Fractal Forms and the Deterioration of Artefacts
David A. Scott
In memoriam Nigel J. Seeley 1942—2004
The deterioration of artefacts may involve a number of different decay mechanisms and produce a
variety of morphologies dependent on the mode of formation of the deterioration product and the
type of artefact concerned. Deterioration can be considered as derived from reconstructive,
epitactic, or topotactic events, but does this categorization help us to explain why it is that some
deterioration processes are random, chaotic, structured or periodic? One of the approaches that
can help to explain the complex structures observed in ancient artefacts is that of fractal geometry.
Examples are given of bronze and iron artefacts whose heavily corroded structures illustrate fractal
morphologies, layered or banded structures, which represent self organizing systems. These
phenomena may produce layers of precipitation, or periodic microstructures whose genesis is not
related to annual or fluctuating burial conditions, but to the type of growth phenomena of which the
Liesegang precipitation is one example. Fractal and related models which help to explain some of
these deteriorated structures are discussed, together with illustrated examples of several different
types of morphologies.
INTRODUCTION
Are the deterioration processes of ancient artefacts random, chaotic, structured or periodic? This
question is of interest to us because conservation seeks to preserve :he object in its present state or
'original' condition as rar as possible, without loss of surface detail or deteriorated parts of the
object, which may now be very fragile and easily lost without proper conservation care. We need to
understand the types of deterioration processes that our objects have suffered over time. It is here
that discussion of the types of processes involved in the alteration of artefacts has relevance for us.
Part of the nature of an 'original' object may now be present, for example, in degraded surface
varnishes and finishes, cracked or mottled ceramic glazes, within onion-like layers of corroded
glass, as gypsum crusts on limestone statues, seemingly amorphous lumps of iron corrosion,
desiccated wooden objects, or heavily corroded bronze surfaces. In some cases, the vestigial
retention of the original structure, of the object is clearly reflected within the deterioration products
which have replaced it. This is the case with a corroded bronze dagger handle from the Middle
Bronze Age of Iran, whose photomicrograph is shown in Figure 1. The structure which is preserved
in the pseudomorph of corrosion is the original dendritic structure of the cast bronze, whose nonequilibrium morphology is preserved over the millennia by this mineralization. We shall return to
discuss the issue of non-equilibrium structures later in this paper, which aims to explore some of the
different types of morphologies found in deteriorated artefacts and to suggest that the analysis of
them should take into account both non-equilibrium events and fractal processes.
The preservation of shape and form of heavily deteriorated artefacts may involve a variety of
different decay mechanisms, some of which may produce very different structures from those of the
original artefact. An important part of the conservation of most objects is the preservation of
original structure or surface detail, which, if derived from a process of transformation, may involve
reconstructive, epitactic, or topotactic events in the process of deterioration [1]. For example, a
gypsum
Figure 1 Part of the surface of a Middle Bronze Age dagger handle from Iran showing
pseudomorphic retention of the cast bronze structure within the patina of the dagger handle.
Magnification x45.
Figure 2 Environmental scanning electron photomicrograph (ESEM) surface view of wood
microstructure preserved by pseudomorphic replacement with copper corrosion products. From the
surface patina of a Greek copper plaque of the eighth to ninth century BC. Magnification x340.
crust on an exposed limestone sculpture arises from the dissolution of the original calcium
carbonate of the limestone surface in air polluted with sulphur dioxide, and the precipitation of the
overlying gypsum surface over the calcite of the original limestone. Normally such events are
reconstructive in nature, the original surface of the limestone sculpture is not preserved within the
gypsum crust in the same way as the dendritic structure of the cast bronze axe shown in Figure 1 is.
In such cases, dissolution—precipitation reconstruction has resulted in loss of the original surface of
the limestone artefact. The remaining micromorphology of the gypsum is not related to that of the
calcite crystals that formed the original limestone object. This problem of loss remains an important
feature of reconstructive events.
In the case of epitactic transformations, there is a very strong morphological similarity between the
micro-structure of the original artefact and the deterioration products that have replaced it, either
through direct correspondence or close similarity in lattice dimensions and types.
For example, Figure 2 shows part of the mineralized surface of a copper plaque from the ninth to
eighth century BC, which preserves one of the earliest known versions of the Greek language.
