
Fractal
Expressionism 

Can
Science Be Used To Further Our Understanding Of Art?
Content
: Richard
Taylor, Adam
P. Micolich and David
Jonas
This question
triggers reservations from both scientists and artists. However,
for the abstract paintings produced by Jackson Pollock in the
late 1940s, scientific objectivity proves to be an essential tool
for determining their fundamental content. Pollock dripped paint
from a can on to vast canvases rolled out across the floor of
his barn. Although recognised as a crucial advancement in the
evolution of modern art, the precise quality and significance
of the patterns created by this unorthodox technique remain controversial.
Here we analyse Pollock's patterns and show that they are fractal
 the fingerprint of Nature. 

In contrast to the broken lines painted by conventional
brush contact with the canvas surface, Jackson Pollock used a
constant stream of paint to produce a uniquely continuous trajectory
as it splattered on to the canvas below [1]. A typical canvas
would be reworked many times over a period of several months,
with Pollock building a dense web of paint trajectories. This
repetitive, cumulative, 'continuous dynamic' painting process
is strikingly similar to the way patterns in Nature evolve. Other
parallels with natural processes are also apparent. Gravity plays
a central role for both Pollock and Nature. Furthermore, by abandoning
the easel, the horizontal canvas became a physical terrain to
be traversed, and his approach from all four sides replicated
the isotropy and homogeneity of many natural patterns. His canvases
were also large and unframed, similar to a natural environment.
Can these shared characteristics be the signature of a deeper
common approach?
Since its discovery in the 1960s, chaos theory [2] has experienced
spectacular success in explaining many of Nature's processes [3].
Could Pollock's painting process therefore also be chaotic? There
are two revolutionary aspects to Pollock's application of paint
and both have potential to introduce chaos. The first is his motion
around the canvas. In contrast to traditional brushcanvas contact
techniques, where the artist's motions are limited to hand and
arm movements, Pollock used his whole body to introduce a wide
range of length scales into his painting motion. In doing so,
Pollock's dashes around the canvas possibly followed Levy flights:
a special distribution of movements, first investigated by Paul
Levy in 1936, which has recently been used to describe the statistics
of chaotic systems [4, 5]. The second revolutionary aspect concerns
his application of paint by letting it drip on to the canvas.
In 1984, a study of a dripping tap showed that small adjustments
could change the falling fluid from a nonchaotic to chaotic flow
[6], and Pollock could have likewise mastered a chaotic flow.

To investigate this possibility,
a simple system can be designed to generate drip trajectories
where the degree of chaos can be tuned. The system consists of
a pendulum which records its motion by dripping an identical paint
trajectory on to a horizontal canvas positioned below. When left
to swing on its own, the pendulum follows a predictable, nonchaotic
motion. However, by knocking the pendulum at a frequency slightly
lower than the one at which it naturally swings, the system becomes
a 'kicked rotator' [7, 8]. By tuning the kick (which can be applied
very precisely using, for example, electromagnetic driving coils),
chaotic motion can be generated. Example sections of nonchaotic
(left) and chaotic (middle) drip paintings are shown in Fig. 1.
Since Pollock's paintings are built from many crisscrossing trajectories,
these pendulum paintings likewise feature a number of trajectories
generated by varying the launch conditions. For comparison, the
right picture is a section of Pollock's dripped trajectories.
Figure 1: Detail of nonchaotic (left) and
chaotic (middle) drip trajectories generated by a pendulum
and detail of Pollock's 'Number 14' painting from 1948
(right) (Yale University Art Gallery, USA).

A
striking visual similarity exists between the drip patterns of
Pollock and those generated by a chaotic drip system. If both
drip patterns are generated by chaos, what common quality would
be expected in the patterns left behind? Many natural chaotic
systems form fractals in the patterns that record the process
[9, 10]. Nature builds its fractals using statistical selfsimilarity:
the patterns observed at different magnifications, although not
identical, are described by the same statistics. The results are
visually more subtle than the instantly identifiable, artificial
patterns generated using exact selfsimilarity, where the patterns
repeat exactly at different magnifications. However, there are
visual clues which help to identify statistical selfsimilarity.
The first relates to 'fractal scaling.' The visual consequence
of obeying the same statistics at different magnifications is
that it becomes difficult to judge the magnification and hence
the length scale of the pattern being viewed. This is demonstrated
in Fig. 2(left) and Fig.2(middle) for Nature's fractal scenery
and in Fig. 2(right) for Pollock's painting.
