Lichtenberg figure

[Roger Bagula]

http://en.wikipedia.org/wiki/Lichtenberg_figure
Lichtenberg figure
From Wikipedia, the free encyclopedia.

Lichtenberg figures are named after the German physicist Georg Christoph
Lichtenberg, who originally discovered and studied them. At the time, it
was thought that their characteristic shapes might help to reveal the
nature of positive and negative electric "fluids". Lichtenberg Figures
are typically formed by the rapid discharge or redistribution of charges
residing on the surface of non-conductors. One way that they can be
produced is as follows: A sharp-pointed needle is placed perpendicular
to a non-conducting plate, such as of resin, ebonite, or glass, with its
point very near to or in contact with the plate, and a high voltage
Leyden jar (a type of capacitor) or a static electricity generator is
discharged into the needle. The electrification of the plate is now
tested by sifting over it a mixture of powdered flowers of sulfur and
red lead (Pb3O4 or lead tetroxide). The negatively electrified sulfur is
seen to attach itself to the positively electrified parts of the plate,
and the positively electrified red lead to the negatively electrified
parts. In addition to the distribution of color thereby produced, there
is a marked difference in the form of the figure, according to the
polarity of the electrical charge that was applied to the plate. If the
charge was positive, a widely extending patch is seen on the plate,
consisting of a dense nucleus, from which branches radiate in all
directions; if negative, the patch is much smaller and has a sharp
circular boundary entirely devoid of branches.

If the plate receives a mixed charge, as, for example, from an induction
coil, a mixed figure results, consisting of a large red central nucleus,
corresponding to the negative charge, surrounded by yellow rays,
corresponding to the positive charge. The difference between the
positive and negative figures seems to depend on the presence of the
air; for the difference tends to disappear when the experiment is
conducted in vacuo. Riess explains it by the negative electrification of
the plate caused by the friction of the water vapour, etc., driven along
the surface by the explosion which accompanies the disruptive discharge
at the point. This electrification would favor the spread of a positive,
but hinder that of a negative discharge. There is, in all probability, a
connection between this phenomenon and the peculiarities of positive and
negative brush or streamer discharges in air. Lichtenberg figures are
fully described in two memoirs Super nova methodo naturam ac motum
fluidi electrici investigandi (Göttingen, 1777-1778).
Modern 3D Lichtenberg Figure in a block of clear acrylic. The fractal
discharge pattern is believed to extend down to the molecular level.
Actual size: 3" x 3" x 2".
Enlarge
Modern 3D Lichtenberg Figure in a block of clear acrylic. The fractal
discharge pattern is believed to extend down to the molecular level.
Actual size: 3" x 3" x 2".

Modern Lichtenberg Figures can also be created within solid blocks of
acrylic plastic (polymethyl methacrylate or PMMA) using a beam of
electrons from a linear electron beam accelerator (or Linac, a type of
particle accelerator). Inside the Linac, electrons are focused and
accelerated to form a beam of high speed particles. Electrons emerging
from the accelerator are moving an appreciable fraction (95 - 99+
percent) of the speed of light (relativistic velocities). The electron
beam is then aimed towards an acrylic specimen. When the relativistic
electrons hit the acrylic, they easily penetrate the surface, slowing
down as they collide with the molecules inside the plastic, and finally
coming to rest deep inside the specimen. Since acrylic is an excellent
electrical insulator, the electrons inside the specimen are temporarily
trapped. Under continued irradiation by the beam, the excess electrons
form a plane of negative charge inside the specimen. As the charge
builds, the effective voltage inside the specimen can reach millions of
volts. Ultimately the huge electrical stress exceeds the dielectric
strength of the plastic, causing it suddenly become conductive in a
process called dielectric breakdown.

Once breakdown occurs, branching tree or fern-like conductive channels
are rapidly formed within the plastic, allowing the trapped charge to
suddenly rush out in a miniature lightning-like flash and bang.
Electrical breakdown of a charged specimen may also be manually
triggered by poking the plastic with a pointed conductive object. The
powerful electrical spark leaves thousands of permanent branching chains
of fractures behind - creating a Lichtenberg figure. Although the
internal charge within the specimen is negative, the actual discharge is
initiated from the positively charged exterior surfaces of the specimen,
and the resulting discharge actually creates a positive Lictenberg
figure within. These rare and beautiful objects are sometimes called
electron trees, beam trees, or lightning trees. As the electrons rapidly
decellerate inside the acrylic, they also generate powerful X-rays.
These X-rays darken the acrylic by introducing defects (color centers)
in a process called solarization. Solarization turns acrylic specimens
an amber or brownish color, although older acrylic blends sometimes turn
a beautiful lime green. The color usually fades over time, and gentle
heating, combined with oxygen, accelerates the fading process.