Associated with this plaque surface are mineralized wood fragments, from which the
photomicrograph of Figure 2 is taken. The structure preserved by replacement in copper corrosion
products shows part of the spiral thickening of one of the cell walls of the original wood, of which
only this cast now remains. The spiral thickening is represented here by a cast in copper corrosion
products of the space formerly formed by the spiral morphology of the wood cells. In
dicotyledonous woods, for example, spiral thickenings occur on vessels and fibres in longitudinal
section and may occur as helical ridges on the internal face of the secondary wall. The epitaxial
replacement of copper or bronze grains with cuprite, which preserves the grain structure of the
original copper alloy, is another well-known pseudomorphic preservation of structure that involves
epitaxial growth. These kinds of structural modifications through replacement are important in
helping to retain the shape of the original artefact or to preserve pseudomorphs of textiles, feathers
and wooden artefacts.
Changes which take place in the solid state to create a completely different lattice structure from the
original material are known as topotaxial transformations, and these may or may not create severe
structural alterations of the original material. In these transformations the 'original surface' of the
object may be displaced or lost as the alteration front consumes the original material and replaces it
with something quite different in composition and lattice dimensions. There are many examples of
this kind of transformation in the process of iron corrosion where the alteration of metallic iron to
goethite or haematite may disrupt the original surface so that only an approximation of the original
shape can be recovered during conservation cleaning. This is one reason why the use of Xradiographs to reveal the shape of a heavily corroded iron object is so valuable, since beneath
concretions the corrosion interface between the original shape of the object and the overlying
corrosion is often hard to detect on mechanical cleaning.
Several reviews dealing with the concept and examination of the original surface or marker layer
within corroded metallic surfaces have been published in recent years [1—4]. This subject will not
be discussed in further detail here, but will be a reference point for the further exploration of shape
and morphology of artefacts.
Beyond this simple description of deterioration in terms of these three events, there are a number of
other morphologies that need to be considered, or which can be examined from different points of
view.
PSEUDOMORPHIC MORPHOLOGIES
We have already discussed some aspects of the pseudomorphic replacement of the chemical
constituents of the original material with deterioration products of the same size and shape as the
original form. The chemical composition may be completely altered in this process or may preserve
vestigial remnants of the original material. For example, mineralized textile fibres may still contain
organic material which can be extracted from the pseudomorphic forms. Iron corrosion products
may preserve microstructural detail of the original phases within a totally corroded matrix. This is
illustrated in Figure 3, which shows a photomicrograph of a Chinese cast iron dmg, or tripod, from
Li County Museum, Gansu Province, probably from the Spring and Autumn Period (770—476 BC).
The grey cast iron matrix contains a substantial amount of the ternary eutectic, steadite,
Figure 3 Corroded microstructure of a Chinese cast iron ding of the Spring and Autumn Period
(770-476 BC). The grey cast iron matrix shows well-preserved steadite in iron corrosion products.
Holes in the steadite structure show as the lighter phase. Magnification x240.
whose microstructure, reminiscent of a Swiss cheese, can clearly be seen preserved within the iron
oxides or hydroxides which have preserved this structure without any alteration of shape. The
lighter regions of iron oxides correspond to the 'holes' that would originally have been present in the
steadite structure, which in the metallic form would have been filled with ferrite. There are many
examples of pseudomorphic retention of shape in a variety of different materials, representing the
imposition of order on the subsequent deterioration product by the original material. But this is by
no means a universal event and some alterations devise their own types of structures, which may be
chaotic and disordered or reveal patterned, branching, cracked, or layered morphologies.
FRACTAL MORPHOLOGIES
Fractal forms are an essential part of chaos theory. Fractal geometry deals with the ability of simple
models or shapes, such as the grain structure of an iron object, to generate irregular or highly
differentiated structures from the starting material, unlike the pseudomorph illustrated in Figure 1.
Relationships between solid iron, soil particles, groundwater solutions, temperature, conductivity,
and dissolved oxygen concentrations may create a series of non-linear relationships forming stable
cycles within fields of chaos. The corrosion that forms here may act as its own feedback mechanism
which impinges on the process of deterioration as it develops. Non-linear feedback is an important
feature of many systems undergoing deterioration.
