Impossible, ever rising staircases like the four-sided one to the left were first published by Lionel and (now Sir) Roger Penrose about fifty years ago. An example is also shown to the left slightly opened up, as if made of rubber, to show how the puzzle arises. Note that as with the impossible objects we looked at in the commentary to picture 13 (also discovered by the Penroses), the effect depends upon seeing parts of the object as connected to one another, whilst in the real world they have to lie at very different distances from the viewer. Imagine looking down on a bird's eye view of one of these stairs. Its ground-plan would be roughly spiral. The trick then depends on seeing the step nearest to the centre of the spiral line up exactly with a step on the very edge of the spiral, so that it seems you could step from one to the other.
The effect requires that we dispense with several of the depth cues we rely on in everyday vision. Perspective diminution is one. The further step would have to be much larger than the near one to appear the same size, though so much further away. In the pictures to the left, I've also added a bit of shading, or aerial perspective, which is another powerful depth cue in everyday vision that must be suppressed for the illusion to work.
A "rubber" version of an oval staircase, like the one that collapses in comic book picture 22, is also shown at left, to demonstrate that it works in the same way. To draw your own versions, note the importance of the contextual detail of the stonework around the oval stair in comic book picture 22, which makes it hard to envisage the discontinuity that seems to stand out prominently in the "rubber" demonstration version.
& Comic Book Picture 22, The Endless Staircase
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Impossible, ever rising staircases like the four-sided one to the left were first published by Lionel and (now Sir) Roger Penrose about fifty years ago. An example is also shown to the left slightly opened up, as if made of rubber, to show how the puzzle arises. Note that as with the impossible objects we looked at in the commentary to picture 13 (also discovered by the Penroses), the effect depends upon seeing parts of the object as connected to one another, whilst in the real world they have to lie at very different distances from the viewer. Imagine looking down on a bird's eye view of one of these stairs. Its ground-plan would be roughly spiral. The trick then depends on seeing the step nearest to the centre of the spiral line up exactly with a step on the very edge of the spiral, so that it seems you could step from one to the other.
The effect requires that we dispense with several of the depth cues we rely on in everyday vision. Perspective diminution is one. The further step would have to be much larger than the near one to appear the same size, though so much further away. In the pictures to the left, I've also added a bit of shading, or aerial perspective, which is another powerful depth cue in everyday vision that must be suppressed for the illusion to work.
A "rubber" version of an oval staircase, like the one that collapses in comic book picture 22, is also shown at left, to demonstrate that it works in the same way. To draw your own versions, note the importance of the contextual detail of the stonework around the oval stair in comic book picture 22, which makes it hard to envisage the discontinuity that seems to stand out prominently in the "rubber" demonstration version.