Figure
2: Photographs of (left) a 0.1m section of snow on the ground,
(middle) a 50m section of forest and (right) a 2.5m section
of Pollock's 'One: Number 31' painted in 1950 (Museum of Modern
Art, USA) 
A second visual clue relates to 'fractal displacement',
which refers to the pattern's property of being described by the
same statistics at different spatial locations. As a visual consequence,
the patterns gain a uniform character and this is confirmed for
Pollock's work in the upper inset of Fig. 3, where the pattern
density P is plotted as a function of position across the canvas.
These visual clues to fractal content can be confirmed by calculating
the fractal dimension D of Pollock's drip paintings [9]. The large
amount of repeating structure within a fractal pattern causes
it to occupy more space than a one dimensional line but not to
the extent of completely filling the two dimensional plane [2,
9, 10]. To detect and quantify this intermediate dimensionality
of fractals, we calculate D using the wellestablished 'boxcounting'
method [2, 9, 10]. We cover the scanned photograph of a Pollock
painting with a computergenerated mesh of identical squares.
The number of squares N(L) which contain part of the painted pattern
is then counted. This count is repeated as the size L of the squares
in the mesh is reduced. In this way the amount of canvas filled
by the pattern can be compared at different magnifications. The
largest size of square is chosen to match the canvas size (L=2.08m)
and the smallest is chosen to match the finest paint work (L=0.8mm).
Within this size range, the count is not affected by any measurement
resolution limits (for example those associated with the photographic
or scanning procedures). D values are then extracted from a graph,
such as the one shown in Fig. 3, using the relationship N(L) ~
L^{D}. The validity of this expression increases in the
small L limit [2,9,10] where the total number of boxes N_{T}
in the mesh is large enough to provide reliable counting statistics.
This condition is satisfied for Fig. 3, where N_{T} varies
from 100 to 4x106 squares over the shown L range. The straight
lines reflect the statistical selfsimilarity of the pattern [9,
10] and D is calculated from the gradients. The accuracy of the
method is confirmed by analysing test patterns consisting of standard
fractals of known dimension.
The two chaotic processes proposed for generating Pollock's paint
trajectories  Pollock's body motions and the dripping fluid motions
 operate across distinctly different length scales. These scales
can be estimated from film and still photography of Pollock's
painting process [11]. Based on the physical range of his body
motions and the canvas size, his Levy flights over the canvas
are expected to cover the approximate length scales 1cm < L
< 2.5m. In contrast, the drip process is expected to shape
the trajectories over the approximate scales 1mm < L < 5cm.
This range is calculated from variables which affect the drip
process (such as paint velocity and drop height) and those which
affect paint absorption into the canvas surface (such as paint
fluidity and canvas porosity). We would therefore expect the fractal
analysis to reveal two D values at different ranges, and Fig.
3 shows this to be the case. We label these as the drip fractal
dimension D_{D} and the Levy flight fractal dimension
D_{L}. We note that systems described by two or more D
values are not unusual: trees and bronchial systems are common
examples in Nature. A consequence of multiple D values is that
each value can be observed over only a limited range of scales.
Recently, it has been stressed that socalled 'limited range'
fractals are no less fractal than ones observed over many orders
of magnitude [12]. Furthermore, a survey of fractals measured
in physical systems indicates that the average range over which
fractals are observed is approximately one order of magnitude
[13]. The range over which D_{D} is measured is 1.1 
1.3 orders (depending on the painting being analysed) and D_{L}
is measured over 2 orders. The length scale L_{T} which
marks the transition between D_{D} and D_{L} occurs
at typically 1  5cm. These ranges are consistent with the values
calculated above from their respective chaotic processes.
