Right-Angle Preference in Impossible Objects and Impossible Motion

There is a class of pictures called pictures of impossible objects. Although they are named “impossible,” they are not necessarily impossible. Some of them can be realized as three-dimensional solid objects. In this study, we show that the impression of impossibility comes from the human preference for right angles. This preference can also explain another visual illusion called impossible motions. . Introduction There is a class of pictures called pictures of impossible objects. Typical examples are Penroses’ triangle and an endless loop of steps [2]. The latter became famous in that it was used in Escher’s “Ascending and Descending” (1960) [3]. Pictures of impossible objects are usually considered as just images that generate an impression of three-dimensional structures and cannot be realized in three-dimensional space. However, some of these so-called impossible objects can be actually constructed as three-dimensional objects. There are two well-known tricks by which this can be contrived [1]. One is to make gaps in depth in such a way that faces look connected when seen from a particular viewpoint. The second is to use curved surfaces that look planar from a particular viewpoint. There is a third trick, in which neither gaps nor curved surfaces are used, but by which three-dimensional impossible objects can still be constructed [4]. This trick is based on an inverse problem of projection, in which we are shown a picture and search for a three-dimensional object whose projection generates the given picture. A similar trick can also be used to generate an impression that we perceive physically impossible motions [5]. When we consider what kind of three-dimensional objects can generate this illusion effectively, our answer is that we should use non-right angles in such a way that they look like right angles when seen from a specific viewpoint. This is because the human visual system prefers right angles much more than other angles. In this paper, we discuss this observation in detail. Impossible Objects and Impossible Motions Let us start our discussion with an example. Fig. 1 shows an example of an impossible object. Fig. 1(a) is a drawing of an impossible object called the endless loop of steps. What is represented by this drawing seems impossible, but it can be constructed as shown in Fig. 1(b). The actual shape of this object is such that, as shown in Fig. 1(c), three of the four walls have normal steps, but the fourth wall has distorted steps that absorb the difference in the heights of the start step and the end step of the normal steps. Proceedings of Bridges 2014: Mathematics, Music, Art, Architecture, Culture