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In the conclusion of my article for the fourth issue of Tout-Fait Journal , I identified a possible theme in the artistic
events of the 1900’s. I'm referring to the gradual emergence, in art,
of important ideas and conceptual themes which also belong to the
grounding kernel of the complexity sciences.
As a first step,
my concern was (and still is) to illuminate some unexpected links,
all of them directly related to some fundamental ideas of complexity,
between some leading figures in twentieth century art, namely Klee,
Duchamp and Escher. This unexpected relationship is even more surprising
considering the radical differences between their personalities and
their artistic results, or at least the retinal (to use a duchampian
term) ones. Furthermore, as far as I know, there is nothing in their
writings that links these artists. Relations between Duchamp, Klee
and Escher cover a huge range of ideas, and the complexity sciences
provides us with a realm in which we can unify them.
At the yearly conference “Matematica e Cultura” in Venice , organized by Prof. Michele
Emmer, I gave a talk titled “Strands of complexity in art: Klee,
Duchamp and Escher”, where I presented some preliminary findings of my research.
This article supplements those preliminary findings with new analogies.
I'll start by summarizing those first ideas; and then I'll introduce
other subjects, such as evolution, topology, impossible 3D objects
and enlarged conceptions of perspective. Finally I'll try to relate
these themes with those of complexity.
1. A summary of preliminary findings
I divided the
common traits between Klee, Duchamp and Escher into three groups,
all of them mathematically relevant and strictly related to each other
and to corresponding complexity themes. They are:
a. Recursion and fractals
b. Feedback loops and self organization
c. Instability and chaos
(Particularly for the a. and b. points, I took the most
part of my argument about Duchamp from my article on Tout-Fait
Journal cited above, where the reader can find some detailed explanations
about the subjects summarized below).
a.
In Escher's work the role of recursion, and the presence of fractal structures
have been well known and accepted since the appearance of Hofstadter’s
classic book .
As far as Duchamp
is concerned, recursive structures underlie not only several individual
works, but also creative processes on a larger scale, involving several
works at once. I also suggested the presence (at least in embryonic
form) of the idea of fractal structures, mainly linked to the typical
duchampian procedure of repetition on a lower (reduced) scale.
Much of Klees
work is based on recursive (iterative) procedures. Klee called them
progressions. They are mainly related to natural processes.
In relation to natural processes Klee’s intuition of abstract mathematical
concepts, like fractal dimensions (ie non-integer dimension),
is notable, especially in relation to the botanical world.
b.
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Figure 1
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Marcel Duchamp, The Bride
Stripped Bare by Her Bachelors, Even, 1915-23
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Here Escher’s
use of tessellation comes to mind. A game of symmetries could be
seen as a complex system, where very simple rules (namely the given
symmetries) exert their reciprocal feedback locally; as well as interacting
to have dramatic, global and complex consequences on the whole tiling
system. This can be (meaningfully) related to concepts regarding morphogenesis:
simple rules can create global complexity, provided that the components
of the system are sufficiently connected to each other.
In most of Klee’s
works we can see feedback loops in action, both negative and positive.
True dynamic systems are the results of these loops. Klee relates
them to morphogenetic processes. Once again, the key point is: local
simplicity coupled with a huge network of connections) can determine
the emergence of global organizational patterns.
Several of Duchamp’s
wordplays show self-organizing properties. In a broader sense there
are similar random self-organizing processes acting in the Glass.
(Fig. 1)
c.
Looking at Escher’s
prints, exposure to conflicting stimuli (such as black-white, concave-convex,
figure-background) destabilizes the observer. This theme has been
already widely discussed by scholars . Also, Escher is particularly
interested in whirling structures that draw together self-reference,
fractal structures and whirling, chaotic motions.
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Figure 2
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Marcel Duchamp, The Bride
Stripped Bare by Her Bachelors Even [The Green Box], 1934
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It is well known
that instability plays a key role in Klee’s compositions. Moreover
Klee was attracted by what nowadays is called deterministic chaos,
ie. unpredictable, irregular behavior, rising from the iterations
of simple deterministic procedures. A number of different figurative
frameworks are borne out of iterative procedures that have been triggered
to behave irregularly. The most interesting thing, however, is that
from such quite chaotic tangles of lines, often perfect vital and
shiny forms emerge. An interesting analogy can be drawn here with
the edge-of-chaos idea of complexity.
Instability
and chaos are quite typical duchampian themes. His wordplays depend
on predetermined lexical conditions; the slightest differences in
either a single syllable or letter or even simply intonation could
cause radical shifting in the meaning of a sentence (here we have
a true sensitive dependence on initial conditions). Furthermore, Duchamp
saw the creative power of instability. In the loosest sense it could
be seen everywhere in the Glass and in the Notes of the
Green Box, (Fig. 2)but more specifically
we see that in the works based on rotatory motion, where highly unstable
planar sets of rotating circles can create the illusion of the stable
third dimension. Here again the creative power of instability has
been exploited which is also powerful edge-of-chaos idea.
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2. Evolving systems
Before the twentieth
century it was physics, not biology, that was the leading area of
scientific endeavour. It was from physics that models and protocols
for science were drawn. The 1900s saw a shift towards biology. This
shift was consistent with the progressive affirmation of the new paradigm
of complexity.
This interest
in biology is reflected in the work of Klee, Duchamp and Escher. Firstly
Klee, whose interest in Natural History (especially in botany) is well known;
like a naturalist he focused (as both artist and teacher) on the central
problem of organic growth. He investigated both morphology (the study
of forms) and morphogenesis (the study of processes leading to form);
his intuitive, biological investigations are notable.
Escher, for
his part, was more attracted by abstract ideas; more mathematical
than biological, but was nonetheless intrigued by the natural world.
He dealt especially with the inanimate world of minerals and crystals.
However biologists drew analogies of his abstract ideas with corresponding
biological concepts; sometimes Escher dealt with biological processes
themselves.
