Inconsistent Images

The University of Adelaide Australia
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Beginning with Escher's Cube,

and building on Cochran's Crate,

this is what I got when I slatified Penrose's triangle.

This led to the simple extension.

So what else could I build onto the slatified Penrose figure? Suppose I wanted to build a chook house (chicken shed in fine old Australian slang). Starting with Cochran's box for the wall frame, I could use Penrose's tri-bar to build a perch for it like so,

or like so,

or like so,

and thanks to Bruno Ernst's ideas for stairs, I can build a roosting box for the hens to lay their eggs in.

Now for the roof of the chook house; first some roof joists,

and a ladder to get up there,

(always handy around a chook house), then a nice roof with a skylight,

(I thought the chickens might like to see the stars).

But then I need a fence to keep the chooks in, so

(multiplied), and lastly the chook yard gate

with a latch like so (much smaller of course),

thanks to Schuster. Put it all together and its done!

A bit rough, maybe, but the chooks won't mind, I'll build the fence around it some Saturday-arvo when they're off watching the footy.

SLATS: comments.

What does the ubiquitous possibility of slatification show? First and most obviously, it enhances the impression that the 3-D object presented to us could really exist out there. It strengthens the illusion that you could build it. It is wonderful what a few nails, woodgrain and knotholes can do! The tradition of impossible images has of course exploited this in a more general way, with skilfully drawn backgrounds integrating the problematic object into an environment to enhance its credibility as an object. Perhaps the most striking examples of this are Escher's masterpieces such as Belvedere, from which the leading cube above is taken, right out of the hands of the little man who is holding it.

In a more theoretical vein, slatification appears to be available for any image whose impossibility derives from a reversed occlusion. Slatification is a way of preserving occlusions (indeed, of making occlusions more apparent). Slats provide "faces with thickness" as it were. But note that they are faces which do not necessarily join to give a boundary of an enclosed solid figure: the enclosed solid is (sometimes) suggested and filled in by the mind, not drawn explicitly.

In this respect, the slatification picture of Ernst's stairs is significant. Slatification omits what it seems to be the problematic stair-face, ambiguous between being a flat or a riser. This poses a problem for the inconsistent mathematical description of the same, in a regulation (vertices, edges, faces, occlusions) theory.

On the other hand, slatification does not add any new basic classification of figures, beyond the four basic constructions so far identified. That being so, slatification is neutral on the conjecture that all impossible figures are "occlusion illusions", i.e. derive their impossibility from reversed occlusion. The conjecture remains unconfirmed and unrefuted.

Steve Leishman, May 2007 (narrative and images)
Chris Mortensen, April 2007 (further comments)

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