If we take a relatively simple equation such as:
x2 + b = r
where x is complex number and b is a fixed complex number, and we continuously feed back into
this equation the result, r, as our new complex number, x, extraordinary graphical shapes can be
produced. At one limit these shapes are known as the Mandelbrot set, named after the French
mathematician, Benoit Mandelbrot, who first elaborated the field of fractal geometry [5].
Fractal geometry is now used to describe or investigate many complex systems, such as turbulent
flow in water, earthquake events, or weather systems. Systems of this type manifest unstable
periodic behaviour that never repeats itself and continues to be influenced by very small differences
in the starting conditions or variables of the system. Fractal morphologies can encompass a -wide
array of different types of shapes which have direct correspondence to the geometry of nature, from
snowflakes and galaxies to trees and river discharges [6— 9]. Mandelbrot devotes a whole chapter
of his 1982 book to the subject of trees, scaling residues and non-uniform fractals [5]. A good
example of this kind of structure within a wrought iron Icelandic blade fragment from the site of
Mosfell in Iceland, dating from the tenth century AD is shown in Figure 4. The iron oxyhydrox-ides
which comprise this branching structure, within a massive matrix of iron oxides, have grown as
tree-like forms, fragmenting the space into ever-finer variations of tree-like branches. This kind of
corroded microstructure displays a typically fractal geometry whose branching shapes could be
described by a mathematical analysis of the type of branching that this example manifests. The
microstructure here is totally dissimilar to that shown in Figure 3, and is not derived in any sense
from the microstructure of its wrought iron progenitor, which is a typical wrought iron with very
low carbon and phosphorus content, consisting of grains of ferrite.
We have to look elsewhere to account for this type of structure, which fractal geometry provides in
its detailed mathematical description of a variety of forms -fractal tree skeletons are one such form
— and which help us to understand the complexity, the order within the chaos, of this type of
massive corrosion whose structure would otherwise be unaccountable [10, 11]. Other examples of
structures similar to those derived from the
Figure 4 Branching iron oxyhydroxides within a massively corroded iron mineralization crust
comprising part of an Icelandic blade fragment from the site of Mosfell, tenth century AD. This
branching, tree-like structure is fractal in form. Magnification x200.
Mandelbrot sets can be seen in Figures 5 and 6, which are from a fragment of the copper Dead Sea
scrolls, dating from the first century AD, found in 1952 in a cave in the Khirbet Qumran, Judean
Desert, Jordan. The fragment is completely mineralized and interestingly displays exceptionally
fine pseudomorphic retention of shape of the cuprite and malachite patina, enabling the text of the
copper Dead Sea scrolls to be read. A detailed description of the features of the cuprite layers
observed in the copper Dead Sea scrolls has already been provided by Bertholon [12]. Bertholon
divides the types of layers into several different categories. He describes three visually demarcated
layers of outer corrosion overlying another three layers principally of cuprite. These cuprite layers
overlie another layer, which marks the region of the interior. This is followed by two further layers
on the other side of the central region, which are also mineralized. One of the remarkable features
of the copper Dead Sea scrolls is that they are more or less totally mineralized and yet retain
excellent pseudomorphic retention of the hammered letters within the completely mineralized
surface. For a more detailed discussion of the corrosion of the Scrolls, and an explanation of the
descriptive terms that are used to describe them, the reader is referred to Bertholon [12: 299-327].
Within this preserved and totally corroded matrix lie complex cuprite morphologies which are
unrelated either to the preservation of the original surface, the worked and annealed alpha-phase
grain structure of the copper scroll itself, or the larger features of the cuprite layers which can be
demarcated and which make up the structural components of the cuprite matrix. Bertholon,
Robbiola and Lacoudre also found that the properties of the cuprite layer changed on both sides of
the sheet surface, between the outer and inner corrosion layers [4]. They were able to demonstrate
by photoelectro-chemistry that the inner cuprous oxide layer (with which we are concerned here)
had type-n semiconducting properties. The microstructure shown in Figure 5 illustrates part of the
very fine cuprite precipitation within the cuprite matrix. The cuprite particles are composed of
successively smaller particles which in their aggregation assume undulating patterns within a field
of massive cuprite. The finely shaped lobes and spacefilling characteristics of this patterned
precipitate are very representative of the fractal shapes derived from the Mandelbrot model. The
distribution of these cuprite precipitates within the field of massive cuprite is not a random
distribution; it is structured in a growth form that is fractal. The edge of a field of cuprite
precipitates within a cuprite matrix from the copper Dead Sea scrolls
Figure 5 Microstructure of part of a totally mineralized fragment of :he copper Dead Sea scrolls
from Khirbet Qumran, Judean Desert, first century AD. The microstructure shows very fine cuprite
orecipitation within a cuprite matrix. This precipitate is lobed and satterned and is a typical
example of the fractal shapes derived from the Mandelbrot model. Magnification x450.
has a more chaotic microstructure with some organization of the structure, and a boundary whose
organization and structure is typically fractal in its geometry, as shown in Figure 6.