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| Figure 3 |
| Marcel Duchamp,
Bride's Domain from the Large Glass, 1915-23 |
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| Figure 4 |
| Marcel Duchamp,
Bachelor Apparatus from the Large Glass, 1915-23 |
Perhaps Duchamp
is the artist who best expressed (both in advance and inadvertently)
the inversion of the relationship between biology and physics: by
distending the laws of physics and chemistry he bypassed the
rigid determinism and the reductionism of those disciplines. By grafting
the organic forms of the Bride (in the higher part of the Glass)
(Fig. 3) onto the mechanical machinery
of the Bachelors (in the lower part) (Fig.
4) Duchamp not only expressed the idea of a marriage between
physics and chemistry (at the bottom, in a three dimensional world),
and biology, (above, in a four dimensional world) but perhaps even
the superiority of the latter: after all the bride is queen.
This new kind
of interest in biology is related to the development in every scientific
field of systemic theories. This began in the 1940’s with cybernetics
and ended up with the establishment of complexity sciences: what better
paradigm of a system is there than an organism? Biology teaches us
that complex systems adapt and evolve, we call them complex adaptive
systems (CAS) Adaptation and evolution are key in all areas of the
complexity sciences. Can artists, that were aware of world-system
complexity, have been unaware of these notions of adaptation and evolution,
at least at some intuitive level? In my opinion no. Being sensitive
to complexity implies having some awareness of evolutionary processes
(not necessarily biological), driven by random probabilistic events
coupled with adaptation, which make the world-system ever changing.
In Klees’ writings we find a number of references to evolutionary biology .
He clearly had some understanding of the subject matter. However,
it is what he did as a painter, more than what he thought as a naturalist,
that is interesting here.
He would lovingly
cultivate mistakes he made, and embed them in his paintings. He would
encourage pupils to draw with their left hand, and to nurture the
irregularities that ensued. He also would introduce subtle and repeated
variations into his work that would form mobile, ever changing patterns.
Klee clearly loved chance.
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| Figure 5 |
Figure 6 |
| Paul Klee, Red fugue,
1921 |
Paul
Klee, Sheet from the town book, 1928 |
Here, two of
Klees’ groups of work are of note: those of Red fugue (1921)
(Fig. 5) and Sheet from the town book
(1928) (Fig. 6). Both groups are based
on repetitive horizontal sequences, which are gradually transformed
by introducing constant, apparently random variation in the repetitions
of the starting shape. The representation of an evolutionary process
could be seen in these paintings, where random mutations seem to be
somehow selected to obtain certain properties of the resulting patterns.
I discussed the subject in some detail in the article already cited
,
and I showed by means of computer simulations that evolutionary algorithms
can produce quite similar patterns.
Let us consider
now Escher’s use of tessellation or tiling (i.e. covering of the surface
by means of repeated tiles, without empty gaps and overlapping), such
as the one signed E15 (1938) (Fig. 7).
Each single piece of tiling contains the complete information necessary
to build the whole surface; of course this holds
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| Figure 7 |
| M.
C. Escher, E15, 1938 |
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| Figure 8 |
| M.
C. Escher, Metamorphosis, late 1930s |
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| Figure 9 |
| M.
C. Escher, Verbum, 1942 |
meaningful analogies
with the idea of complete genetic information contained within each
cell of an organism. Parallels between Escher’s tessellation and mechanisms
in biochemistry have been developed by Edward Whitehead .
Interestingly,
Escher’s tiling often depicts a process which gradually transforms
the structure of the tiles. This transformation is rendered infinite
by the introduction of a circular narrative pattern, which leads it
back to its starting point. The strips named Metamorphosis
(the process which turns the larva into insect) are examples of this
(Fig. 8) which Escher created in the
late 30’s. Such transformation can be seen to some extent as a metaphor
for evolutionary process. This, at least, was the opinion of Nobel
chemist Melvin Calvin on Escher’s Verbum (1942) (Fig.
9) .
Let us thirdly consider Duchamp.
The theme of
the dichotomy mother - egg, and the paradoxes that lie therein
are worthy of investigation. I discussed the subject in the already
cited article .
Duchamp used
objects and moulds to signify the idea of mother and egg. The mould
represents the egg, the object the mother. He used these in a number
of different contexts, including the Malic Moulds of the Glass.
(Fig. 10)The object and the mould are
self-perpetuating and codependent : the object is used to cast the
mould and the mould to shape the object. Similar ideas are identifiable
in Three Standard Stoppages (1913-14) (Fig.
11)and Tu m’ (1918) (Fig. 12).
For the latter he made three wooden templates, for transferring the
outline of the threads contained in the Stoppages onto the
oil painting. In Tu m’ these templates appear again, depicted
in the bottom-left corner; their respective threads in the right hand
corner. We have the old threads; their templates, and the new threads…
The two elements (thread-Mother and template-Egg) are present in both
the Stoppages and Tu m’. (En passant: notice
that in Tu m’ the representations of templates and threads
stand at the opposite sides of the picture, as we said above; in the
middle, the psychological epopee of Bride and Bachelor is abridged,
maybe as the necessary step to link egg and offspring).
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| Figure 10 |
Figure 11 |
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| Marcel
Duchamp, Nine Malic Molds, 1914-15 |
Marcel Duchamp,
Three Standard Stoppages, 1913 |
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Figure 12 |
Marcel Duchamp,
Tu m', 1918 |
Now the key
question is: is this chain deterministic? Could this cyclic process
repeat itself unchanged, giving rise to ever equal objects? It couldn’t.
Indeed, remember that Duchamp explicitly connects the idea of mould
with the idea of infra-thin difference:
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| Figure 13 |
| Marcel Duchamp,
Cemetery of Uniforms and Liveries, No. 2, 1914 |
Infra-thin
separation. 2 forms cast in the same mould (?) differ from each other
by an infra thin separative amount. All “identical” as identical as
they can be, (and the more identical they are) move toward this infra
thin separative difference. (Note posthume).
