Inconsistent Images

Impossible pictures are so-named not because the two-dimensional picture itself is impossible (for then there would be no picture!), but because what is depicted, an apparently three-dimensional thing, is impossible. This raises a significant puzzle: how can it be that one is able to draw a picture of a thing which cannot exist because to do so would violate the laws of logic or mathematics? One surely cannot draw a picture of a standard contradiction, such as "Snow is white and snow is not white." But impossible pictures are different: there is an experience "I see it but I don't believe it" which cries out for explication.

The history of impossible pictures goes back to Pompeii. Later, there are isolated medieval altarpieces, Piranesi's Carceri contains some strange-looking stairs, and Marcel Duchamp drew a peculiar bed. Rene Magritte was responsible for at least one fine image. However, impossible pictures were not drawn in any systematic way until Oscar Reutersvärd began his career in Stockholm in 1934, drawing over 4,000 pictures in the subsequent decades and being honoured by the Swedish government in the 1980s. M.C.Escher and the Penroses followed from 1955 onward, Escher in particular producing masterpieces such as Ascending and Descending, Belvedere, and Waterfall.

Impossible pictures should be distinguished from pictures which permit more than one gestalt, such as the duck-rabbit or the candlestick-faces. In the present study, these are classified as incomplete, not inconsistent. The property of incompleteness is a logical dual to inconsistency in more then one sense. Since this duality is well-known, this means that mathematical treatment of incomplete pictures is readily available once it has been worked out for impossible pictures. However, the latter is a hard problem, not yet solved satisfactorily.

A research project conducted in the Discipline of Philosophy aims to address the issue on several fronts. First, impossible pictures need to be described mathematically. This requires the tools of inconsistent mathematics and paraconsistent logic, that is logic which is tolerant of inconsistencies. The general idea is on viewing an impossible picture, the brain encodes an inconsistent theory. This is somewhat analogous to the way that the brain encodes projective geometry as a projection of a three-dimensional reality, except that the "virtual" three-dimensional reality is inconsistent. Clearly, this has connections with cognitive science: it is hardly being suggested that there is an inconsistent reality "out there", rather it is a matter of the brain's capacity to represent in an inconsistent fashion. A start has been made on the mathematics, but much more needs to be done. Second, in particular there is an issue of classification into various types here, types which seem not to be reducible to one another. These types ought to reflect different mathematical theories. Third, Reutersvärd's own program of drawing different pictures is being extended by Steve Leishman and others. Fourth, there is the prospect of virtual reality itself. There has been a conjecture by Bruno Ernst to the effect that one cannot rotate an impossible picture. This is now known to be false: Mortensen demonstrated this in principle at the 1999 Australasian Association for Logic Annual Conference (Melbourne), and Peter Quigley has now implemented it in detail, discovering more than one way of doing so with impossible Necker cubes. It is apparent that this animation is a preliminary to virtual reality, wherein one has the prospect of being able to wander through a whole impossible environment.

Peter Quigley has provided some further discussion and examples, including animations.
Steve Leishman has created a gallery of impossible pictures (best viewed with Internet Explorer 6, Netscape 7 or Firefox).
John Mercier has created another gallery of impossible pictures (thanks to Peter Quigley).

Preliminary Classification of Impossible Forms

In considering the types of forms, there seem to be no more than four basic forms. These are exemplified by:

These can be labelled respectively (left to right) as: Necker cubes, Reutersvärd triangle, Schuster pipes or fork, and Ernst stairs. There may be other basic forms, but they are not in the existing literature and it has proved impossible to date to construct them. Nomenclature is not without interest. Necker cubes derive from the more familiar Necker cube which does not have broken lines. The device of broken lines signifying occlusions is necessary for the inconsistent effect (see impossible knots). The Reutersvärd triangle is perhaps better known as the Penrose triangle. However since Reutersvärd predated Penrose by twenty years it is proper to give appropriate acknowledgement. The Schuster fork is also called the Devil's fork. Bruno Ernst seems to have been the first to display the stairs.

There are numerous variations of these. These include:

• number of sides (edges and faces),
• the figure can itself broken up into cubes ("cubified"), as Reutersvärd pioneered; or simply solid as in the triangle above,
• the figure can be presented as slats,
• the figure can be presented with or without perspective,
• number of twists in the corners,
• mirror images of whole or part of figure.

Conjecture: Define an "occlusion illusion" to be any inconsistent figure that can be rendered consistent merely by changing the occlusion. Then it is claimed that Necker cubes and Reutersvärd triangles are occlusion illusions, but Schuster's fork and Ernst's stairs are not.

Attention is drawn to the fact that all four basic forms can be rotated (see Quigley's animations).

Many of these themes come together in a visual essay, The Impossible University.

For an instructive moral tale, see Sylvan's Box Gefunden.

Chris Mortensen 2005, revised 2007 Contact email: chris.mortensen@adelaide.edu.au This project is conducted within the Department of Philosophy, University of Adelaide

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