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ASCS09: Proceedings of the 9th Conference of the Australasian Society for Cognitive Science
Visual paradox and cognition
Chris Mortensen ([email protected])
Department of Philosophy
The University of Adelaide, SA 5005, Australia
Abstract
This paper offers a systematic approach to describing the socalled visual paradoxes. The particular example treated is
what is known somewhat incorrectly as the Penrose triangle.
The analysis is in three parts: an inconsistent logical theory
which demonstrates the existence of a paradox, a cognitive
explanation of how the logical theory is internalised, and a
mathematical model which validates the logical theory.
Keywords: visual; paradox; inconsistent; Penrose triangle;
cognition.
Visual Paradoxes
There has been much activity on the subject of so-called
“impossible images”. Almost all of it has been in the
twentieth century, mostly by artists (notably Oscar
Reutersvaard and M.C.Escher), some by psychologists,
mathematicians and philosophers. Impossible images, better
termed visual paradoxes, are impossible in the sense that
the visual system projection to 3-D is of an apparently
contradictory object. Such a thing cannot exist. This is the
key question: what is the 3-D content in virtue of which we
judge it impossible? Clearly the answer is that the visual
system constructs an inconsistent theory. Only then can it be
proved that it is a paradox, ie. a contradictory theory. This
takes the tools of symbolic logic to prove, as we see below.
The second task is to explain why it looks to us in this
way; that is, what cognitive structures as exemplified in us
in virtue of which we identify it as impossible. This goes
along with the classification problem, namely these images
come in natural kinds. See Thro (1983). When are they
essentially the same principle, and when are they different?
The internalized cognitive tendencies must have something
to do with this.
Finally, mathematics can make a
contribution to the classification problem by providing
models which validate the sentences of the internalized
inconsistent theories, and not other sentences. This part
takes some mathematical development, and is not strictly
relevant to the cognitive story, so it is omitted here.
The Triangle is a Paradox
Consider the triangle, discovered by Reutersvaard in 1934,
re-discovered by the Penroses in 1958. See Fig 1, (with
letters added).
Article DOI: 10.5096/ASCS200937
Figure 1: The triangle (with letters)
Axes x and y are the usual axes in the plane of the paper,
the z axis is normal to the plane of the paper. Now one key
intuition about Fig 1 is that it constantly receded from the
observer as the eye traverses the image anticlockwise. For a
closed loop to do this is impossible. To show this
rigorously, we adapt some psychological findings by Cowan
and Pringle (1978).
The corner A, considered in 3-D, tilts into the page when
it is traversed anticlockwise. This means that points further
along the face AB are further away in the z-direction than
points not so far along. Thus we may deduce that:
(1) A is closer in the z-direction than B
Traversing the corner B anticlockwise, the face BC
similarly tilts into the page. Thus:
(2) B is closer in the z-direction than C
By similar reasoning:
(3) C is closer in the z-direction than A
We now make the natural assumption, that the relation
closer is transitive.
(4) Trans (closer).
It follows that:
(5) A is closer in the z-direction than A
But clearly, by observation (or by definition of closer):
(6) A is not closer (in the z-direction) than A.
The statements (5) and (6) contradict one another. This is
the proposed proof of the paradoxicality of the triangle.
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ASCS09: Proceedings of the 9th Conference of the Australasian Society for Cognitive Science
The Triangle is an Occlusion Paradox
Now we proceed with a first step on the classification
problem, by proving that it is an occlusion paradox. This
will take several sections. An occlusion paradox is defined
as a paradox which can be rendered non-paradoxical by
reversing one or more occlusions. Not all paradoxes are
occlusion paradoxes. The proof that the triangle is an
occlusion paradox requires two reversals of occlusions.
The first reversal is on the bottom left hand corner. Fig 2
shows the reversal and those edges whose occlusions are
reversed
not the same as the previous proof of inconsistency, the
previous one won’t work. This is confirmation that these are
2 different types of triangular form.
Keeping x and y axes as usual, the z-axis is chosen as a
normal to the AB face. This allows us to say:
(1) A is the same distance in the z-direction as B
Now the B corner still turns in, when traversed
anticlockwise. So:
(2) B is closer in the z-direction than C1
Now, by inspection, C1 is part of the same plane as the
AB face, therefore with the same normal. Hence:
(3) C1 is the same distance in the z-direction as A
Now we invoke a weaker form of transitivity, a form of
functionality, namely that if A is the same distance as B
and B is closer then C1, then A is closer than C1. So:
(4) A is closer in the z-direction than C1. But this implies
the negation of (3).
This is a contradiction. It follows that the second triangle
is likewise paradoxical.