Thus the process
contains an important random event (possibly corresponding to biological
mutation). A parallel could be drawn between evolutionary process
and the cyclical alternation of the object and the mould(Mother-Egg).
At the very least, this observation would be consistent with Shearer’s
idea that the cemetery of liveries (Fig.
13) (the Malic Moulds) could be seen as the place
where scientific knowledge is recorded .
An empty livery is a repository for a scientific idea as well as being
a mould that produces new ideas on which are encoded new theories,
thence new liveries and so on.
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3. Topology
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| Figure 14 |
Figure 15 |
| M. C. Escher, Moebius band I, 1961 |
M. C. Escher, Moebius band II, 1963 |
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| Animation 1 |
| Animation of Moebius Strips |
Klee, Duchamp
and Escher were all three attracted by topologically interesting figures.
Several of Escher’s
works deal with topological figures, such as knots (Knots,
1965) and Moebius strips (as in Moebius band I, 1961
(Fig. 14) and Moebius band II,
1963 (Fig. 15)).
Moebius
strips (see Animation 1 which explains
a possible genesis of the strip starting from a cylinder) exhibit
a number of interesting properties; I’ll recall and briefly explain
some of them.
First, unlike
the cylinder, which has both an internal and external surface, the
Moebius strip has only one surface; This can be easily verified by
mentally painting its whole surface, without lifting the brush from
the strip.
Second, whilst
a cylinder has two edges (lower and higher) the Moebius strip has
one only edge; once again you can follow this edge completely with
the finger without having to lift it from the edge.
Furthermore,
if one cuts the cylinder longitudinally, two distinct cylinders will
be obtained, whereas by cutting the Moebius strip the same way, one
only new strip is yielded.
Escher carefully
showed these properties in his prints. In Moebius band I he
cut the strip longitudinally and obtained three snakes eating each
other’s tail, while in Moebius band II nine ants in line walk
on the strip, so as to highlight the single edge and single face concepts.
Escher was interested
in the circularity of his knots and strips. He drew them together
in a monograph
under the chapter heading spatial circles and spirals. His knots,
Moebius strips, as well as planar and spherical spirals, were all
drawn together in this section. The knots follow circular pathways
which end up at the starting point after torsions and self-intersections.
The same holds for the Moebius strips.
Klee too, was
interested in knots, and had been since childhood. Several of his
early drawings show knotted worms hanging from a fisherman’s hook.
We see the same worms, now abstract knots, in later drawings such
as Ways Toward the Knot (1930) (Fig. 16).
2D knots are also drawn according to even more essential forms, like
the infinity-shaped motif (and its polygonal variants) shown in his
pedagogical sketch (see Sketch 1). We
see a huge collection of similar patterns in pictures like Dynamically
polyphonic group (1931) (Fig. 17),
which is based on a feature of those 2D knots Klee was interested
in (see Animation 2): a hatch follows
the course of the knot with continuity, but always remaining on the
same side of the line; in so doing, the hatch highlights the inside
of one half of the motif, and the outside of the other half. How is
it possible to pass from inside to outside, while remaining on the
same side of the line? It is due to the self-intersection of the 2D
knot (corresponding with the torsion in the Moebius strip),
which allows passage from an inner to an outer region, without
passing from one side of the line to the other. As we saw before,
Moebius strip has a similar property. Let us return to Klee’s 2D knots.
He amplified them, to form complex, perpetual pathways, once again
formed by uninterrupted, closed, self-intersecting lines (often polygonal
instead of curved) and always returning to the starting point. This
is typical of Klee’s drawings of the late 20’s and the early 30’s.
An example is the drawing Mechanics of an Urban Area (1928)
(Fig. 18).
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| Figure 16 |
Figure 17 |
Sketch 1 |
| Paul Klee, Ways Toward
the Knot, 1930 |
Paul Klee, Dynamically polyphonic group,
1931 |
Pedagogical sketch by Paul Klee |
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| Animation
2 |
Figure 18 |
| Animation
based on the 2-D knots |
Paul
klee, Mechanics of an Urban Area, 1928 |
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| Figure 19 |
| Paul Klee, Excited, 1934 |
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| Sketch 2 |
| Folding
recursive process |
We have other
evidence of the special topological meaning of Klee’s images, such
as labyrinths. Labyrinthine lines and signs are indeed among the most
important patterns in Klee’s late style. Several hypotheses could
be made to explain the genesis of such patterns, and in my opinion
they are often linked to morphogenetic processes . There are, however, some drawings, such as Excited (1934) (Fig.
19) and all the others, based on the same framework, which
particularly show the underlying presence of the folding recursive
process (see Sketch 2). Similar (but
reversed) processes are sometime used for classifying labyrinths in
topology
unrolling them, to obtain the simpler equivalent form. But, interestingly,
similar folding processes can also give rise to fractal and/or chaotic
structures
which in turn can be connected with the corresponding themes highlighted
above.
With reference
to the use of Duchamp’s topological figures, I have already underlined
the importance of the Kleinian bottle, along with some related Moebius-strip-like
structures in his writings and works ,
and I have already stressed a possible meaning of this circular self-penetrating
and self-encompassing figure, this is recommended further reading.
Now I’ll focus
on further interesting links between Klee, Escher and Duchamp with
respect to the use they made of the well-known topological properties
of those surfaces.
The analogy between the infinity-shaped motif of Klees’ and Eschers’
Moebius strips is further reinforced by observing some other prints
of Escher’s, which came 10 or 15 years before his Moebius strips;
Horsemen (1946) (Fig. 20) and
Predestination (1951) (Fig. 21)
show the planar infinity-shaped motif.
Let us take
Duchamp’s Steeplechase (Fig. 22):
it is a self-made racing course, for a childish horserace game in
which there is a clear connection with both Klee’s infinity-shaped
motif and Escher’s prints Horseman and Predestination.
This could be seen as an antecedent of Sculpture for Travelling
(1918). (Fig. 23)
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| Figure 20 |
Figure 21 |
| M.