Lichtenberg figures may also appear on the skin of lightning victims.
These are reddish, fernlike patterns that may persist for hours or days
on survivors. They are also a useful indicator for medical examiners
when trying to determine the cause of death in a victim. Lichtenberg
figures appearing on people are sometimes called Lightning Flowers, and
they are thought to be caused by the rupture of small capillaries under
the skin due to either the passage of the lightning current or the shock
wave from the lightning discharge. A lightning strike can also create a
large Lichtenberg Figure in the grass surrounding the point hit by the
bolt. These are sometimes found on golf courses or in grassy meadows.
Fulgurites may also be created as sand and soil is fused into glassy
tubes by the heat of the high current lightning discharge.

The branching, self-similar patterns observed in Lichtenberg figures
exhibit fractal properties. Lichtenberg figures often develop during the
dielectric breakdown of solids, liquids, and even gases. Their
appearance and growth appear to be related to a process called Diffusion
Limited Aggregation or DLA. A useful macroscopic model that combines an
electric field with DLA was developed by Niemeyer, Pietronero, and
Weismann in 1984, and is known as the Dielectric Breakdown Model (DBM).
Although the electrical breakdown mechanisms of air and PMMA are
considerably different, the branching discharges turn out to be related.
So, it should not be surprising that the branching forms taken by
natural lightning also have fractal characteristics.
[edit]

[Twittering One]

"Lichtenberg figure ~
The branching, self-similar patterns observed in Lichtenberg figures
Exhibit fractal properties ..."
~ Dr. AE

"... but what
Of Lister?"
~ Twittering

"... or this twister?"
~ Folly

"Are you a Wizard,
Or merely a chicken gizzard?
Or, lost in a snow blizzard,
Your companion, a smiling reptile, a lizard?"
~ Merryvale

[Roger Bagula]

http://www.minresco.com/fulgurites/fulgurites.htm
FULGURITES

Lechatelierite Var. Fulgurite is the varietal name given to fused
Quartz, Si02, which has been fused by the action of lightning striking
the Earth and locally melting the sand. The best known Fulgurites are
found in Quartz sands, where the Fulgurites take the form of tubes,
sometimes exceeding a half inch or more in diameter. This type of
formation is called a Sand Fulgurite. As the lightning strikes the Earth
and courses downward through the sand, the sand is instantly super
heated (i.e., melted and fused). After cooling, glass-like hollow tubes
(Fulgurites) can sometimes be located beneath the surface of the sand,
generally decreasing in diameter and sometimes branching as they
descend, sometimes extending for several feet. The outer surfaces sand
fulgurites are often rough with adhering, unfused Quartz sand grains.
The inner surfaces and openings of the tubes are usually smooth and
glassy, in some specimens resembling an applied glaze, sometimes with
blister-like bubbling present. Rock Fulgurites are formed when
lightning strikes the surface of a rock, melting and fusing the surface,
and sometimes the interior of the rock. The melting point of Si02 is
2950oF. The color of the glassy, fused Si02 varies from pale gray, to
smoky gray, to shiny black. The name Fulgurite is from the Latin: FULGUR
(lightning).

In addition to having been found in terrestrial Fulgurites,
Lechatelierite (natural fused Silica, Si02) has also been identified in
the glass-like particles found in Tektites. This includes Australites,
Darwin Glass, Indochinites, Javanites, Libyan Desert Glass, Moldavites,
Philippinites, Bediasites, Muong Nong-type glass, and Aouelloul crater
glass (Impactite). Lechatelierite is not present in Obsidian, a
terrestrial glass-like material associated with volcanic activity

http://www.sciencemall-usa.com/fulgurites2.html

Post by Twittering One
"Lichtenberg figure ~
The branching, self-similar patterns observed in Lichtenberg figures
Exhibit fractal properties ..."
~ Dr. AE
"... but what
Of Lister?"
~ Twittering
"... or this twister?"
~ Folly
"Are you a Wizard,
Or merely a chicken gizzard?
Or, lost in a snow blizzard,
Your companion, a smiling reptile, a lizard?"
~ Merryvale

[Roger Bagula]

Theorems and Axioms of Non-Euclidean Primes:

1) The Primes are a finite Cantor dust set of the Integers.
2) Infinity in terms of the integers is a composite number which
contains a product of all the primes.
3) The dimension of the Cantor dust set is near Log[2]/3.
4) The numbers of primes decreases as n get larger at a rate dependent
of the increased formation of the composites
until no more primes are possible.
5) The gaps in the primes are one and all the even numbers 2*n until
2*n becomes infinite while the composite gaps are only {1,2,3}.
6) The thermal miodel of the prime gaps is:
Prime[n+1]-Prime[n]=1/T[n]
where T[n] is the temperature and T[n] -->0 as n-->Infinity.
7) The composite numbers are an infinite set ( a set that includes
Infinity).
8) The composite gap limit is one:
Limit[CompositeGap[n],n->Infinity]=1
Theorem 1 :
There are a finite number of pair primes.

