plane with which it operates in the same way as in the previous examples; that is, it gives directionality (coordinates) to the plane. Directionality, being neutral, plays the role of the ambivalent space "a." So that, as in Figs. 2 and 4, directionality inverts the figure-ground relationship, since the additional bodies change the black space, which was originally figure, into the ground (background). I t is interesting to note the analogy between this effect and the mechanism of abstract reasoning applied to the double negative where and So that in this case what is impossible or absurd in the combination of two inverses seems to become even more distinct than a 'normal' figure with the appearance of the space defined by the new body in 'la."
Impossible Penetrations Another kind of impossible figure is represented in Figs. 9 and 10 where apparently large volumes are contained inside narrow gaps. Considering the extreme simplicity, it is surprising that until now nobody appears to have noticed this illusory possibility offered by classical perspective. I consider that in the interpretation of these impossible penetrations, we should refer to a classical problem in the representation of space on a flat surface: the calculation of apparent size. One of the coherent solutions of the impossibdity of Fig. 9 would be to imagine the transversal slot being of huge size but seen by the viewer at an extremely oblique angle of vision. This revives the Euclidean idea rejected by Renaissance perspective, according to which apparent size is directly related to the angle of vision. Fig. 11 tends to corroborate this: here objects of very different sizes have very similar apparent projections on to the same plane but the apparent angle of vision is different.
THOUGHTS ON IMPOSSIBLE FIGURES
FIGS. 1, 2, 3, 4. Examples of impossible figures
J. V. MOLINS
FIGS.5 , 6 , 7, 8. Examples of impossible figures
THOUGHTS ON IMPOSSIBLE FIGURES
FIGS.9, 10, 11, 12.
Examples of impossible figures
J. V. MOLINS
15
FIGS.13, 14, 15. Examples of impossible figures
T H O U G H T S ON IMPOSSIBLE FIGURES
113
Also related is the effect shown in Fig. 12, wherein the tendencytowards grouping of the transversal bars is much stronger than the separation effect produced by the large cube apparently placed between them. Another speculative interpretation of the illusion aims at an approximation of a linguistic analogy. We can imagine that the body penetrating the slot is not solid, but a plane representation of a body (e.g., a piece of wood or paper on which a cube has been drawn). Then we would have a kind of logical metaproblem, the plane representation of a plane representation, which produces the illusionary effect of reality.
Symmetry in Impossible Figures At first glance it may seem axiomatic that impossible figures, just like any other representation of a solid on a plane surface, give figures which are symmetrical with respect to an axis. However, the analysis of these symmetries makes it clear that it is conceptually correct to talk about two impossible triangles, and so on, as can be seen in Figs. 13, 14, and 15. I n Fig. 13 I return to the strip, the simplest impossible shape, to point out that the inverses fuse when we perform an imaginary rotation. I n symmetrical figures this imaginary rotation would be clockwise in the figure on the right of the symmetry axis and counterclockwise in the figure on the left. Perceptually, one can note that clockwise rotations in volumetric figures produce a movement of the figure into the paper, while counterclockwise turns move them outwards, i.e., towards the observer. I n Fig. 15 we see the movements which the clockwise and counterclockwise triangles of Fig. 14 would make were they not closed off and made into impossible figures: the clockwise triangle is continuously (and infinitely) rotating inwards, while the counterclockwise triangle does the reverse. We could perform the same operation on any sirmlar figure. I n closing, I would like to repeat that the intention of this paper is solely to describe the observations I have made, without wishing to preempt more rigorous subsequent work. A theory of impossible figures has proven to be a difficult undertaking and so far lacks a universally accepted Formulation. Richard Gregory (1968) semantically resolved the issue when he pointed out that any representation in a plane of a three-dimensional object is in fact an illusion. This statement is certainly true; however, the human effort involved in this representation comprises the whole history of perspective, and the millions of 'unreal' images to which we are exposed every day make it important to study representation itself, including all the illusions which it produces by its very nature. For this reason, I am convinced that a proper study will need to be interdisciplinary, since it relates to subjects such as geometry, and ultimately, even to the theory of knowledge and semeiotics. REFERENCES GREGORY, R. L. Visual illusions. Scienhfic American, 1968, 219, 66-76. PENROSE,L. S., & PENROSE,R. Impossible objects: a special type of illusion. Britirh Journal of Psychology, 1958, 58, 49-138.
Accepted July 2, 1391
J , V, hlOLINS
APPENDIX'
-Although u7e are not able to reproduce the striking colors of the original prints, we find the effects.-Eds. Author's more complex figures present highly interesting