C. Escher, Horsemen, 1946 |
M.
C. Escher, Predestination, 1951 |
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| Figure 22 |
Figure 23 |
| Marcel
Duchamp, Steeple-chase cloth,
ca. 1910 |
Marcel Duchamp, Sculpture
for Travelling, 1918 |
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Jean Clair
suggested an interesting analogy between the kleinian bottle and some
alchemic symbols, such as the one of the pelican devouring itself,
and then he connected it with Duchamp’s Air de Paris (Fig.
24). Look now at Escher’s preparatory sketch (Sketch
3) of a Pelican; although in the definitive print
he substituted the pelican with a dragon (Dragon, 1952) (Fig.
25), he maintained however the same idea of a self-penetrating
and self-eating animal (after all, the snakes eating each other’s
tail in Moebius band I refer to the same theme). As far as
Klee is concerned, we saw similar self-eating structures, though abstract,
in the meandering lines of the drawings around 1934, use the same
techniques as the previously mentioned Excited: the basic motif
(see Sketch 4) is formed by two curves
penetrating one another, the end of the one into the belly of the
other.
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| Figure 24 |
Figure 25 |
| Marcel
Duchamp, Air de Paris, 1919 |
M.
C. Escher, Dragon, 1952 |
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| Sketch 3 |
Sketch 4 |
| M. C. Escher, preparatory sketch of a Pelican |
Two curves penetrating one another |
In his latest
style Klee used a typical pattern whose genesis and meaning we can
better understand by looking at a detail of the drawing The fugitive
is Looking Back (1939) (Fig. 26);
the body of the fugitive is based on branching curved lines, the one
starting from the back of another. The head too is formed by a similar,
curved line, but it is branching from its own back, in a circular,
self-referential scheme. This motif is further amplified in innumerable
drawings and paintings around 1939-40, where we find a lot of self-embedded,
self-encompassed figures, such as in Fastening (1939) (Fig.
27).
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| Figure 26 |
Figure 27 |
| Paul Klee, The
fugitive is Looking Back, 1939 |
Paul Klee, Fastening,
1939 |
What is the
significance of this trend for using topological figures? What relationship
can we establish between that and complexity?
First, with
Klee, Duchamp and Escher there is a tendency to represent very complex
things, where the parts are widely connected to each other, interacting
with non-linear pathways, often looping and returning to some crucial
points. Thus the tangled intricacy of some knots or labyrinths visually
and effectively expresses the corresponding intricacy of the components
of their complex systems.
Second, such
intricacy of connections within a system often produces unexpected
outcomes, which in turn imply new unexpected outcomes, and so on.
Thus in the complex system represented in their works by our artists,
it is difficult to discern clearly causes and effects, because of
the network of their reciprocal feedback. The unexpected, often strange
and sometimes paradoxical outcomes rising from systems subjected to
circular feedback and self-referential loops have corresponded with
the strange and paradoxical properties of figures such as knots, the
Moebius strip or the Kleinian bottle, due to their circularity, their
self-intersections or self-penetrations. The same could be said for
those figures discussed above, often used by the late Klee, which
are self-encompassing.
Particularly in the case of Duchamp, as I have already shown , the topological properties of the kleinian bottle were used to express
the paradoxical identity Egg-Mother (or Bride-Glass).
This was discussed in the previous section, and in general to express
the autopoetic properties of the duo Glass-Box.
4. Enlarged perspective and Impossible 3D objects
| click to enlarge |
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| Figure 28 |
| Marcel Duchamp,
Apolinère Enameled, 1916-17 |
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| Figure 29 |
| Paul Klee, Chess, 1931 |
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| Sketch 5 |
| Trihedral junction versus Dihedral
T-junction |
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| Sketch 6 |
| Trihedral
junction versus Dihedral
T-junction in Apolinère Enameled |
Rhonda Shearer
thoroughly discussed the relationship between some of Eschers’ and
Duchamps’ works, based on 3D impossible objects. She documented how
Duchamp’s Apolinère Enameled (1916-17) (Fig.
28) predates by forty years the seminal
paper of Lionel and Roger Penrose on impossible 3D figures .
She also stresses the bond of friendship between Duchamp and Roland
Penrose, a close relative of Lionel and Roger. The cited Penrose article
is the professed source of inspiration of Escher’s famous impossible
figures, so that the reading of Shearer’s article cements a direct
link between Escher and Duchamp via the Penroses.
But, what about
Klee’s impossible 3D objects? We shall discuss some works, which are
representative of corresponding frameworks, all of them developed
in about 1930 and deeply linked with one another.
The first we
shall consider is Chess (1931) (Fig. 29).
I have elsewhere already examined this painting, its genesis and its
possible meaning .
Here I want only to recall that the bare, empty room in the background
is an impossible 3D object (as a matter of fact, many other elements
in the painting are spatially inconsistent, but here we shall confine
ourselves to the background only).
The walls of
the room are joined to each other by means of vertical edges, three
of which are explicitly traced, whilst the fourth (the dotted one
in Sketch 5) is only suggested by the
left side of the paler rectangle in the upper right hand corner of
the painting. Three of those vertical edges have mutually inconsistent
junctions at the opposite extremities: one end shows a trihedral
junction, where three distinct edges converge, while at the opposite
end, two of the three edges line up one another, giving rise to a
dihedral T-junction. Thus, the background is an impossible,
puzzling 3D object, and the checkerboard covering over the scene may
suggest something like a chess problem, just to emphasize the spatial
enigma posed by the background.
Look now at
Apolinère Enameled: one among the ingredients for making this
3D object impossible, is just the same as for Klee’s Chess:
mutually inconsistent ends of a edge, highlighted in Sketch
6.
Klee used to
express the concepts and the ideas he was interested in, by means
of graphic simplifications, focusing his attention on only the essential
parts. He would discard irrelevant and non-essential details, that
might mislead the observer and would especially avoid repetition and
redundancy. If necessary they wold just suggested.