Logically, since the Integer Infinity is a composite number it can not be a
Prime. There are, thus, no Infinite Primes and the set of Primes is finite.

This axiom set is as consistent as the Euclidean Primes.
Roger L. Bagula { email: ***@sbcglobal.net or ***@yahoo.com }

11759 Waterhill Road,
Lakeside, Ca. 92040 telephone: [phone]
\

[Roger Bagula]

Hey Roger, are you talking to yourself ?
Or is this a sophisticated SPAM-variety ?
Or are you a robot who populates empty news-space ?

I reply to other posts:

Cassandra
The world is a snide place
Run by rude, stupid men
I have my Cassandra Complex
An Army of 12 monkeys in my head
The mathematical model of history
The disease of War
The time series of our ignorance
Our inability to change the chaos
That unfolds around us
While we powerless live in it
Knowing that wars will happen
That our friends and loved ones will die
That eventually I will die as well

By Roger L. Bagula 25 Oct. 1999©

[G Ribouille, Ph.D.]

Post by Twittering One
"Are you a Wizard,
Or merely a chicken gizzard?
Or, lost in a snow blizzard,
Your companion, a smiling reptile, a lizard?"
~Merryvale

Now may I hazard
An answer most haphazard?
Most likely, a smirking volatile, a buzzard!

G. Ribouille, PhD

[m***@aol.com]

"Got gutted mind?
Got Mind Shaft shafted, wondering around, all alone, another time,

Got know map, got know way home,
Got over ~ charged, too? Too much negative, all quantum foam,

No positive ~ A deficit, for sure, short ~ changed,
Deranged, rearranged, harangued,

Left where even kangaroos don't roam,
Never venture, but the vultures do?

Pain, your prescription ~ Your endless subscription ..."
~ White Chalk

[m***@aol.com]

"Queen Anne o' Ponders o' The Forest, right,
Off ~ center,

Off pitch ~ Stirrers o' Glances, affixed, o' Pewee over there,
Right by that vulture, right over there ~ Who

Sings The chuffin' ~ Muses, 'ear, right, 'ere, right, the bloomin'
forest, right,
Whole, right, knows, wirrs, o' the bleedin' scrapings, spoons roll
O' trap o' denouement o' glibness ~ A murder

Of crows
Knows ~ !"
~ Raven Noir

"Infinite divisibility
Of a magnifying defying glass ~ !"
~ Folly

[m***@aol.com]

"Musical Selections ~

11. Haphazard, or Buzz Off Buzzard Bait
12. Do, or Wait & See
13. Call for Help, or Just Try Harder

~ * ~
Ideas for Hors d' Oeuvres
Afterward ~ !"
~ Black Chalk

[m***@aol.com]

"Are you a Wizard,
Or merely a chicken gizzard?
Or, perhaps you're aimlessly lost in a snow blizzard,

Roaming randomly ~

Your cheerful reptile companion,
A smiling chartreuse lizard ~ ?"
~ Merryvale

"O, are you A Twisted Sister, from Twittering's opera
~ * Noir ! Twisted Notes * ~ ?"
~ Folly

[m***@aol.com]

"If the rage is a reaction to external conditions beyond your locus
of
control ~ then, it's outrage ~ not 'rage'

Sorry to be so nitpicky but, one has to differentiate outrage any
reasonable person would feel in certain conditions from pathological
rage."
~ Linda

"Screaming for your life, losing, screaming louder,
Being ignored, pounding on doors, pounding on cars, 4 am, Boston
South End ~

No ears ...

Rage or outrage?"
~ Twittering

"Your locus of
Control ~ ?

What's that. Your locus,
Same as mine ~ ? O, Know ~ I think knot ~ !"
~ Folly

[Roger Bagula]

http://www.physicstoday.org/pt/vol-53/iss-11/p36.html
Diffusion-Limited Aggregation: A Model for Pattern Formation
Recent insights from this well-studied model have led to many new
applications--from river networks to oil recovery, and from
electrodeposition to string theory.

Thomas C. Halsey

Nature confronts us at every turn with patterns--whether the stately
spiral shapes of galaxies and hurricanes or the beautiful symmetries of
snowflakes and silicon. A host of processes can play a role in forming
natural patterns, though they usually involve an interaction between the
transport and the thermodynamic properties of the matter and radiation
involved.

Typically, convection dominates the transport, in both terrestrial and
astrophysical contexts. A classical example is Rayleigh-Bénard
convection. The instabilities and patterns generated in a fluid that is
convectively transporting heat have implications in contexts as
far-flung as laboratory fluid dynamics and solar physics.