The analysis of Fig. 3 produces D_{D} = 1.63. For comparison,
we note that typical D values for natural fractal patterns such
as coastlines and lightning are 1.25 and 1.3. Our analysis also
shows that Pollock refined his dripping technique, with D_{D}
increasing steadily through the years [14]. 'Composition with
Pouring II', one of his first drip paintings of 1943, has a D_{D}
value close to 1. 'Number 14' (1948), 'Autumn Rhythm' (1950) and
'Blue Poles' (1952) have values of 1.45, 1.67 and 1.72. In each
case, the value of D_{L} was closer to 2 than D_{D},
indicating a more efficient spacefilling fractal pattern at the
larger scales. How did Pollock construct and refine his fractal
patterns? In many paintings (though not all), he introduced the
different colours more or less sequentially: the majority of trajectories
with the same colour were deposited during the same period in
the painting's evolution. To investigate how Pollock built his
fractal patterns, we have therefore electronically deconstructed
the paintings into their constituent coloured layers and calculated
each layer's fractal content. We find that each of the individual
layers consist of a uniform, fractal pattern. As each of the coloured
patterns is reincorporated to build up the complete pattern,
the fractal dimension of the overall painting rises. Thus the
combined pattern of many colours has a higher fractal dimension
than those of the individual coloured contributions. The layer
he painted first plays a pivotal role  it has a significantly
higher D value than subsequent layers. This layer essentially
determines the fractal nature of the overall painting, acting
as an anchor layer for the subsequent layers which then fine tune
the fractal dimension. A comparison between the anchor layer and
the corresponding complete painting is shown in Fig. 4.
Fig. 4: A comparison of (left) the black anchor layer and
(right) the complete pattern consisting of four layers (black,
brown, white and grey on a beige canvas) for the painting
'Autumn Rhythm: Number 30' (2.66m by 5.30m) painted in 1950
(The Metropolitan Museum of Art, USA). The complete pattern
occupies 47% of the canvas surface area. The anchor layer
occupies 32%.
Pollock
died in 1956, before chaos and fractals were discovered. It
is highly unlikely, therefore, that Pollock consciously understood
the fractals he was painting. Nevertheless, his introduction
of fractals was deliberate. For example, the colour of the anchor
layer was chosen to produce the sharpest contrast against the
canvas background and this layer also occupies more canvas space
than the other layers, suggesting that Pollock wanted this highly
fractal anchor layer to visually dominate the painting. Furthermore,
after the paintings were completed, he would dock the canvas
to remove regions near the canvas edge where the pattern density
was less uniform. He also took steps to perfect the 'drip and
splash' technique itself. His initial drip paintings of 1943
consisted of a single layer of trajectories which, although
distributed across the whole canvas, only occupied 20% of the
0.35m2 canvas area. By 1952 he was painting multiple layers
of trajectories which covered over 90% of his 9.96m2 canvas.
This increase in both canvas size and density of trajectories
was accompanied by a rise in the pattern's fractal dimension
from 1 to 1.72. Finally we note that, because D follows such
a distinct evolution with time, the fractal analysis could be
employed as a quantitative, objective technique to both validate
and date Pollock's drip paintings [15]. In conclusion, Pollock's
contribution to the evolution of art is secure. He described
Nature directly. Rather than mimicking Nature, he adopted its
language  fractals  to build his own patterns. 
[1] E.G. Landau 1989 Jackson Pollock Thames and Hudson, London
[2] E. Ott 1993 Chaos in dynamical systems Cambridge University
Press, Cambridge
[3] J. Gleick 1987 Chaos Penguin Books, New York
[4] C. Tsallis 1997 Levy distributions Physics World 10 4345
[5] J. Klafter, M.F. Shesinger and G. Zumofen 1996 Beyond Brownian
motion Physics Today 49 3339
[6] R. Shaw 1984 The dripping faucet as a model chaotic system
Aerial Press, Santa Cruz
[7] R.P. Taylor 1998 Splashdown New Scientist 159 3031
[8] D. Tritton 1986 Ordered and chaotic motion of a forced spherical
pendulum European Journal of Physics 7, 162
[9] B.B. Mandelbrot, 1977 The fractal geometry of nature W.H.
Freeman and Company, New York
[10] J. Gouyet 1996 Physics and fractal structures SpringerVerlag,
New York
[11] P. Falkenberg and H. Namuth 1950
[12] B.B. Mandelbrot 1998 Is nature fractal? Science 279 783784
[13] D. Avnir, O. Biham and O. Malcai 1998 Is the geometry of
nature fractal? Science 279 3940
[14] R.P. Taylor, A.P. Micolich and D. Jonas 1999 Fractal Analysis
of Pollock's Drip Paintings Nature 399 422
[15] RPT acknowledges James Coddington, Chief Conservator at
the Museum of Modern Art, New York, for useful discussions on
this subject.
Article
originally appeared as a feature article in
Physics World magazine, October 1999.
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