Rotating the Impossible Rectangle
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Impossible figures convey the impression of a three-dimensional object; this strongly implies that one might be able to rotate such an object and view it from different angles. Three-dimensional (3D) models of Escher’s Belvédère [1,2] have been built by Shigeo Fukuda, and models of the impossible tri-bar and crazy crate have been built by Mathieu Hamaekers [3]. Sugihara has described the generation of polyhedra from impossible figures and how to draw their unfolded surfaces [4]. However, all these models must be viewed from a particular direction, as they typically contain a number of gaps or twists. Once revealed, these gaps and twists destroy the illusion of impossibility. Thus, a model of an impossible object has to be handcrafted to suit the desired viewpoint. If the viewpoint changes, the model must be adjusted. We can solve this problem by using a computer model of the object rather than a physical one. The computer can continuously modify a 3D model so that, as the viewpoint changes, its twodimensional (2D) projection continues to satisfy the various properties that make the figure impossible. A computer needs some rules by which it can construct a model of an impossible object. We have observed that a wide class of impossible objects can be represented in terms of complementary halves [5,6]. For example, the impossible rectangle can be divided into two halves, which when viewed independently correspond to a 3D object that is globally consistent and “possible.” Each complementary half presents a very different view of an object; when joined they produce a globally inconsistent figure (see Fig. 1). One complementary half can be obtained from the other via two sequential reflections across perpendicular axes in the image plane. Examples of impossible figures that can be constructed via complementary halves include impossible tori, the impossible stall (upon which Escher’s Belvédère is based), and the crazy crate or impossible cuboid. Being able to divide an impossible figure into complementary halves greatly simplifies the task of constructing a 3D model of an impossible object. All we have to do is construct a 3D model representing one half of the impossible object (in the case of the impossible rect-
[1] M. C. Escher,et al. The graphic work of M.C. Escher , 1960 .
[2] Kokichi Sugihara,et al. Three-dimensional realization of anomalous pictures--An application of picture interpretation theory to toy design , 1997, Pattern Recognit..