Inconsistent Images

The University of Adelaide Australia
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The Stairs

There are a number of different ways in which stairs can be impossibly configured. One notable one is seen in Escher’s painting Ascending and Descending, which in turn follows the photograph and drawing in Penrose and Penrose’s 1958 British Journal of Psychology paper. But these seem to have the character of the Reutersvard-Penrose triangle.

Irreducibly different are Ernst’s stairs, which seem to have first been drawn by Bruno Ernst 1984, titled The Wearisome and the Easy way to the Top, or gazebo for short. Bruno Ernst is the pseudonym of the Dutch artist J.A.F.Rijk, and this picture is reproduced on page 66 of his excellent book The Eye Beguiled (Taschen 1986).

The technical problem of providing a mathematics for the stairs is unsolved, though I conjecture that the stairs have the nature of the Sorites Paradox. (The Sorites is the paradox that argues from the two premises that someone with many hairs is hairy, and that removing just one hair from a hairy head leaves it hairy, to the conclusion that bald is hairy). Despite its antiquity, in my view the Sorites itself lacks a satisfactory logico/mathematical explanation.

We content ourselves here by displaying a number of stills and animations, including some photographs, which exploit this theme.

Ernst's original gazebo is depicted below. Notice that the gazebo has two steps on the left, three in the middle, and five steps on the right.

Click above to see animation of bouncing balls.Click above to see Escher's man walking through Ernst's Gazebo.

The west wing of the Bonython Hall at The University of Adelaide can be treated in the same way as Ernst's gazebo. This time, instead of two steps into three into five, we simply have have three steps into two. Click on the above right animation to see Escher's man walking through what has become an inconsistent building.(This animation is titled Escher Meets Ernst, by Quigley, Leishman, and Mortensen, with thanks also to Andrew Kelleher for his timely contribution.)

Even statues can exhibit Ernst's stairs inconsistency.

Click on this image of Sir Thomas Elder to see a 3D stereo anaglyph of three stairs into two. To see an anglyph correctly you will need a pair of glasses with a red lens mounted in the left frame and a cyan lens mounted in the right.

Text: Chris Mortensen   Images: Peter Quigley
Revised Feb 22-2007.

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