There are many other examples of these types of structure within bronze and iron corrosion
products which display fractal characteristics. That this should be
Figure 6 Microstructure of part of a totally mineralized fragment of the copper Dead Sea scrolls
from Khirbet Qumran, Judean Desert, first century AD. The microstructure shows a boundary edge
where one form of cuprite precipitation merges with another more massive /ariety. Magnification
x400.
the case is hardly surprising given the complexity of the events which may transform a solid artefact
into something completely different from the starting material and which may produce structures
that are chaotic or disordered.
Cellular structures or tessellations can also be analysed using fractal geometry. The polygonal
networks of cracks and fractures that can be seen in ceramic glazes, old layers of varnish, mud
figurines, limestone sculptures, metallic objects and pigment layers may all be regarded as
examples of fractal geometric deterioration. Bucklow has published two useful accounts which
illustrate types of craquelure seen in fourteenth- and fifteenth-century AD Italian panels, fifteenthand sixteenth-century Flemish panels, seventeenth-century Dutch paintings on canvas and
eighteenth-century French paintings on canvas [13, 14]. These studies showed that claims by
connoisseurs regarding the significance of surface cracking in the attribution of paintings to certain
schools did have validity due to the complex relationships between craquelure and the various
factors that affect the aging of support, ground, media and pigment which make up these works of
art.
Within any medium subject to volumetric decrease, one of two different crack systems can be
generated depending on the homogeneity and plasticity of the material. Within inhomogeneous
materials which behave plastically, an orthogonal system of cracks develops whose angle of contact
is close to 90 degrees. In such systems, cracks form at loci of low strength or high stress
concentrations, but they are not propagated simultaneously. An example of this type of cracking can
be seen in Bucklow [13: Figure 17]. In non-orthogonal systems, homogenous media which behave
in a non-plastic fashion are involved, with cracks propagating laterally until a limiting velocity is
reached when the cracks bifurcate at obtuse angles. Unlike the orthogonal system of cracks, all
elements of non-orthogonal intersections are generated virtually at the same time. These structures
are aggregates of non-overlapping cells which, along with their boundaries, can cover whole
surfaces of artefacts. Each cell of the aggregate is a polygon. Edges are common to two cells,
vertices are common to three or more. In the ideal case the three edges meet at each vertex. An
example of this type is seen in Bucklow [13: Figure 7]. In the deterioration of glazed surfaces the
crack pattern may sometimes be random orthogonal, while the cracking observed in many aged
picture varnishes may be of a variety of different forms.
Figure 7 Banded structure in the cross-section of a bronze rod or pin fragment from the Middle
Bronze Age (MBA II) of Iran which shows a fine succession of Liesegang-type periodic
precipitation phenomena. Note that both the cuprite and malachite layers in this cross-section are
banded. Magnification x280.
SELF-ORGANIZING SYSTEMS OF DETERIORATION
Scientists have expressed a great deal of interest in the examination of ordered or patterned systems
that occur when one might have supposed there would be no reason for any special order or pattern
to take place. Nature is full of examples of these, from periodic colours of seashells to subtly
banded green strata in a malachite mineral specimen.
A useful theoretical review of ordered or patterned systems existing outside of an equilibrium series
of conditions is provided by Cross and Hohenberg [15], from the perspective of the physicist, which
can be rather impenetrable to the ordinary reader. These
authors point out that many non-equilibrium spatial arrangements can be classified according to the
linear instabilities of an infinite spatially uniform system. These instabilities arise when the system
evolves away from a thermal equilibrium by the increase of a certain function called the control
parameter. Linear instabilities are categorized into three broad classes according to the values of the
characteristic wave vector and/or the characteristic frequency of the system which appear at the
instability threshold. The discussion is rich in mathematical detail and the interested reader is
referred to this authoritative review for further information. According to Cross and Hohenberg
[15], it is possible to describe the periodic or ordered system using a phenomenological model set of
equations which have the same linear instability as the experimental system, but which are
analytically or numerically more traceable than the starting equations.