That’s the reason
why we find traces of other impossible 3D objects in a very simplified
form; as is the case for The Conqueror (1930) (Fig.
30). Look at his banner. Though a banner is essentially
a flat object, at first sight we actually perceive something like
a cube, a solid figure; but counting the peripheral sides of the overall
silhouette, we find that there are five, not six, as we would generally
expect (Sketch 7). Something here is
wrong: as soon as we accept the hypothesis of a possible 3D vision,
we immediately recognize that it is inconsistent with some details
of the motif. There is something missing. To better understand what
really is missing, let us examine a further simplified versions of
the same motif in another of Klees’ pictures: Six species (1930)
(Fig. 31). Look at the flower displayed
in Sketch 8. To make it spatially plausible,
we have to mentally add a missing edge to form a trihedron; the same
holds of course for each other flower in the painting. Without the
addition of the missing edge we perceive something oscillating between
a dihedron and a trihedron, which leads us back to the analogous ambiguity
we saw in Chess.
| click to enlarge |
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| Figure 30 |
Figure 31 |
| Paul Klee, The
Conqueror, 1930 |
Paul Klee, Six
Species, 1930 |
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| Sketch 7 |
Sketch 8 |
| The
impossible 3D object |
An
edge is added to form a trihedron |
Notice now that
Duchamp was interested in exactly the same ambiguity. Look indeed
at the recto side of the Hershey Postcard note (circa 1915)
(Fig. 32), or even at the miniature reproduction
of Why Not Sneeze Rose Sélavy? in the Boite-en-Valise
(1941) (Fig. 33).
click to enlarge |
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| Figure 32 |
Figure 33 |
| Marcel Duchamp,
Note on Hershey Postcard, circa 1915 |
Marcel Duchamp,
miniature version of Why Not Sneeze Rose Sélavy? (1921), in Boite-en-Valise
(1941) |
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Returning now
to Klee’s Conqueror, it is easy to see similar treatments in
its banner. Especially in this case, as we said above, the perception
oscillates continually between the 2D and 3D: no sooner have we arrived
at a 2D hypothesis, then we are pushed to reject it and embrace 3D
one, and vice versa. The relevance of some of Duchamp’s and Escher's
ideas is here clear, for it is well-known that the conflict between
surface and space is one of the most important among their themes.
| click to enlarge |
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| Figure 34 |
| Paul Klee, Soaring,
Before the Ascension, 1930 |
Let’s now turn
our attention to Klee’s Soaring, Before the Ascension (1930)
(Fig. 34) which is representative of
several paintings based on the same framework, worked out in the years
we are considering. The framework is based on rectangles freely soaring
over the whole surface of the work, connected to each other with colored
bars.
At the first
glance we realize that the whole is spatially inconsistent, though
the local details are not. Particularly, it happens that focusing
our attention on a couple of connected rectangles at once, there is
no problem; but considering three or more connected rectangles at
once, in the most cases it yields spatial inconsistencies, that prevent
the observer from seeing which are the closest or the farthest planes
(unless one admits the bars could make a hole in the rectangles and
pass through them).
In Soaring
Klee used several skewed perspective boxes at once, like the ones
in his pedagogical sketch (Sketch 9).
Here we are confronted with the desired effect of spatial ambiguity,
for a face (the red one in see Sketch 10)
might simultaneously belong to several boxes, each of them suggesting
a different perspective; thus, that face has an ambiguous spatial
collocation. We can easily see the practical effects of such a strategy
in Sketch 11, which displays several
of the possible simultaneous perspectives contained in a single detail
of Soaring. Interestingly, because of their shared surfaces,
the perspective boxes used by Klee form a wide network of connected
elements. Notice: not just a linear chain of elements, but a true
net, which allows a multiplicity of possible circular courses .
| click to enlarge |
 |
 |
 |
| Sketch 9 |
Sketch 10 |
Sketch 11 |
| Pedagogical sketch by Klee |
Detail from
Klee's pedagogical Sketch 9 |
Possible
simultaneous perspectives contained in a single detail of Soaring
|
| click
to enlarge |
 |
 |
| Sketch 12 |
Figure 35 |
| Hypercube |
Marcel Duchamp, Poster for the
Third French Chess Championship, 1925 |
This kind of
construction makes me think to something like the hypercube displayed
in Sketch 12and this of course recalls
the Duchamp’s pet; the fourth dimension. Thus, look at Poster for
the Third Chess Championship (1925) (Fig. 35),
where Rhonda Shearer
showed several analogous spatial inconsistencies.
One of the most
famous 3D impossible objects of Escher’s is Ascending and Descending
(1960) (Fig. 36): on the roof of a building
we see an endless staircase. Once again we have a circular course
ever returning to its starting point. It is well-known, and Bruno
Ernst
explained it carefully, that the building, which has the impossible
staircase on its roof, has a strange perspective structure, shown
in Sketch 13. More than any verbal explanation,
animations 3 and 4 help us understand the key reason for this. Animation
3 is a perspective sketch with one only vanishing point.
It starts by showing three distinct parallel planes. They are perspectively
represented with three closed polygonal lines (namely three rectangles)
whose edges are of course not connected with each other. But
by slightly rotating one of the edges of the optical pyramid around
the vanishing point, we get a spiraling polygon, which joins in a
single connected line the edges of several planes. The same holds
if perspective has three vanishing points: look at Animation
4, which explains the perspective structure of Escher’s
impossible building. Here is the surprise. Look at Sketch
14: the impossible room in Klee’s Chess is based
just on the construction presented in animation 3, thus it is deeply
linked to the impossible building of Escher’s Ascending and descending.
(Further explanation for this can be found in the article cited above
).
| click
to enlarge |
 |
 |
 |
| Figure 36 |
Sketch 13 |
Sketch 14 |
| M.