In many natural settings, however, convection simply cannot occur. In
those cases, diffusion usually dominates the transport. Consider the
formation of river networks, frost on glass, or veins of minerals in
geologic formations. Similarly, convection plays no role in many
patterns in laboratory settings--for example, during ion deposition,
electrodeposition, or other solidification processes.

The patterns occurring in this type of system have some general
features, which are captured by a number of simple models. The most
famous of these models is diffusion-limited aggregation.1 DLA was
originally introduced by Tom Witten and Len Sander as a model for
irreversible colloidal aggregation, although they and others quickly
realized that the model is very widely applicable. Recent progress in
our understanding of DLA has hinged on scaling studies in nonequilibrium
statistical physics. Those studies have advanced dramatically in recent
years, due in no small part to innovative applications of
renormalization group techniques. Yet, many aspects of DLA remain
puzzling to specialists.

The basic concept

To understand the basics, consider colloidal particles undergoing
Brownian motion in some fluid, and let them adhere irreversibly on
contact with one another. Suppose further that the density of the
colloidal particles is quite low, so one might imagine that the
aggregation process occurs one particle at a time. We are then led to
the following model.

Fix a seed particle at the origin of some coordinate system. Now
introduce another particle at a large distance from the seed, and let it
perform a random walk. Ultimately, that second particle will either
escape to infinity or contact the seed, to which it will stick
irreversibly. Now introduce a third particle into the system and allow
it to walk randomly until it either sticks to the two-particle cluster
or escapes to infinity. Clearly, this process can be repeated to an
extent limited only by the modeler's patience and ingenuity (the
required computational resources grow rapidly with n, the number of
particles).



The clusters generated by this process are both highly branched and
fractal. The cluster's fractal structure arises because the faster
growing parts of the cluster shield the other parts, which therefore
become less accessible to incoming particles. An arriving random walker
is far more likely to attach to one of the tips of the cluster shown in
figure 1a than to penetrate deeply into one of the cluster's "fjords"
without first contacting any surface site. Thus the tips tend to screen
the fjords, a process that evidently operates on all length scales.
Figure 1b shows the "equipotential lines" of walker probability density
near the cluster, confirming the unlikelihood of random walkers
penetrating the fjords.

The example of Hele-Shaw flow

The preceding model is quite interesting, but its general relevance is
not immediately apparent, even for colloidal aggregation at finite
concentration. To illustrate the model's generality, let us consider a
very different problem: Hele-Shaw fluid flow.2

In a thin cell, or in a porous medium, a fluid's velocity is
proportional to the pressure gradient,

(1)

where k is the permeability in a porous medium and m is the viscosity of
the fluid. If the fluid is incompressible, then taking the divergence of
equation 1 yields the Laplace equation,


(2)

Suppose that into such a fluid we inject a second, immiscible fluid of
much lower viscosity--the result is Hele-Shaw flow. An example beloved
of the oil and gas industry is the injection of water into highly
viscous oil in a porous rock (such as sandstone), which is a practical
form of secondary oil recovery. Because of its low viscosity, the
injected fluid's pressure can be set to a constant. Then the flow of the
more viscous fluid is determined by equation 2 with a constant-pressure
boundary condition, and its velocity is given by equation 1--which thus
also determines the velocity of the interface between the two fluids.



An experimental realization is displayed in figure 2. A high-viscosity
light-colored hydrophobic fluid (2.5% hexadecyl end-capped polymer) was
confined to a space 0.4 mm thick between two glass plates 40 cm across.
Water (colored dark) was then injected. The branched structure clearly
resembles a smeared-out version of the DLA simulation shown in figure 1.
Remarkably, the mathematical descriptions of the two problems are almost
identical. For Hele-Shaw flow, the pressure field satisfies the Laplace
equation with constant-pressure boundary conditions, and the velocity of
the interface between the two liquids is proportional to the gradient of
the pressure. For DLA, the probability density of the randomly walking
particle satisfies the Laplace equation, with the cluster's surface
providing a surface of constant probability density. In this case, the
probability of growth (not the growth rate) at the surface is given by
the gradient of this probability density. Thus DLA is a stochastic
version of the Hele-Shaw problem.

The relation between Hele-Shaw and DLA is even more subtle than this,
however. In 1984, Boris Shraiman and David Bensimon analyzed the growth
of the surface in the Hele-Shaw problem in two dimensions, and reached
the surprising conclusion that the problem is, in a mathematical sense,
ill-posed.3 An arbitrary initial surface will generate singular cusps
within a finite time after the initiation of growth, a mathematical
reflection of the so-called Mullins-Sekerka instability in
solidification. Thus, one must add some other physical effect, such as
surface tension, to our model of the Hele-Shaw problem to hold these
mathematical singularities at bay. In DLA, by contrast, the finite
particle size prevents the appearance of any such singularities.



