The Penrose Illusion

  

     Sir Roger Penrose's Illusion is illustrated here in its simplest form.  It appears to be three bars of square cross section joined to form a triangle. If you cover up any one corner of this figure, the three bars appear to be fastened together properly at right angles to each other at the other two corners–a perfectly normal situation. But now if you slowly uncover a corner it becomes apparent that deception is involved. These two bars which connect at this corner wouldn't even be near each other if they were joined properly at the other two corners

 
  The Penrose illusion depends on 'false perspective',  the same kind used in engineering `isometric' drawings. Some artists refer to it as `Chinese perspective', because traditional Chinese art often used it. This kind of picture displays an inherent ambiguity of depth, which we will call isometric depth ambiguity.'
 

    

     Isometric drawings represent all parallel lines as parallel, even if they are tilted with respect to the observer. An object tilted away from the observer by some angle looks the same as if were tilted toward the observer by the same angle. A tilted rectangle has a two-fold ambiguity, as demonstrated by Mach's figure (right) which may be seen as an open book with pages facing you, or as the covers of a book, with the spine facing you. It may also be seen as two symmetric parallelograms side by side and lying in a plane, but few people describe it that way.

The Thiery figure (above) illustrates the same idea.
        
     
 Schroeder's reversible staircase illusion is a very `pure' example of isometric depth ambiguity. It may be perceived as a stairway which one could ascend from right to left, or as the underside of a stairway, seen from below. Any attempt to draw this with vanishing points would destroy the illusion.


     This simple design looks like three faces of a string of cubes, seen either from the outside, or the inside. If you put your mind to it, you can see them as alternating: inside, outside, inside. But it's very hard, even if you try, to see at as simply a pattern of parallelograms in a plane.


     Here some areas have been blackened. The black parallelograms at the top are seen either as from below, or from above. Try as hard as you can to see them as alternating, one from below, one from above, and so on, left to right. Most people can't. Why should we be unable to do this? This is, I think, one of the most baffling of the simple illusions.


      
   The other classic Penrose illusion is their impossible staircase. It is often rendered as an isometric drawing, even in the *Penrose paper. This version is identical to that of the Penrose paper, except for its lack of shading.

*In the Penrose Paper, Dr.Penrose asserts that human behavior is not a function of pure mathematics and cannot be reduced to an algorithm.

 

      
The Escher Waterfall is based on the Penrose illusion, sometimes called a `tribar illusion.'

     Penrose Parallelograms is a tiling system invented in 1974 by English physicist, Sir Roger Penrose. Penrose was asking how can a tiling not repeat? How can shapes completely fill up an infinite space, but never have a repeating pattern? Penrose went on to ask, "What is the smallest number of shapes that could create an non-repeating infinite tiling?" Asking that question is remarkable. Answering it is astounding. Discovering that the answer is only two is both shocking and delightful.

See Examples of Saxe-Patterson's Penrose Tilings