C. Escher, Ascending and Descending, 1960 |
Sketch shows
the strange perspective structure of Escher’s
Ascending and Descending |
The
impossible room in
Klee’s Chess |
| |
 |
 |
| Animation 3 |
Animation 4 |
| One
vanishing point perspective, with iterative spiralling
motion |
Three
vanishing points perspective, with iterative spiralling
motion |
|
Thus, in these
cases both Klee and Escher conceived perspective in terms of an iterative
process, whose outcome is the spiraling, growing motion we saw in
their buildings, as well as in a nautilus shell; thus they thought
of the vanishing point as a sort of attractor of a dynamic system.
| click
to enlarge |
 |
Figure 37 |
| Marcel Duchamp,
Completed Large Glass, 1965 |
Can we see anything
of this in Duchamp’s work? Not exactly the same, but in a way the
answer is: yes, there are.
One of the major
achievements of Duchamp on perspective is of course the lower half
of the Glass (we shall consider the Completed Large Glass,
1965 (Fig. 37)). Thus, look at the Slide,
a perfect perspective box which contains the rotatory element named
the Water mill. Many other rotatory elements can also be found
in the lower part of the Glass, such as the Chocolate grinder
or the Oculist chards, but particularly the pathway described
by the Sieves or the Toboggan have the feature of a
spiral shell we are interested in.
The analogy
between these elements and the perspective spirals we saw above is
admittedly weak. But look now at Rotary demisphere (1925) (Fig.
38). Animation 5 can help
visualize the surprising perspective depth effect one yields once
a similar device is rotating. This is quite close to Klee's and Escher's
idea of considering the perspective vanishing point as a sort of attractor
of an iterative process which implies spiral motions.
| |
 |
 |
| Figure 38 |
Animation 5 |
| Marcel Duchamp,
Rotary Demisphere, 1925 |
Fac Simile of
the spiralling motion visible as the Rotary Demisphere is rotating |
####PAGES####
| click
to enlarge |
 |
 |
| Figure 39 |
Figure 40 |
| M.
C. Escher, Another world II, 1947 |
Paul klee, Perspective
with inhabitant, 1921 |
What I said
so far shows clearly that the theme of the impossible 3D objects and
the one of an enlarged conception of the perspective are tightly linked
to each other. Thus I want just to recall some of Escher’s experiments
on perspective which Klee and Duchamp also did with similar outcomes.
In Escher’s
Another world II (1947) (Fig. 39)
the only vanishing point (roughly in the center of the print) must
be considered at once on the horizon, or at the zenith or at the nadir,
depending on which bird (and which wall) we are considering. The same
holds for Klee’s Perspective with inhabitants (1921) (Fig.
40).
In Relativity
(1953) (Fig. 41) Escher needed three
distinct vanishing points to represent a world with three different
gravitational fields, where people can walk on the walls as well as
on the floor or the ceiling. In Klee’s Arab town (1922) (Fig.
42) we see a similar effect: the ground plane containing
the floor of the higher part of the painting is the same plane containing
the back walls in the lower part.
The fluid perspective
in Escher’s House of stairs (1951) (Fig.
43) could be also explained by Klee’s idea of a stray
viewpoint, and the final outcome could be compared with the one
of Soaring.
| click
to enlarge |
 |
 |
 |
| Figure 41 |
Figure 42 |
Figure 43 |
| M.
C. Escher, Relativity, 1953 |
Paul Klee, Arab
Town, 1922 |
M.
C. Escher, House of Stairs, 1951 |
Finally, as
far as Duchamp’s perspective experiments are concerned, Shearer showed a quantity of different perspective
tricks devised by Duchamp, ranging from stray viewpoints, multiple
vanishing points, fluid perspective, photographic overlapping, and
so on .
Of course, they maintain an high degree of similarity with the ones
of Escher and Klee.
Let’s now return
to the impossible objects, and see them from another viewpoint.
| click to enlarge |
 |
| Figure 44 |
| M.
C. Escher, Waterfall, 1961 |
| |
| Sketch 15 |
| The
simple tribar underling the building of Waterfall |
What do they
represent? Especially in Escher’s case it is evident that they are
variation on a leitmotiv: the one of perpetuum mobile. Look
at Waterfall (1961) (Fig. 44)
or even at Ascending and Descending, which explicitly show
endless motions. But look even at the simple tribar (Sketch
15) which underlies the building of Waterfall. As
we go with the eye along its bars, we perceive a sense of depth, we
feel we are leaving the plane where the bars are actually drawn, to
enter in the third dimension; and the pathway is really endless because,
once the turn is completed, we can repeat it again and again; every
time we find ourselves at the starting point.
Thus the impossible
objects belong to a world which isn’t subjected to the law of thermodynamics:
here entropy doesn’t increase, but reduces itself, to allow perpetual
motions, such as in Waterfall.
A similar overturning
is exactly what happens in complex systems with self-organization,
which is one of the key concept in complexity sciences. Self-organization
means that a system, provided certain conditions (one of them being
the complexity of the system itself) spontaneously reduces its entropy,
by introducing new levels of order among its elements. The slogan
coined by Stuart Kauffman ,
Order for free, effectively captures the essence of the stunning
and seemingly paradoxical discover of a self-established order.
Now, if Klee,
Duchamp and Escher guessed something about self organization as I
believe and as I tried to highlight, then the impossible objects in
a way could express with their paradoxical properties the surprising
order-for-free nature of complex systems.
After all, something
similar has been already said by Jean Clair about the Glass:
Michel Carruoges
noticed that the intricate machinery of the Glass shows several
analogies with other imaginary machines and engines, devised in same
period by Jarry, Roussel, Kafka… Several years after, Jean Clair
recalled Carrouges’ statement, and further deepened the parallelism,
including in the list a quantity of pseudoscientific inventions which
were popular in those years. One of the leitmotivs of those peculiar
machines (Glass included) was that they produce more energy
than they use, said Clair, thus they are variations on the theme of
the perpetuum mobile. In fact they overturn the reality principle
(the second law of thermodynamics) into the pleasure principle (the
dream of an energy completely free and available).