Interestingly, it was the famous British mathematician, Alan Turing, who showed in 1952 that two
simple components or ingredients could lead to a wide range of pattern-forming instabilities. The
tendency towards pattern formation in the process of solidification is demonstrated by the instability
of a plane front of the solid phase growing into a supercooled liquid. This is known to physicists as
the Mullins-Sekerka instability and can be understood from an examination of the enhanced
diffusion in front of a bulge of the surface into the diffusion field of temperature or impurity
concentration that limits the growth rate. This enhanced diffusion results in a local increase in the
growth velocity and in turn to further growth of the perturbation. The resulting state is what is
known to metallurgists as a collection of dendrites, growing outwards, according to Cross and
Hohenberg [15] along crystal symmetry directions. However, the pseudomorphically preserved
dendrites (an example of which is illustrated in Figure 1) are not generally derived from crystal
symmetry considerations, but simply grow at various rates and sizes to fill the spaces of the grain
area as the bronze alloy cools down, and may possess very different morphologies even within the
same binary alloy system.
There are many ways of looking at these patterned or ordered systems. One way of doing so is to
consider the proposition that the development of self-organizing systems is a feature of the growth
of some deterioration processes. An example is shown in Figure 7, which represents part of the
corrosion now comprising a bronze rod or pin fragment from the Middle Bronze Age (MBA II) of
Iran, from the collections of the Iran Bastian Museum, Tehran, dating from the early centuries BC.
The features that can be seen in Figure 7 represent very finely banded deposits of cuprite, the
alteration of colour from red to yellow here being influenced by the grain size of the cuprite
precipitates. Very finely crystalline cuprite can appear yellow in colour rather than red, and the
presence of some tin compounds within the corrosion crust may also influence the colour ot this
cuprite deposit, which can be associated with hydrous tin oxides within the corrosion crust. Bands
of malachite can also be seen towards the top of the photomicrograph, which appear green by
pleochroism under the crossed polar illumination used to record Figure 7. This structure, which may
be related to the Liesegang phenomena, is completely unrelated to the structure of the original
material and there are a number of different possible mechanisms by which it could have developed.
The discussion of this particular example as it relates to Liesegang phenomena is discussed in detail
elsewhere [16], and will not be repeated here. There was always some doubt about how valid a
completely Liesegang-derived interpretation would be for such layered systems if they did not
precisely fit the mathematical model proposed for their formation. However, secondary systems
were also discussed [16] as a possible mitigating factor in attempting to analyse this system
mathematically. These self-organizing systems have a number of characteristics; they are open
systems and are part of their own environment (such as banded copper corrosion products or the
layered structures under discussion here), which may create a microstructure unrelated to the
dendritic or grain structure of the original object and maintain this non-equilibrium morphology
indefinitely as they grow. Such systems, in their creation of a new order, are sometimes claimed to
run counter to the second law of thermodynamics, which would normally hold that disordered states
are more favourable in terms of their energy levels than ordered systems, and that everything would
tend to become more disordered upon corrosion and dissolution. The second law really applies to
closed systems, but open systems still have a tendency to create disorder, which in the case of the
fractal morphologies we are considering here is not the case.
Formation of patterned or layered deterioration products could be created by self-organizing
systems in which reaction-transport and feedback mechanisms result in structured formations.
There are a variety of models which could be applied to account for the type of structure illustrated
by the photomicrograph in Figure 7 [17—24].
(1) Reactive-infiltration instability
This kind of instability can occur when a deterioration product or corrosion crust contains
components that are partially soluble in rainwater or groundwaters and is infiltrated by such
solutions. Small local differences in reactivity create fingering or scalloping of the reactiondissolution front [6]. If this front constitutes the 'original surface' of the object, it may be subject to
some degree of alteration during this process, creating shallow ripples, fingering or scalloping of
this interface.
