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Penrose Stairs seem to be a locally valid but globally inconsistent contraption. I have a couple of questions:

  1. Is it physically realizable? In other words, is it possible to build a 3-D structure of this kind? I have seen a couple of images on similar illusions made of Lego blocks on Google, though I am not sure whether they are real.

  2. Is there any explanation behind the effect?

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Sure - imagine if the steps are not actually "flat", but sloped. Then you can get something which looks like this. Remember, a picture is a 2D projection of some object. –  Thomas Andrews Feb 6 '13 at 18:08
    
The effect is really more about optical illusions, which is a neurological thing rather than a geometric/mathematical thing, I'd say. –  Thomas Andrews Feb 6 '13 at 18:09
    
@ThomasAndrews: There must be a more precise location of the source of the paradox than assuming that humans are wired this way. –  dexter04 Feb 6 '13 at 18:20
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I'm sure there is a more precise reason, but I do not think it is a mathematical one. When we look at 2D drawings, our brains try to make sense of them as 3D objects, and we use a lot of visual queues to determine "this is actually connected, this is flat, etc." –  Thomas Andrews Feb 6 '13 at 18:21
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Sorry to not be more help. My point is, there is no mathematical paradox or contradiction in the picture. It is only in the attempt to interpret the picture as a 3D object based on our intuition that the "steps" are flat and parallel to the ground, that we get anything like a paradox. That interpretation is neurological more than mathematical. –  Thomas Andrews Feb 6 '13 at 18:36
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2 Answers

up vote 4 down vote accepted

Concerning the question "2. Is there any explanation behind the effect?":

Look at the following figure:

enter image description here

Locally, in other words: when you look only at a sectorial part of the figure, you see a staircase, and there is no visual paradox (other than that it is not clear whether going clockwise goes up or down, but that is another matter). It is only when you look at the figure as a whole that the idea of a staircase is no longer sustainable.

The main cause of the illusion is the fact that our brain is automatically interpreting the two-dimensional input of a part of this figure as an image of a scene which is in reality three-dimensional.

Note that in the case of a real three-dimensional staircase even our two eyes looking at it would not be able to resolve the front-back ambiguity alluded to in the sentence in parentheses above.

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Isn't it obvious that if you built this object, the landing on the far left would be simultaneously higher and lower than the one on the right?

This would have to be true if the stairs were assumed to have level steps. But we don't have any assurance that they are that way. The base of the staircases are also equally disorienting, since we have no way to know for sure they are drawn to be parallel with the step surfaces.

When we look at it we are tempted make certian assumptions that just aren't so. That's a big part of making the illusion.

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