In short, in
my way of seeing things, we can group impossible objects and these
machines together.
| click
to enlarge |
 |
| Sketch
16 |
| Necker
Cube |
But a further
connection can also be found, particularly referred to the Necker
Cube (Sketch 16), which underlies several
impossible objects, such as the Impossible Bed of Apolinaire
enameled. Indeed, as Rosen
observed, the Necker Cube encompasses within itself, thus we can connect
it to the same themes we discussed for the self-encompassing topological
figures: hence feedback looping and self-reference.
5. A unifying reading perspective
In conclusion,
I would like to delimit the context in which this paper should be
read.
Often it happens
that many people at once, unconsciously, independently and following
different courses, elaborate the same new ideas and concepts, and
help cement them into the Zeitgeist. In fact they contribute to the
emergence of new sensibilities and new ways of observing, interpreting
and understanding the world. This was the case for Klee, Duchamp and
Escher: I believe they expressed (being in advance on their time)
and contributed towards ideas that would grow and be affirmed. Nowadays,
this new paradigm, this new way of seeing things is expressed by the
so called complexity sciences.
Reiner Hedrich
stressed some salient traits in the development of complexity sciences.
Here I’m interested in two of them. First, the gestation period of
the new theories has been very long, about one hundred years. This
long latency period, necessary to find a solid mathematical theory
useful to describe dynamic systems behavior generously covers the
lifetimes of Klee, Duchamp and Escher. Thus, in a way, they were immersed
in a stream of ideas and concepts which were still under construction
and organizing in theories. I’m talking about a nascient stream in
the cultural subconscious, not one that was flowing on the surface
of the well established scientific culture of those years; thus I
don’t think that our artists could have been directly influenced by
those scientific ideas; generally speaking, even admitting the possibility
of such an exchange of views, in my opinion it is easier to think
it happened in the opposite direction. With a few exceptions: for
some aspects (I think of concepts such as instabilities and chaos)
scholars acknowledge Duchamp has been influenced by the reading of
Poincaré; but they are just only some aspects of his complex thought;
the same holds for Klee’s possible understanding of (evolutionary)
biology and natural sciences, or for the mathematical readings of
Escher, which moreover in some cases he admitted to be unable to understand.
The second feature in the development of complexity sciences stressed
by Hedrich is that the grounding ideas of the new paradigm, the kernel
of complexity, didn’t deal with a specific disciplinary field: instead,
they are a sort of conceptual foundation, a shared background for
any empirical discipline dealing with complex systems. This fact makes
it conceivable that there was, to some extent, a widespread and unconscious
emergence of such ideas, even though in purely qualitative and intuitive
terms. In other words, I’m talking about quite general concepts, and
not about specific subjects or details of a well-defined disciplinary
field: it makes possible that the same ideas could have been grasped
by someone in more intuitive forms, which is what I’m suggesting for
Klee, Duchamp and Escher.
The concepts
I’m talking about are closely related to each other, they form a tangle
of interconnected ideas, that bringing one of them to the light, mostly
implies that many (or even all of) the others could also somehow come
out. Really, it is impossible to examine thoroughly one of them without
revealing a cascade connection with each other. Indeed, the idea of
cyclic dynamism of a system entails feedback, recursion and self-reference;
in turn self-organization, fractals and chaos are entailed, and again
emergence, dynamic and unstable equilibria, (co)evolution; further,
the visual expression of such a complexity needs new and different
ways to conceive the space, where new and more complex relations may
occur between its elements; thus, the represented space has strange
topological properties.
It is well-known
that complexity introduces several relevant changes in the way we
used to know the world. This is not the place to discuss them: I’ll
limit myself to summarizing some of them.
First: mainly
due to both deterministic chaos and sensitive dependence on the initial
conditions which characterize the dynamical systems, we have to accept
two weaker versions of both the causality principle and determinism.
| click
to enlarge |
 |
| Sketch
17 |
| Diagram
explaining concept of complexity |
Second: because
of the concept of emergence, the reductionist approach is not more
suitable to study complex systems. This is sometime expressed by opposing
reductionism and holism. I like the way Chris Langton expresses the
concept by means of Sketch 17 which I
took from Roger Lewin .
To effectively
explain the radical change in the way of seeing the world implied
by complexity, a set of dichotomies is often used, which confronts
old paradigms with new:
simplicity-complexity,
reductionism-holism, determinism-uncertainty, quantity-quality, necessity-contingence,
predictability-unpredictability, reversibility-irreversibility, repeatability-unrepeatability,
causality-randomness, law-chaos… Interestingly, if we attempt to describe
Klee, Duchamp and Escher by choosing either of the poles in such dichotomies,
we always have to choose the second one.
As far as Duchamp
is concerned, the discussion about the crisis of determinism and reductionism
has been already widely discussed, particularly in relation to the
ideas of Poincaré’s.
Coming to Klee,
I want just recall his essay titled Exact experiences in the field
of art ,
where he expressed a concept which is one of the leitmotiv of his
activity as both artist and teacher: the freedom and the intuition
of the artist act in the space between law and unpredictability,
but always remembering the necessity of both:
Oh, don't let
the eternal spark become completely smothered by law's measure! Take
steps in time! But don't go away from this world completely.
Escher expressed
a similar ambivalence with his prints, in the paradoxical coexistence
of extreme formal rigor and uncertainty, mathematical exactness and
instability, rigorous application of exact principles and unpredictability
in the outcomes.
On the other
hand, only a few words are needed to recall that Klee, Duchamp and
Escher built their work as organisms, composed by a number of interacting
elements, where the whole is always greater than the sum of its parts.
In their works complex processes are mostly represented, whose outcomes
can be emergent, unexpected properties. We can dissect them, but in
so doing we always lose something. This is the true essence of their
holistic art.
Notes
 1. Roberto Giunti, “R. rO. S. E. Sel. A. Vy”, Tout-Fait : The
Marcel Duchamp Studies Online Journal 2.4 (January 2002) Articles
<http://www.toutfait.com/duchamp.jsp?postid=1240&keyword=>.