(2) The supersaturation—nucleation—depletion cycle
This is the mechanism for the periodic precipitation phenomena observed in the hydrous layers of
ancient glass and in metallic deterioration processes. First observed by Liesegang in 1896 [23] as a
series of banded precipitates within a homogenous diffusing chemical system, they have been the
object of considerable scientific curiosity ever since.
In its simplest form this cycle produces a precipitate after the concentration product of a barely
soluble component, such as silica gel, cuprite, or malachite, attains a level of supersaturation. By the
fast generation of nuclei and their subsequent growth, the surrounding concentration in the liquid
phase is rapidly depleted and must be built up again before another generation of nuclei can be
precipitated. The precipitated phases are then laid down in rings or parallel structures, as shown in
Figure 7, which is a typical Liesegang type of structure.
(3) The autocatalytic particle growth cycle
This model is based on Ostwald ripening of precipitates producing a competitive particle growth
which may result in further differentiation of the Liesegang precipitation bands, or the creation of
patterns from a continuous layer of precipitation. The hydrated onion-like layers found in ancient
glass could also be explained by Ostwald ripening [24].
The competitive particle growth model is based on a mechanism of spatial instability in an Ostwald
ripening process. Initially, it may comprise two dimensionless variables, such as the local particle
radius, Ψ, and the degree of supersaturation, σ. In the case of the Liesegang ring formation we can
model the formation of cuprite bands as a result of:
where k1 is a velocity constant, II represents solid cuprite, A represents cuprous ions, and B the
oxygen or hydroxide ions. Derived from these, a and b would be their dimensionless concentrations.
From these and other variables, complex partial differential equations can be derived which help to
model the kind of periodic precipitation observed in systems of this kind. The interested reader is
referred to the work of Krug et al. [17] for the mathematical model used successfully here in
simulating the development of periodic precipitation structures in a two-component system in which
the variables employed are further defined.
Such systems can operate in iron corrosion processes, where both epitactic and topotactic
relationships exist between different iron oxides and hydroxides. This can be seen from Figure 8,
which is a banded periodic structure found in the same Mosfell iron blade as shown in Figure 4, and
which occurs contiguously with the previously described structure, showing that periodic
precipitation bands and fractal structures can be closely related to each other within the same object
and corrosion crust. Liesegang-type structures are not often reported for iron corrosion processes,
but samples in the author's collection display features of this type, and indeed were found in another
region of the Chinese cast iron ding previously discussed in this paper.
(4) The diffusion-limited aggregation model
This model was formulated in 1981 by Witten and Sander whose work is reviewed in Korvin [7:
350], These authors proposed a simple rule for one form of fractal growth, whose morphology is
related to that of the dendritic growth in ancient copper alloys and in naturally occurring rocks and
minerals. These 'figured stones', Lapides figumti or Lapides idiomorphk, which were of
considerable interest throughout the Middle Ages [7], were described by the natural scientist and
alchemist Athanasius Kircher (1602-1680), as some of the structures were thought to resemble trees
and animals. Examination of similar rocks, such as the structure of Rhaetic Cotham marble from
southern England, has suggested to modern researchers [7] that the growth of these dendritic and
lobed shapes is fractal in form. There are many examples of dendritic growth and cellular growth in
ancient and historic metallic alloys which represent non-equilibrium structures, and which can be
analysed and modelled by fractal geometry.
In the original model of diffusion-limited aggregation the rules of successive growth are as follows:
first an initial seed particle of copper (for example) is defined at
Figure 8 Successive layers of precipitation of iron oxyhydroxides within a massively corroded
iron mineralization crust comprising part of an Icelandic blade fragment from the site of Mosfell,
tenth century AD. This layer of iron corrosion products overlies the branching, treelike structure
illustrated in Figure 4. Magnification x200.
a lattice point. Another copper particle is allowed to walk at random on the lattice from far away
until it arrives at a lattice site adjacent to the occupied site. Here it is stopped, and a new particle
arrives and is halted when contiguous to either of the two occupied sites, and so the process
continues.
