Inconsistent Images

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Another Paradox?


Consider the image below, which I will call Three To Five.
 
 
 

 It is constructed from a Rubic Cube, which has 27 sub-cubes. How then is it possible to have a tower of 5 cubes stretching from the bottom front to the top back? Where did the two extra translucent cubes come from? This appears to be different from Reutersvaard?s Opus One, discussed elsewhere in these pages. If the sides of Opus One are filled in, we get the Penrose triangle, which is paradoxical, but not of the same kind. So it may well be a hitherto undiscovered type.

Attempting to simplify the effect, consider the next image, Two to Three.

 The black latticework on top serves to make the top surfaces of the two cubes on top, to be coplanar. But the black latticework is not essential:

 This seems to have a certain ambiguity, and therefore is automatically suspect as a paradox, for as we have seen elsewhere in these pages, ambiguity does not generate contradiction but rather incompleteness (eg. the duck-rabbit isn't impossible, only under-determined). Ambiguity abounds for Necker-like structures, of course.

  But the ambiguity is itself curious. A consistent way to see this image, is as two sub-cubes with the top square face of the bottom front cube co-planar with the bottom square face of the top back cube. Then two lines in this plane join the corners of the two coplanar squares. The ambiguity arises because there is another way to see this, namely as a tower of three cubes with a translucent cube in the middle position. This makes it like a simplified or cut-down version of Three To Five. But whereas this interpretation does not seem forced upon us for Two To Three, it appears more robustly forced on us in Three To Five. This seems to be because while we could stack the three sub-cubes of the latter, their faces are not in a single plane but two, and so joining the corresponding vertices of the squares would not produce a single tower. If this diagnosis of ambiguity for Two To Three is accepted, then the ambiguity is between a consistent interpretation and an inconsistent interpretation. The fact that the image continues to look strange when this is pointed out, reinforces the thought that one of the disjuncts of the ambiguity is impossible.

  Smaller, proto-versions exist, such as One To Two and Zero To One.

 

 The right hand image is the simplest. It shows a square out of which a cube grows, or a top cube suppressed to be coplanar with the top of the bigger square .

There is also a recipe, therefore, for stacking further levels on top: just add a Zero To One. It follows therefore, that the paradox here does not depend on the superstructure being cubic, nor are translucent cubes a necessary part. Thus there is what we can call Three To Four:

 

 

The tower structure is surprisingly like Brancusi's sculpture Endless Column below.

 


The towers can also be made to go in different directions:

 

Tesselations with many directions are possible:

 

Structures which more obviously manifest ambiguity abound in this area. Consider the following image:


This is ambiguous between (1) a spline going across the middle, and (2) two cubes from the top row plus a cube sticking out of the right hand end. This is an extension of the well-known ambiguity of Necker's cube. These images make for interesting paradoxical effects when rendered.

 

The following is called The Little ImpossibleTheatre.

 

 

Another form of the paradox exploits the other three planar symmetries instead of the four diagonal ones. The extra cubes in theses cases appear as the orthogonal splines of the three-cube, emerging as the middle most cube on each face.

For a sketchbook of images exploiting this other kind of paradoxicality, click here.

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Text: Chris Mortensen

Little Impossible Theatre variation by Peter Quigley

Other images and gallery: Steve Leishman and Chris Mortensen

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December 2009.

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This project is conducted within the Department of Philosophy, University of Adelaide, and funded by The Australian Research Council.