Organizing the properties of impossible figures

The empirical study of impossible figures requires that independent variables be well-specified. This paper provides an aid for such a task by developing a classification system which partitions cornered torus figures into meaningful subsets. These can be used to identify both formal and psychological properties of known as well as unknown possible and impossible figures in much the same manner as with the periodic table of chemical elements. The classification is polytopic (for demensional) and is formally, not empirically, derived. Coherent three-dimensional and two-dimensional forms of the system are possible without loss of information.     

Some thoughts on impossible figures

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.     

Preference for impossible figures in 4-month-olds

Can infants use interposition and line junction cues to infer three-dimensional (3D) structure? Previous work has shown that in a task that required 4-month-olds to discriminate between static two-dimensional (2D) pictures of possible and impossible cubes, infants exhibited a spontaneous preference for displays of the impossible cube but left open the question of whether they did so on the basis of purely local “critical regions” or whether they were able to employ more global clues. Here infants were presented with possible and impossible cubes in which the strictly local cues that could have derived from exterior binding contours were deleted. Results showed that infants were still able to discriminate possible cubes from impossible cubes, suggesting that longer looking infants are sensitive to global properties and that the capacity to integrate pictorial information to perceive aspects of global 3D shape may develop earlier than demonstrated previously using reaching tasks.     

Concept formation and categorization of complex,asymmetric, and impossible figures

Impossible figures are striking examples of inconsistencies between global and local perceptual structures, in which the overall spatial configuration of the depicted image does not yield a coherent three-dimensional object. In order to investigate whether structural “impossibility” is an important perceptual property of depicted objects, we used a category formation task in which subjects were asked to divide pictures of shapes into groups that seemed most natural to them. Category formation is usually unidimensional, such that sorting is dominated by a single perceptual property, so this task can serve as a measure of which dimensions are most salient. In Experiment 1, subjects received sets of 12 line drawings consisting of six possible and six impossible objects. Very few subjects grouped the figures by impossibility on the first try, and only half did so after multiple attempts at sorting. In Experiment 2, we investigated other global properties of figures: symmetry and complexity. Subjects readily sorted objects by complexity, but seldom by symmetry. In Experiment 3, subjects were asked to draw each of the figures before sorting them, which had only a minimal effect on categorization. Finally, in Experiment 4, subjects were explicitly instructed to divide the shapes by symmetry or impossibility. Performance on this task was perfect for symmetry, but not for impossibility. Although global properties of figures seem extremely important to our perception, the results suggest that some of these cues are not immediately obvious or salient for most observers.     

Supplementary report: Braids,side segments,and impossible figures

Impossible figures and their braid analyses are reconsidered from the point of view of their side segments rather than their corners.     

Believing the impossible

A version of this paper was read at the Western Division meetings of the APA in April 1990. I am indebted for the comments of my commentator on that occasion, Stephen Hetherington. I am also indebted to Ruth Marcus for helpful correspondence, and to Steven Luper-Foy for comments. Work on the paper was supported by an academic leave from Trinity University.     

Thinking the impossible

The article looks at the structure of impossible worlds, and their deployment in the analysis of some intentional notions. In particular, it is argued that one can, in fact, conceive anything, whether or not it is impossible. Thus a semantics of conceivability requires impossible worlds.     

Rock's cognitive theory of illusory figures: a commentary.

Contrary to a previously published opinion, the 'cognitive' theory of illusory contours is rich in detail and empirical support.     

The rising fastball: baseball's impossible pitch

Batters in professional baseball are confronted with pitches that appear to curve, dip, wobble, or rise. The rising fastball is a pitch where the ball appears to hop up as much as a third of a meter with a sudden increase in speed. Physics experiments confirm that many reported trajectories are possible, but not the rising fastball. The present paper shows how the apparent rise may be explained as a perceptual illusion due to the hitter underestimating original speed of the pitch.     

Putting order in the impossible

The class of visual illusions called 'impossible figures' (illusory spatial interpretations of pictures) is analyzed in order to introduce an ordering into the great variety of such figures. Such an ordering facilitates reference, unifies terminology, and establishes a conceptual framework for further investigations of the subject, making easier the choice and systematic generation of various types of figures (for example, in systematic psychological experiments). First, the notion of 'impossible figure' is defined and certain other related classes of figures (so-called 'likely' and 'unlikely' figures) are distinguished. Second, the fundamental 'impossibility sources' are identified as elementary 'building blocks' of all impossible figures. Finally, two broad classes of impossible figures, multibars (or 'impossible polygons') and striped figures, are briefly described.