``Impossible figures'' and interpretations of polyhedral figures (Q795770)

''Impossible figures'' such as Escher's endless staircase provide a proving ground for theories of perception. In this paper the author describes a theoretical basis for determining whether a figure is impossible. The theory is based upon the fact a figure is ''possible'' if it is a two- dimensional representation of a virtual three-dimensional scene. Given a two-dimensional figure that may or may not represent a three-dimensional scene one can associate to it an adjacency graph which, in addition to information on the way that faces meet, carries information on the way that faces may occlude one another. This graph is used to construct for each point M in the space another graph called the ordering graph at M. The ordering graph describes the relationship between the faces as they might be seen from the point M. An adjacency graph is called locally consistent if it does not give rise to ordering graphs that contain cycles leading to an inconsistent division of the three-space by the faces making up the cycle. This notion while it captures some forms of impossible figures fails to detect the impossibility of the Escher staircase. To cover this case the author introduces a notion of comprehensive consistency based on the interrelationships of faces that are interpreted as parallel. A comprehensively consistent interpretation is said to be completable if for any pair (triplet) of the faces one can construct a line (point) corresponding to their intersections. The main result of the paper gives a necessary condition for a completable figure to be feasible (meaning that it can be realized as a three-dimensional figure). In the general case the condition requires that there exist four faces which make up a tetrahedron that can be seen from some point as flat. In the special case where all faces are either concurrent or parallel the tetrahedron condition must be replaced by a condition involving cross-ratios.