In colloidal aggregation, the particles diffuse, while in Hele-Shaw
flow, the fluid's pressure diffuses. In each case, the growth of the
interface is sufficiently slow that we can use the Laplace equation
rather than the diffusion equation to model the diffusing field. This
suggests that the Laplacian model might be useful for general pattern
formation problems in which diffusive transport controls the growth of a
structure. This is indeed the case: DLA, or some variant of DLA, has
been used to model phenomena as diverse as electrodeposition, surface
poisoning in ion-beam microscopy, and dielectric breakdown.4 Figure 3
shows a mineralogical example, in which a deposition process on a rock
surface has led to beautiful dendritic patterns.

DLA, fractals, and multifractals

DLA clusters are among the most widely known and studied fractal
objects. The fractal dimension D connects the number of particles n with
the size r of the cluster: n = rD. In two dimensions, one finds D »
1.71, and in three dimensions, D » 2.5. Numerical simulations have
determined D in up to eight spatial dimensions, with the result5 that in
high numbers of spatial dimensions d, the cluster fractal dimension D ®
d - 1.

However, in two dimensions, where DLA has been most completely studied,
its fractal nature is curiously fragile. For example, the fractal
dimension is sensitive to the lattice structure of the problem. Thus, if
one performs the succession of random walks, and grows the cluster
without an underlying lattice, one obtains the aforementioned D = 1.71.
However, if one studies precisely the same problem on a square lattice,
one finds,6 for large clusters, that D crosses over to a value of 3/2.
One of the few rigorous results on the fractal properties of DLA is the
bound D ³ 3/2 in two dimensions, proved by Harry Kesten.7

In addition, the fractal dimension of DLA appears to depend weakly on
the geometry of the simulation. The result D = 1.71 is obtained for
radial growth from a seed. However, for growth from a surface, or in a
channel, one obtains a result closer to D = 1.67, a small but robust
difference from the radial growth case that seems to persist to the
asymptotic growth limit.8 DLA clusters also exhibit "multifractality," a
property of the growth probabilities on the surface of the cluster.9
Consider a cluster of n particles. The ith particle has a probability pi
that the next particle to arrive at the cluster will attach to it. The
probability measure defined on the surface of a 2D cluster by the {pi}
is termed the "harmonic measure," due to its relationship with the
theory of analytic functions. The probabilities pi are distributed over
a wide range, being relatively large at a cluster's outer tips and quite
small deep within the fjords. It is thus natural to examine the scaling
of the moments of this probability distribution. We can define a scaling
function s(q) as an exponent,


(3)

The existence of a nontrivial function s(q) implies multifractality,
which can be interpreted as each particular range of growth probability
dp being associated with a different fractal dimensionality.
Nevertheless, the cluster as a whole still has a unique fractal
dimension--the maximum over the fractal dimensions of all the possible
ranges of growth probability.

For deterministic problems, the multifractal scaling function s(q) often
exists even for negative values of q. In those cases, the sum over
probabilities in equation 3 is dominated by the very small values of pi.
For a stochastic problem such as DLA, one might be skeptical about the
existence of such negative-q scaling behavior, which can be easily
disrupted by fluctuations. Several researchers have explored the
breakdown of scaling for negative values of q; in general, the precise
manner of the breakdown depends on the details of averaging the
summation over the stochastic ensemble of DLA clusters.10

The multifractal exponents corresponding to the harmonic measure have
recently been computed exactly by Bertrand Duplantier, using quantum
gravity techniques, for a variety of "equilibrium" fractals in two
dimensions, such as percolation or Ising clusters, and Brownian walks.11
The results agree qualitatively with the scenario envisioned for DLA
clusters, including the breakdown of the formalism at sufficiently
negative values of q. Alas, there is no indication as yet that these
techniques can be extended to nonequilibrium problems.

Scaling laws for DLA

Multifractality is an interesting formal property in its own right, but
its special interest for DLA lies in the existence of scaling laws
connecting the multifractal properties of the probabilities to the
fractal dimension of the cluster.12 The first, and best established, of
these laws was found by Nikolai Makarov. He showed that for any
continuous curve in two dimensions, the harmonic measure has an
"information dimension" of one. Translated into our notation, this
implies that


(4)

which is in good agreement with numerical results.

A second scaling relation was proposed by Leonid Turkevich and Harvey
Scher. Consider the particle of the cluster that is farthest from the
center. One might expect that the cluster radius will grow only if a
particle attaches to this "tip" particle; a process for which the next
arriving particle will have a probability ptip. Since in this event the
maximum radius rmax will grow by roughly the particle size a, it follows
that drmax/dn ~ ptipa. Given ptip as a function of either r or n, and
the supplementary assumption that all radii--including the maximum
radius--of the cluster scale in the same way with n, this equation can
be integrated to give the dependence of r on n.