2. < http://www.mat.uniroma1.it/venezia2005/>.
3. Roberto Giunti, “Percorsi
della complessità in arte: Klee, Duchamp ed Escher”, in: M. Emmer (ed.) Matematica
e Cultura 2003 (Milano, Springer Verlag – Italia, in print)
4. Douglas R. Hofstadter, Godel, Escher, Bach: an Eternal Golden Braid
(New York, Basic Book, 1979)
5. Teuber M. R. “Perceptual theory and Ambiguity in the Work of M. C.
Escher against the background of 20th Century Art”, , in
H. S. M. Coxeter, M. Emmer, M. L. Teuber, R. Penrose (ed.) M. C.
Escher: Art and Science, (North-Holland, Amsterdam, 1986)
6. Giunti, R. “Una linea ondulata lievemente vibrante. I ritmi della
natura nell’opera di Paul Klee”, Materiali di Estetica No.
2, (2000)
7. Giunti R. [6]
8. Whitehead E. P. “Symmetry in Protein Structure and Functions”, in
H. S. M. Coxeter, M. Emmer, M. L. Teuber, R. Penrose (ed.) M.
C. Escher: Art and Science (North-Holland, Amsterdam, 1986).
9. Calvin M. “Chemical Evolution”, Oregon State System of Higher Education, Eugene, Oregon, 1961
10. R. Giunti [1], p. 13, <http://www.toutfait.com/duchamp.jsp?postid=1240&keyword=>.
11. Shearer, R.R. “Marcel Duchamp’s Impossible bed and Other Not Readymade
Objects: A possible route of Influence from Art to Science (Part
I and II). “ Art & Academe, 10:1 & 2. (Fall 1997 and Fall
1998) <http://www.marcelduchamp.org/ImpossibleBed/PartI/>
and <http://www.marcelduchamp.org/ImpossibleBed/PartII/>
12. Escher M. C. The Graphic work (Bendikt Taschen Verlag, Koeln, 1992)
13. Giunti R. “Paul Klee on Computer. Biomathematical models help
us understand his work” in M. Emmer (Ed.) The Visual Mind 2, (The
MIT Press, Cambridge MASS, in print)
14. Tony Phillips “The topology of Roman Mozaic Mazes” in M. Emmer
(Ed.) The Visual Mind (The MIT Press, Cambridge MASS, 1993).
15. D. J. Wright, Dynamical Systems and Fractals Lecture Notes, <http://www.math.okstate.edu/mathdept/dynamics/lecnotes/lecnotes.html>.
16. R. Giunti [1], p. 11, <http://www.toutfait.com/duchamp.jsp?postid=1240&keyword=>.
17. J. Clair, Duchamp at the turn of the Centuries, ToutFait Journal, Issue 3. <http://www.toutfait.com/duchamp.jsp?postid=877&keyword=>.
18. Giunti R. [1], p. 13,<http://www.toutfait.com/duchamp.jsp?postid=1240&keyword=>
19. Shearer R. R. [11]
20. Penrose L.S., Penrose R. “Impossible Objects: a Special Type
of Visual Illusion”, Brit. Journal of Psycology, vol. 49, 1958
21. Giunti R. “Analysing Chess. Some deepening on the concept of
Chaos by Klee”
<http://www.mi.sanu.ac.yu/vismath/giunti/00Chess.htm>
or <http://members.tripod.com/vismath/pap.htm>
22. Indeed, Klee gradually passed from a first conception, where
things are mechanically enchained to each other in a rigid,
linear successions, with a well defined cause-effect relation
(look at the drawing Parade on the track, 1923) Fig.
45 to a final conception where each thing is connected with
each other in a complex network, and causes and effects are
not clearly distinguished: look at the pedagogical sketch (sketch
18). Its caption is says: «Building of an higher organism:
the assembling of parts viewing at the overall function».
| |
 |
 |
| Figure
45 |
Sketch
18 |
| Paul
Klee, Parade on the track, 1923 |
Pedagogical
sketch by Klee |
The framework
of Soaring is just the first important achievement of
such a creative course, which will lead in the late works to
the theme of morphogenesis.
23. Shearer R.R. “Examining Evidence: Did Duchamp simply use a photograph
of “tossed cubes” to create his 1925 Chess Poster?” Tout-Fait
Journal, issue 4, <http://www.toutfait.com/duchamp.jsp?postid=1375&keyword=>.
24. B. Ernst, Der Zauberspiegel des M. C. Escher (Taco, Berlin, 1986)
25. Giunti R. [21]
26. Shearer R. R. “Why the hatrack is and/or is not Readymade: with
interactive software, animations and videos, for readers to
explore”, Tout-Fait Journal, Issue 3, <http://www.toutfait.com/duchamp.jsp?postid=1100&keyword=>
27. Kauffman S. At Home in the Universe. The Search for the Law
of Self- Organization and Complexity (Oxford University Press, 1995)
28. Carrouges M. Les Machines célibataires (Arcanes, Paris, 1954)
29. Clair J. Marcel Duchamp ou le grand fictif (Galilée, Paris, 1975)
30. Rosen, S. M. “Wholeness as the Body of Paradox”. 1997 <http://focusing.org/Rosen.html>.
31. Hedrich R. "The Sciences of Complexity: A Kuhnian Revolution in Sciences?" Epistemologia XII.1 (1999)
<http://www.tilgher.it/epiarthedrich.html>
32. Lewin R. Complexity. Life at the edge of chaos
(The University of Chicago Press, Chicago, 1999)
33. The essay is contained in: Klee P. Das Bildnerische Denken
(Basel: Benno Schwabe & Co., 1956)
34. P. Klee, Tagebücher von Paul Klee 1898-1918 (Köln:Verlag
M. Dumont Scauberg, 1957), note 636, 1905
Figs. 1-2
©2003 Succession Marcel Duchamp, ARS, N.Y./ADAGP, Paris. All rights reserved.
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