Korvin [7: 351] writes: '. . . the model can be generalized for diffusion-limited aggregation on a line
or surface. . . one [can] start with a whole straight line (or surface) of occupied sites as "seed". The
statistical properties of this process are usually studied by Monte Carlo simulation. A typical result
expresses the connection between the root-mean-square thickness of the deposit grown on a line and
the number of particles
These types of fractal analyses have been used to account for the growth of disequilibrium textures
in igneous rocks, the suture-like structures seen in fossil ammonites and the dendritic-like growth of
river networks. In a modification of the Witten-Sander theory, instead of dealing with particle—
cluster aggregations, one can also examine cluster—cluster aggregations, both of which have
proved useful in the science of fractography. Fractography explores the nature of the topography of
broken surfaces, or surfaces that have undergone brittle fracture so that the nature of the break can
be examined in detail, to determine how failure has occurred or the mode of breakage of the
material described and documented.
CONCLUSION
We know that some forms of deterioration of works of art give rise to chaotic structures that have
no relationship to the original artefact and create problems in the preservation of a variety of
different types of artefacts. This review suggests that there are a large number of cases in which
feedback and self-ordering systems at work in deteriorating artefacts lead to structures or
micromorphologies that can be considered to be fractal. Self-ordered systems are situated at the
delicately balanced edge between order and disorder, and small perturbations in the parameters of
the system may have major consequences for the types of structures that are observed. Scientists
differ on the relative value of fractal analysis in their examination of different types of nonequilibrium phenomena. Many patterns and morphologies have been identified as fractal, but
despite this success, its actual utility is often debated. A valid criticism is that the most important
consideration is not the resultant morphology of the deterioration process as such, but the
underlying mechanistic description. Only in cases where fractal dimensions and other scaling laws
can be related to a specific mechanism has there been viable progress made in relating the type of
structures observed to the processes that are involved in a causal manner. It seems probable that
there may be many different mechanisms at work, which result in the types of structures illustrated
in this paper. The more detailed application of fractal geometric analysis may be useful in helping
to understand how some of these structures arise.
The deterioration of artefacts may proceed in ways which give rise to totally random structures,
which preserve vestiges of the original surface or micromorph-ology of the object, or which create
entirely new structural forms. The layered structures observed in weathered crusts found on ancient
glass, for example, were already an object of scientific study in 1853 when Sir David Brewster first
began to carry out research into their nature. They later led some researchers to suggest that
corroded glass could be dated by counting the number of weathered layers, by analogy to the rings
of a tree [25]. We can now explain these layers by the types of mechanisms discussed here which
are unrelated either to the age of the glass or to the original microstructure of the vessel or window
glass from which it evolved.
In discussing chaotic structures, the notion of self-organization and of the Liesegang phenomena,
we are able to place our understanding today on a much sounder theoretical footing, thanks to the
developments
and understanding that fractal geometry provides. The further elaboration of these types of decay
mechanisms will undoubtedly give rise to greater insight into the deterioration of works of art in the
future. In order to examine a quantitative mathematical model, a considerable amount of
manipulation of the primary data is required in order to pursue a fractal analysis of the kind of
morphology shown in Figure 5. In the opinion of Didier Sornette, who is a professor in the Institute
of Geophysics and Planetary Physics at the University of California, Los Angeles, and an expert in
the field of fractals [26], this appears to represent a good example for further study. Funding tor this
kind of study is not yet available, but should it become so, further work on this interesting topic can
then be undertaken.
ACKNOWLEDGEMENTS
Thanks are due to Chris de Brer, research assistant, for helping with the digital transmission and
printing of the images in this paper, and to the late Dr Nigel J. Seeley, for expanding horizons nonlinearly many years ago.