Let us suppose that ptip is the maximum over the set of all growth
probabilities of the particles in the cluster. Then its scaling can be
extracted from the multifractal behavior of the growth probability
distribution, connecting the asymptotic behavior of this distribution
with the fractal dimension. The result is the Turkevich-Scher scaling
relation,


(5)

Relaxing our assumptions leads to an inequality, in which the dimension
is greater than or equal to the right-hand side of equation 5. But
therein lies a puzzle: It is the inequality, not the equality, that is
satisfied by numerical results.

An additional scaling relation is the "electrostatic" scaling relation
that I proposed. It originates in a formula for the change in the
capacitance C of a surface with small changes in the surface geometry.
Since the growth of a cluster by the addition of particles results in a
succession of relatively small changes in the surface, one can convert
this formula into a form relevant for DLA:


(6)

In two dimensions, this yields Si p3i µ 1/n. Equivalently, comparing
with equation 3, s(3) = 1. In higher dimensions, this argument yields a
modified scaling relation connecting D, d, and s(3). That relation
agrees with numerical results.

Theoretical approaches to DLA

Naturally, the richness of DLA has attracted a number of theoretical
attempts at a comprehensive analysis. Challenges and puzzles, however,
abound. One difficulty facing all such attempts has been the absence of
an easily identifiable small parameter that would allow a perturbation
analysis. DLA seems to yield fractal structures in which fluctuations
are important up to arbitrarily high spatial dimensions; there is no
upper critical dimension, above which mean-field theory would be valid.
In fact, mean-field theory for DLA predicts D = d - 1, which only
appears to be true in the limit of infinite spatial dimensionality.
Also, as a nonequilibrium problem, DLA has no obvious relationship to
the class of problems--mostly related to equilibrium statistical
mechanics--that can be solved in two dimensions by conformal
field-theory techniques.

The self-similarity of DLA clusters suggests that their structure might
be determined by a renormalization group approach. Several proposals for
applying real-space renormalization methods to DLA have indeed been
made. Probably the most sophisticated and successful has been the "fixed
scale transformation" of Luciano Pietronero and his coworkers.13
Although not a real-space renormalization group in the classical sense,
it is based on the enumeration of real-space configurations, and uses a
transformation between scales. It gives good results for the fractal and
multifractal properties of DLA (and a number of other statistical
physics problems) in two dimensions. It also shares with real-space
renormalization groups the lack of a small perturbative parameter.

My coworkers and I have taken an entirely different approach.14 A
noticeable feature of DLA is the way that branches screen one another
simultaneously on a variety of length scales. In two dimensions without
a lattice, DLA typically has four or five large branches, which are more
or less stable. At smaller length scales, however, branches compete in a
never-ending vicious cycle of precarious survival. In fact, for any two
neighbors among these smaller branches, at most one will survive as the
cluster grows. The death of branches as they are screened by their
neighbors is balanced by the creation of new branches via microscopic
tip-splitting processes.

This picture of DLA growth led to the "branched growth model," in which
the competition--on all length scales--between branches is represented
as a dynamical system. The overall cluster dynamics is then represented
as a large family of coupled dynamical systems running simultaneously.

This approach allows approximate but quite detailed solutions for the
cluster dynamics and fractal properties in all dimensions. Results for D
from the branched growth model are in excellent agreement with numerical
results, especially in high dimensions. This approach also allows one to
compute multifractal properties; those results also agree with
simulations. Finally, this approach is especially well suited for
computing the topological self-similarity of the clusters (see the box
on page 39).

The Hastings-Levitov approach

Recent work on DLA has been dominated by a new formulation of the
problem in two dimensions, due to Matthew Hastings and Leonid Levitov.15
Although it has always been known that the Hele-Shaw problem in two
dimensions has a natural conformal representation, Hastings and Levitov
were the first to generate an elegant representation of DLA growth as a
problem in iterated conformal maps. Their formulation has revived
interest in the DLA problem, by making available the powerful tools of
analytic function theory.

Consider a cluster of n particles. The Riemann mapping theorem assures
us that there exists a conformal map, w = Fn(z), that maps a unit circle
in the complex z-plane onto the surface of the cluster in the physical
w-plane. If the exterior of the unit circle is mapped onto the cluster
exterior, then it follows that Fn is an analytic function in the
exterior of the unit circle. Conformal mapping then tells us that the
angular distance between two points on the circle circumference in
z-space is proportional to the total growth probability along the arc
connected by the images of those two points in the physical space.



