order in the impossible
Institute of Biocybernetics
and Biomedical Engineering. 00-818 Warsaw Poland
[Published in: Perception, 1987, vol. 16, pp. 201-214]
The class of visual illusions called 'impossible figures'
(illusory spatial interpretations of pictures) is analyzed in order
to introduce an ordering into the great variety of such figures.
Such an ordering facilitates reference, unifies terminology, and
establishes a conceptual framework for further investigations of
the subject, making easier the choice and systematic generation
of various types of figures (for example, in systematic psychological
experiments). First, the notion of 'impossible figure' is defined
and certain other related classes of figures (so-called 'likely'
and 'unlikely' figures) are distinguished. Second, the fundamental
'impossibility sources' are identified as elementary 'building blocks'
of all impossible figures. Finally, two broad classes of impossible
figures, multibars or ('impossible polygons') and striped figures,
are briefly described.
Since the publication of the short paper by Penrose and Penrose
(1958), so-called 'impossible figures' have increasingly attracted
the attention of psychologists, computer scientists, and mathematicians.
Psychologists study them as a new type of visual illusion (Penrose
and Penrose 1958; Gregory 1970; Young and Deregowski 1981) providing
information about mechanisms of visual perception, in particular
the spatial interpretation of pictures (for example, Gregory 1970;
Huffman 1971; Cowan and Pringle 1978; Young and Deregowski 1981;
Kulpa 1983; Térouanne 1983; Thro 1983). For similar reasons
they draw the attention of computer scientists who work in the field
of computer vision and try to create computer models of human perceptive
abilities (Huffman 1971; Sugihara 1982a, 1982b; Kulpa 1983). Finally,
for mathematicians they represent new types of abstract structures
to be formally studied (Huffman 1971; Cowan 1977; Térouanne
1980, 1983). It should also be mentioned that these effects are
of increasing interest to the theory and practice of visual arts
and graphic design (Ernst 1976, 1985, 1986; Yturralde 1978; Reutersvärd
and Kulpa 1984), and that they are also discussed in the contexts
of logic and the philosophy of language (Cresswell 1983).
presentation of just a few examples of the most well-known impossible
figures (figure 3) is sufficient to demonstrate the considerable
diversity of types that exists. Many more examples could be given,
and it is therefore important to find (or impose upon) these figures
some ordering principles and classification schemes.
3. Four examples of impossible figures: (a) Penrose's tribar; (b)
devil's fork; (c) Thiéry's figure; (d) impossible ring
this paper I propose and briefly discuss several basic ordering
schemes. The presentation starts, in section 2, with the definition
of an impossible figure (Kulpa 1983) and a general classification
that relates impossible figures to possible figures, and also to
special classes of figures called 'likely' and 'unlikely' (Huffman
1971; Kulpa 1983). The problem of various degrees of impossibility
is also introduced. In section 3 the fundamental 'impossibility
sources' are identified (Cowan and Pringle 1978; Kulpa 1983; Thro
1983) as elementary 'building blocks' of all impossible figures.
Finally, in sections 4 and 5, two broad classes of impossible figures,
namely multibars (Cowan 1977; Draper 1978; Térouanne 1980;
Kulpa 1981) and striped figures (Robinson and Wilson 1973), are
classification schemes described in this paper are based partially
on my own investigations (Kulpa 1981, 1983), and partially on the
work of others, especially Huffman (1971) and Cowan (1977). Certain
other classification schemes, unrelated, in principle, to those
I discuss below, have also been proposed. Guiraud and Lison (1976)
tried to systematize, from the graphic designer's point of view,
the multitude of ambiguous figures based on the principle behind
Thiéry's figure (Thiéry 1895; see figure 3c). On the
other hand, Sugihara (1982a, 1982b) suggested the general classification
of so-called "labelable figures" (which include many possible
and impossible figures) according to features significant for certain
types of computer algorithms of automatic scene analysis.
Various kinds of impossible figures (Robinson and Wilson 1973; see
figure 4a) can be interpreted as a sequence of stripes, that is,
elongated areas separated by straight parallel edges, according
to the general scheme of figure 17b, where the symbols I1, I2, ...,
In and J1, J2, .... Jn denote appropriate (spatial) interpretations
of the stripes as suggested by pictorial contexts at their respective
endings. When interpretations I and J of a stripe are different
a contradiction occurs, and perhaps eventually produces an impossible
4. (a) Four examples of striped figures, (b) general scheme of a