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AUTHOR
DAVID A. SCOTT: BSC in chemistry from the University of Reading, BA in conservation from the
Institute of Archaeology, London, and PhD from University College, London in 1982. He was
awarded FRSC in 1991 and FIIC in 1994. From 1981 to 1987 he was a Lecturer in Conservation at
University College, Institute of Archaeology, Department of Conservation and Materials Science,
and from 1987 to 2003 he was Head of the Getty Museum Research Laboratory, Getty
Conservation Institute. He has been Professor in Art History and Archaeology, and Chair of the
UCLA/ Getty Program in Archaeological and Ethnographic Conservation since 2003. His principal
interests are the analysis of museum objects, the characterization of pigments, ancient metals and
microstructure, the teaching of conservation, and the archaeometallurgy of pre-Hispanic Colombia
and Ecuador. Address: The Cotsen Institute of Archaeology, Room A410, University of California,
Los Angeles, 405 Hilgard Avenue, Los Angeles, California 90095, USA. Email: dascott@ucla.edu
Résumé — La détérioration des objets peut mettre en jeu un certain nombre de mécanismes de
dégradation et produire une variété de morphologies qui dépendent du mode deformation des
produits d'altération et du type, d'objet en question. La détérioration peut être considérée comme
dérivant d'événements reconstructifs, épitaxiaux ou topotactiques, mais cette catégorisation peutelle nous aider à expliquer pourquoi tel ou tel processus de dégradation est aléatoire, chaotique,
structuré ou encore périodique? La géométrie fractale peut permettre d'expliquer la complexité des
structures observées dans les objets. On donne ici des exemples d'objets en bronze et enfer dont les
structures, fortement corrodées, présentent des morphologiesfractales, en couches ou en bandes,
qui représentent des systèmes auto-organisés. Ces phénomènes peuvent produire des couches de
précipitation ou des microstructures périodiques dont la genèse n'est pas reliée à des variations
annuelles ou à des conditions d'ensevelissement qui fluctuent, mais plutôt au type de croissance,
dont la précipitation selon le modèle de Liesegang est un exemple. On présente les fractales et
autres modèles reliés permettant d'expliquer quelques-unes de ces structures dégradées, ainsi que
des exemples illustrant plusieurs types différents de morphologies.
Zusammenfassung — Beim Zerfall von Artefakten finden in Abhängigkeit von der Art des Objektes
und dem Zerfallsprozeß viele verschiedene Abbaumechanismen statt, die zur Entstehung zu eine
großen Anzahl von Morphologien führen. Es kann dabei angenommen werden, dass der Zerfall von
verschiedenen epitaktischen oder topotaktischen Vorgängen und der Rückumwandlung des
Korrosionsproduktes abhängt, doch bleibt die Frage ob uns diese Charakterisierung beim
Verständnis hilft, warum manche Abbauprozesse zufällig, chaotisch, strukturiert oder periodisch
vorkommen. Ein anderer Ansatz diese komplexen an Artefakten beobachteten Prozesse zu
verstehen, ist die fraktale Geometrie. Es werden Beispiele von Bronze- und Eisenobjekten gezeigt,
deren stark korrodierte Strukturen fraktale Morphologien zeigen, Schichten oder verbundene
Strukturen, die selbstorganisierte Systeme repräsentieren. Diese Phänomene können Schichten oder
Ausfällungen produzieren, oder auch periodische MikroStrukturen, deren Genese nicht in Bezug zu
im Jahresrhythmus sich ändernden Bedingungen gebracht werden können, wohl aber zu
Phänomenen des Wachstums von denen die Liesegangschen Ringe ein Beispiel sind. Fraktale und
ähnliche Modelle, die bei der Erklärung einiger dieser Strukturen des Zerfalls helfen, werden
anhand illustrierter Beispiele der verschiedenen Typen und Morphologien diskutiert.
Resumen — El deterioro de los objetos puede llevar implícito un determinado número de
mecanismos de alteración así como producir una amplia variedad de morfologías, esto depende
tanto de la manera de formarse el producto de deterioro como del tipo de artefacto del que se trate.
El deterioro puede considerarse derivado de factores reconstructivos, epitácticos o topotácticos,
pero ¿puede esta clasificación ayudarnos a explicar por qué ciertos procesos de alteración se
producen aleatoriamente, caóticamente, de manera estructurada o periódica? Uno de las teorías
que pueden ayudarnos a explicar estructuras complejas observadas en objetos antiguos es la
geometría fractal (de fracciones). Se exponen ciertos ejemplos en casos de objetos de bronce y
hierro, cuyas estructuras altamente corroídas ilustran la morfología fractal, por medio de
estructuras en forma de capas o de bandas, las cuales representan sistemas autoorganizados. Estos
fenómenos pueden producir capas de precipitación o microestructuras periódicas, cuya génesis no
está relacionada tanto con condiciones variables de enterramiento (anuales o fluctuantes) como
por el tipo de desarrollo de los fenómenos, de los cuales la precipitación de Liesegang es un
ejemplo. Se discuten en este trabajo tanto el modelo fractal como otros relacionados que pueden
ayudar a explicar algunas de las estructuras deterioradas; también se incluyen ejemplos
ilustrativos de diferentes tipos de morfologías.