Hastings and Levitov gave a simple algorithm for the construction of the
function Fn(z) corresponding to a given cluster. Suppose that a
function, fl,q(z), giving a "bump" on the unit circle, corresponds to
the attachment of one particle of size l at an angular position q in the
z-plane (see figure 4). Then if Fn is the map for an n-particle cluster,
the map for an (n + 1)-particle cluster, where the last particle is
added at the image of the angular position q, is given by Fn( fl,q(z)).
Iteration then allows the determination of the cluster map from the
"one-particle" maps f. This is a concrete realization of what
mathematicians refer to as a "stochastic Loewner process." Curiously,
such processes have recently been used in the rigorous proof of some of
Duplantier's results for multifractal scaling.

Hastings used the conformal representation of DLA growth to perform a
momentum-space renormalization group calculation for the DLA dimension
in two dimensions. Although the result, D = 1.7, was highly accurate,
the calculation suffered from the ever-present defect of perturbative
approaches to DLA: It was based on a small parameter that wasn't small.

The Hastings-Levitov algorithm lets us reproduce not only DLA, but also
more general models. If the growth positions, q, in z-space (the
pre-image of the physical space) are chosen randomly, we get DLA. But
other choices are possible. An interesting choice is qn = 2pWn for the
angle qn of the nth particle, with W a constant. Although perhaps
unphysical, this fully deterministic choice allows one to explore the
significance of randomness in the DLA model. Benny Davidovitch and his
colleagues16 have shown that for irrational values of the parameter W,
the Hastings-Levitov model leads to branched structures qualitatively
similar to DLA, but with significantly higher values of the fractal
dimension D, as shown in figure 5. Scaling functions for the
Hastings-Levitov model were computed, both for the stochastic (DLA) case
and for the quasiperiodic (W irrational) case; the functions gave
numerically accurate results for the dimensions of both types of cluster
in two dimensions.

DLA, string theory, and beyond



The most surprising recent development suggests a possible relationship
between Hele-Shaw growth and string theory.17 The starting point for
this development is the remarkable fact that Hele-Shaw growth conserves
the harmonic moments of the exterior domains. Those moments are defined by


(7)

where the integral is over the exterior of the growing structure, z = x
+ iy is the ordinary complex variable, and divergences in the integral
are suitably regularized. Of course, C0 varies as the Hele-Shaw pattern
grows, but all of the other Ck are fixed during the growth. Thus, the
problem of determining the patterns created by Hele-Shaw growth is
equivalent to determining the families of curves with different values
of C0 but fixed values of the other Ck.

This problem, in turn, can be related to a set of equations known as the
"integrable Toda hierarchy," which also appear in 2D quantum gravity,
and hence in string theory. In this relation, the parameters Ck become
the degrees of freedom of this integrable hierarchy. Furthermore, it is
known that a particular solution of the Toda hierarchy is related to the
statistical mechanics of Hermitian N ´ N matrices (which, in the large-N
limit, is also believed to reproduce the scaling behavior of 2D quantum
gravity). It is precisely in the N ® ¥ limit that the Toda hierarchy
maps exactly onto the pure Hele-Shaw problem; this suggests a strong,
yet still obscure mathematical relationship between the latter problem
and string theory.

Next year marks the 20th anniversary of the Witten-Sander model, which
opened the door to the wonderful physics of diffusion-limited
aggregation, and revived interest in the classical problem of Hele-Shaw
growth. Those beautiful structures seemed, at the outset, likely to be
understood by then-conventional techniques. Instead, there remains a
certain amount of mystery. Our understanding of the phenomenology of DLA
has certainly become quite sophisticated, and the new techniques of
Pietronero, Hastings and Levitov, and others have afforded new insights.
In addition, it appears that there are deep connections between
Hele-Shaw growth--and thus DLA--and 2D quantum gravity. Such newly
discovered connections to other problems of theoretical physics suggests
that the next 20 years are liable to be full of surprises.

References
1. T. A. Witten Jr, L. M. Sander, Phys. Rev. Lett. 47, 1400 (1981). P.
Meakin, Phys. Rev. A 27, 1495 (1983). T. Vicsek, Fractal Growth
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2. P. G. Saffman, G. I. Taylor, Proc. R. Soc. London, Ser. A 245, 2312
(1958).

3. B. Shraiman, D. Bensimon, Phys. Rev. A 30, 2840 (1984).

4. R. Brady, R. C. Ball, Nature 309, 225 (1984). L. Niemeyer, L.
Pietronero, H. J. Wiesmann, Phys. Rev. Lett. 52, 1033 (1984). J.
Nittmann, G. Daccord, H. E. Stanley, Nature 314, 141 (1985). See also J.
M. Garcia-Ruiz et al., eds., Growth Patterns in Physics and Biology,
Plenum, New York (1993).

5. R. Ball, M. Nauenberg, T. A. Witten, Phys. Rev. A 29, 2017 (1984).

6. R. C. Ball, R. M. Brady, G. Rossi, B. R. Thompson, Phys. Rev. Lett.
55, 1406 (1985). P. Meakin, R. C. Ball, P. Ramanlal, L. M. Sander, Phys.
Rev. A 35, 5233 (1987).

