Inconsistent Images

The University of Adelaide Australia
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Reutersvard’s first opus was in cubes, essentially a cubified version of what is widely, albeit somewhat misleadingly, called the Penrose triangle. The following image by Peter Quigley draws on, but reconstructs, Reutersvard's first opus.

But Reutersvard then rapidly saw and drew the simplest or limiting case, namely the corner elements: three cubes. Thus we have the corner elements from Quigley's reconstruction.


For comparison, below we see the original 1940 drawing Opus 2B, by Oscar Reutersvard ((reproduced with permission of its owner, Bruno Ernst.). Clearly, as well as being the corner elements of Reutersvard's original triangle, it is the limiting case of Reutersvard's triangle when reduced to 3 blocks.


But are these two images really inconsistent? Before you say yes too rapidly, look at the following photograph.


This has not been tinkered with, save to remove the hand which was holding the 3 blocks (the original with hands). The blocks can be moved about and rotated for instance. What then is the difference from Reutersvard's 3 blocks? If you answer that there is no difference, then since the photo is of a consistent situation, perhaps Reutersvard's 3 blocks are after all consistent too? But how could that be: the 3 blocks are just the limiting case of more blocks, and such images with more blocks are incontestably inconsistent. The answer seems to be that Reutersvard's blocks are indeed inconsistent because the faces of each pair of blocks are lined up so that they are coplanar. On the other hand, this is not so in the latter photograph: there is no consistent way to make the outside faces of the 3 cubes coplanar; but because of the way the blocks are presented this is hard to see, the blocks look like they have coplanar faces.

Cubification is also interesting in that it suggests some sort of mathematical analysis in terms of chains of Neckers.

Many images can be presented in cubified form. For example, further cubified images, drawn by John Mercier, can be seen elsewhere on these pages.

Text: Chris Mortensen: June 2007
Images: Peter Quigley, Andrew Kelleher, Steve Leishman, Chris Mortensen, Oscar Reutersvard.

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