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 Impossible Triangle
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     What's wrong with this figure? 

 

So What's Going On?

     Although the impossible triangle certainly looks possible at each corner, you will begin to notice a paradox when you view the triangle as a whole. The beams of the triangle simultaneously appear to recede and come toward you. Yet, somehow, they meet in an impossible configuration! It is difficult to conceive how the various parts can fit together as a real three-dimensional object.

    It is not the drawing itself that is impossible, but only your three-dimensional interpretation of it, which is constrained by how you interpret a pictorial representation into three-dimensional mental model. Given the chance to interpret a drawing or image as three-dimensional, your visual system will do so. It does not generally take a perspective drawing and reinterpret it as flat, because there is a spatial paradox.

    There are many ways that this figure can be perceived as a possible as a misperceived object. In other words, it is possible to construct a physical model of the impossible triangle that looks impossible from only one angle. See the example below. The true construction is revealed in the mirror.

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    Your visual system, however, is very constrainted by how it interprets two-dimensional pictorial images into three-dimensional mental representations. It is with the help of such constraints that your visual system assigns depth to each point in an image.  Furthermore, it is more important for your visual system to adhere to these constraints than to violate them because you have encountered something that is paradoxical, unusual, or inconsistent. It would lead to biological disaster if you were blind to the unusual, inconsistent, or paradoxical.

    In particular, certain elements of an image correspond to similar elements in your three-dimensional interpretation. For example, one constraint is that two-dimensional straight lines should be interpreted as three-dimensional straight lines. Likewise, two-dimensional parallel lines should be interpreted as three-dimensional parallel lines. Continuous straight lines are interpreted as continous straight lines. Acute and obtuse angles are interpreted as 90° angles in perspective. External lines are viewed as the boundary of the shape. This external boundary is extremely important in defining your overall mental image of the shape.

    This can be summed up in "The Generic View Principle," which states that your visual system assumes that you are viewing something from a non-accidental point of view. This holds unless there is information to the contrary.

    Let us see how this applies to the impossible triangle.

 

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     The figure above represents the top vertice of the impossible triangle. The scene, however, is visually ambiguous. For example, the lines abb'b''a'' can define the boundary of a limb whose extension is occluded by the boundary of surface a''b''b'bcc', which is part of the right limb . There are many other possibilities. Another example can be seen in the photograph above.

     In this case, the information is supplied by what are known as T-junctions. This is a junction where the lines meet. Two of the lines are collinear, forming the top of a T. T-junctions are good (but not entirely infallible) clues to depth and occlusion. The top of the T is usually the occluding contour. The stem of the T presumed to continue behind.

    Occlusion, however, is a special case for your visual system. Locally, there are no cues that suggest occlusion. It is straight-forward to interpret lines abc and a'b'c' as continuous straight lines, not as abrupt breaks. Therefore, lines abcc'b'a' define the boundary of one continuous surface.

     This would be the case with all three verticies.

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    These constraints operate at different levels, first local and then global. When you examine a drawing of an impossible triangle, you first build up a global image by examing the local parts.

     Each corner in itself is consistent with spatial perspective, although each corner suggests a different angle of the object. Rejoin the triangle and your global perception produces a spatial paradox.

     Cover any corner (in the right figure) with your hand so that you can only see 2/3rds of the impossible triangle (as in the left figure). It does not matter what corner you occlude. The result is non-paradoxical.  You perceive a "broken limb" model of the impossible triangle; however, notice that your perception will change about the nature of the triangle's shape. Take your hand away and the paradox resumes. It is only by joining the third limb to the other two limbs that forces the spatial paradox.  If you separately occluded the three different corners, you can perceive three different consistent broken limb models.

     You can also break the global paradox by occluding  or closing off, any part of the parallel lines that form the boundary of a limb. For example, occlude the middle of any limb. The paradox disappears when the straight lines no longer perceived as forming a continuous surface boundary.

     The Generic View Principle explains why your visual system does not seize upon the infinite variety of possible, but not probable, three-dimensional solutions. The generic-view principle is so strong at a local level that it recovers a surface representation of an image that is literally impossible from the point of view of object knowledge and spatial perspective.

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History of the Impossible Triangle

     In 1934 Swedish artist Oscar Reutersvärd made the first recognizable impossible triangle out of a peculiar arrangement of cubes. While a number of artists have unintentionally created impossible figures, Reutersvärd was the first to realize that he had entered into a new land of imagination. From that day on, he has created thousands of impossible figures, and today is universally recognized as "the father of impossible figures." In 1980 the Swedish government honored Reutersvärd's impossible triangle by commissioning it for a Swedish stamp. The stamp was produced in 1982 and was issued for about two years.

 

 

 

     In 1954 physicist Roger Penrose, after attending a lecture by the Dutch graphic artist M. C. Escher,  rediscovered the impossible triangle and drew it in its most familiar form, which he published and popularized in a 1958 article, co-authored with his father Lionel Penrose, that appeared in the British Journal of Psychology. Penrose, who was stimulated by Escher's work, wanted to create something that illustrated an impossibility in its purest form. In 1954 Escher had not yet created his three impossible prints: "Belvedere," "Ascending and Descending," and "Waterfall." Penrose was also unfamiliar with the work of Reutersvärd, Piranesi, and others who had created impossible figures previously.

     Penrose's impossible triangle, unlike Reutersvärd's earlier version, was drawn in perspective, which added an additional size paradox to the triangle.

 

 

 

     In 1961 the Dutch graphic artist M.C. Escher, inspired by Penrose's version of the impossible triangle (he was sent a copy of the article by the Penroses) incorporated it into his famous lithograph "Waterfall."

     Since that time, the impossible triangle has reappeared in countless versions. Because of its popularity, many people consider the impossible triangle to be THE impossible figure, and are astonished to find that there are an infinite number of impossible figures possible.

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