7. H. Kesten, Stoch. Proc. Appl. 25, 165 (1987).

8. B. B. Mandelbrot, A. Vespignani, H. Kaufman, Europhys. Lett. 32, 199
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9. T. C. Halsey, P. Meakin, I. Procaccia, Phys. Rev. Lett. 56, 854
(1986). C. Amitrano, A. Coniglio, F. di Liberto, Phys. Rev. Lett. 57,
1016 (1986).

10. M. E. Cates, T. A. Witten, Phys. Rev. A 35, 1809 (1987). T. C.
Halsey, in Fractals: Physical Origin and Properties, L. Pietronero, ed.,
Plenum Press, London (1989). B. B. Mandelbrot, C. J. G. Evertsz, Nature
348, 143 (1990). S. Schwarzer, J. Lee, S. Havlin, H. E. Stanley, P.
Meakin, Phys. Rev. A 43, 1134 (1991). R. C. Ball, R. Blumenfeld, Phys.
Rev. A 44, 828 (1991).

11. B. Duplantier, Phys. Rev. Lett. 82, 880 and 3940 (1999); 84, 1363
(2000).

12. N. G. Makarov, Proc. London Math. Soc. 51, 369 (1985). L. Turkevich,
H. Scher, Phys. Rev. Lett. 55, 1026 (1985). T. C. Halsey, Phys. Rev.
Lett. 59, 2067 (1987).

13. L. Pietronero, A. Erzan, C. J. G. Evertsz, Phys. Rev. Lett. 61, 861
(1988). A. Erzan, L. Pietronero, A. Vespignani, Rev. Mod. Phys. 67, 545
(1995).

14. T. C. Halsey, M. Leibig, Phys. Rev. A 46, 7793 (1992). T. C. Halsey,
Phys. Rev. Lett. 72, 1228 (1994). T. C. Halsey, B. Duplantier, K. Honda,
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15. M. B. Hastings, Phys. Rev. E 55, 135 (1997). M. B. Hastings, L. S.
Levitov, Physica D 115, 244 (1998). B. Davidovitch, H. G. E. Hentschel,
Z. Olami, I. Procaccia, L. M. Sander, E. Somfai, Phys. Rev. E 59, 1368
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16. B. Davidovitch, M. J. Feigenbaum, H. G. E. Hentschel, I. Procaccia,
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Thomas Halsey is a senior staff physicist at ExxonMobil Research and
Engineering Co in Annandale, New Jersey.

[legalize+ (Richard)]

[Please do not mail me a copy of your followup]

This seems to be just a rehash of the work discussed in "Fractal
Growth Phenomena" by T. Fiscek, 1989. Is there something new here?

[zZ^2->Rc]

Very cool, thanks Rodger. But, I wonder, if you freeze lightning or
take any other snapshot of reality, is it fractal? Important definition
of fractal is in the dynamical nature of things. "Aggregation" and
"pattern" might not be the physics of the pattern, but the physics of
perception (snapshot).

[Stewart Robert Hinsley]

Post by zZ^2->Rc
Very cool, thanks Rodger. But, I wonder, if you freeze lightning or
take any other snapshot of reality, is it fractal? Important definition
of fractal is in the dynamical nature of things. "Aggregation" and
"pattern" might not be the physics of the pattern, but the physics of
perception (snapshot).

Fractals need not have to do with dynamics. For example a definition of
the attractor of an iterated function system is a region of a space
which is the union of the images of that region under the functions of
the IFS.

[zZ^2->Rc]

Thanks for the response. I think that your usage of "mathematical"
comes no where near my "physical", although "aesthetic" probably
outperforms them both.


THE intricate spiral patterns seen on the heads of sunflowers, and
thought to occur only in plants, have now been spotted on
nanoparticles. Zexian Cao's team at the Chinese Academy of Sciences in
Beijing was studying stresses in spheres about 10 micrometres in
diameter. The spheres had a core of silver and a 150- nanometre-thick
shell of silicon oxide. When cooled, the shell shrank more than the
core, and bumps appeared on the shell as a result. The bumps were
arranged in two sets of spirals, running in opposite directions.
Curiously, the numbers of spirals in each pair of spiral sets were
always adjacent members of the Fibonacci series, in which each number
is the sum of the previous two (1, 1, 2, 3, 5, 8, 13 and so on). Cao
found sets with 5 and 8 spirals, and 8 and 13 spirals, for example
(Science, vol 309, p 909). "We immediately thought of those amazing
patterns of sunflower seeds, daisy florets and pineapple scales," says
Cao. These patterns help plants deal with the stress of growth. "Simply
minimizing the stress energy leads to the formation of a Fibonacci
pattern," he says.

From issue 2512 of New Scientist magazine, 13 August 2005, page 19