Both Triangles are Occlusion Paradoxes
Figure 2: First reversal of occlusions
This reversal produces an intermediate triangular form. It
is not the same as the first triangle, however. See below.
Returning to our proof that the first triangle is an occlusion
paradox, it suffices to show that the second is also. To do
this, we reverse the occlusions in the top corner B.
A Second Impossible Triangle
Figure 4: Second Occlusion Changes
The outcome is now a consistent triangle, as is evident by
inspection.
Figure 3: A Second Triangle.
This triangle is impossible, but intuitively less so. This
intuition has a natural explanation. The second triangle can
now be rendered consistent with occlusion-changes in just
one corner. The triangle we started takes two sets of
occlusion-changes, involving two corners, so it is further
away from consistency. We register, then, a classification
into two different impossible kinds on the basis of these two
examples.
But it is still a paradox, ie. inconsistent. This then also
deserves proof: that is the logician’s interest. The proof is
Article DOI: 10.5096/ASCS200937
Figure 5: A Consistent Triangle
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ASCS09: Proceedings of the 9th Conference of the Australasian Society for Cognitive Science
Thus we can conclude that both triangles are occlusion
paradoxes, but they are of differing kinds determined by the
different number of occlusion reversals necessary to reduce
to consistency.
From Logic to Cognition
Opposite parities in a figure cancel out, whereas contortions
accumulate.
The results were unmistakeable: mean scores of parity
correlate significantly with ranking in degree of
inconsistency, mean scores of contortions do not. It follows
that parity is the significant variable here, and not
contortion.
This brings me to the topic of cognition. We would like
independent evidence that the premisses of the argument for
the inconsistency of the first triangle, are those which we
internalize, and in virtue of which it looks impossible to us.
Cowan and Pringle (1978) provide just the sort of
experimental data relevant to the claim. They work with 4sided figures, but 3 sided figures are an obvious special
case. They identify 27 different rectangles.
Figure 7: Cowan and Pringle’s Results
Figure 6: Cowan and Pringle, 27 rectangles
Three groups of judges were used, each trained on a
different 9 rectangles, and asked to rate the remainder by
degree of possibility from 1 to 10. Clearly degree of
possibility is related inversely to degree of impossibility. It
is conceivable that there could have turned out to be little or
no correlation between scores after training effects were
corrected for. In fact there was a strong correlation between
individuals. This indicates the presence of an objective
phenomenon fairly constant across the population.
We can do even better with these results. Cowan and
Pringle postulate two different features of these images
which could serve to explain the judgements of degree of
impossibility. One explanation was that impossibility is
related to the contortion of the figure. Contortion is the
number of twists in the sides necessary for the corners to
appear as they do, while the whole figure remains coplanar.
The second explanation was that impossibility is related to
the parity of the figure. They define the parity of a corner to
be +1, -1, or zero, depending on whether, traversed
anticlockwise, the corner turns into, out of, or coplanar with
the plane of the page. The relation between contortion and
parity is that the latter registers the direction of the twist.
Article DOI: 10.5096/ASCS200937
Another explanation raised was whether judgements of
impossibility were affected by stereopsis. Stereoscopic
images of impossible objects are available, and the
experience is well worth having. There are standard
methods for inducing stereopsis. But when the experiment
was run again in those conditions, not only was there no
significant correlation with stereopsis, but the same parityinduced order was found with stereopsis. Hence, no extra
effect.
Conclusion
But now we recall from the first section, that this was
exactly the description employed in the proof of the first
paradoxical triangle. The logical analysis depends on it.
Parity registers into and out of, and it is intuitive
calculations of parity that lead to judgements of
impossibility, in the indicated fashion. In short, it is the
cognition of parity that explains judgements of relative
possibility and impossibility.
Bibliography
Cowan, T., and Pringle, R., (1978) An Investigation of the
Cues Responsible for Cube Impossibility. Journal of
Experimental Psychology: Human Perception and
Performance, 4, 112-120.
Penrose, L&S., Impossible Objects, a Special Kind of
Illusion. British Journal of Psychology, 38, 49.
Thro, E.B., (1983), Distinguishing Two Classes of
Impossible Objects. Perception, 12, 733-751.
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ASCS09: Proceedings of the 9th Conference of the Australasian Society for Cognitive Science
Citation details for this article:
Mortensen, C. (2010). Visual paradox and cognition. In W.
Christensen, E. Schier, and J. Sutton (Eds.), ASCS09:
Proceedings of the 9th Conference of the Australasian
Society for Cognitive Science (pp. 245-248). Sydney:
Macquarie Centre for Cognitive Science.
DOI: 10.5096/ASCS200937
URL:
http://www.maccs.mq.edu.au/news/conferences/2009/ASCS
2009/html/mortensen.html
Article DOI: 10.5096/ASCS200937
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