Spatio temporal modeling of snow flake crystals using packardªs cellular automata

IJRET: International Journal of Research in Engineering and Technology ISSN: 2319-1163
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Volume: 01 Issue: 04 | Dec-2012, Available @ http://www.ijret.org 558
SPATIO-TEMPORAL MODELING OF SNOW FLAKE CRYSTALS USING
PACKARD’S CELLULAR AUTOMATA
Shanthi.M1
, E.G.Rajan2
1
Research Scholar, Mysore University, Hyderabad, India,shanu_shivak@yahoo.com
2
Director, PRC Ltd, Hyderabad, India, rajaneg@yahoo.com
Abstract
Cellular automata (CA) modelling is one of the recent advances in spatial–temporal modeling techniques in the field of growth
dynamics. Spatio-temporal modeling of growth patterns has gained more importance in the recent years especially in the field of
crystal growth, urban growth, biological growth etc. It has become an interest for researchers to study the model on spatial and
temporal dynamic behavior. This paper aimed at modeling crystal growth using cellular Automata, which have dynamic capabilities
to handle spatio-temporal phenomenon for better and efficient growth process. Cellular Automata models are used to simulate the
process of crystal growth and to generate various patterns that the crystals create in nature. CA’s do quite easily reproduce the basic
feature of the overall behavior that occurs in real world.CA models have been successfully used to simulate different growth behavior
of crystals, since cellular Automata and crystals have similar structure.
Index Terms: Crystal Growth, Snowflake Crystal, Cellular Automata, and Types of crystal
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1. INTRODUCTION
The spatial dimension plays a key role in many social
phenomena. Spatial dynamics refers to the sequence of
changes in space and time. The changes which takes place
with respect to space is called spatial process, the latter is
called temporal process. The spatial and the temporal process
are one and the same and they cannot be separated. This
spatiotemporal process is used in planning, urban
development and issues related to geographical phenomenon.
All geographical phenomena are bound to have a spatial and a
temporal dimension. The aim of modeling is to abstract and
represent the entity being studied. Modeling can be
conceptual, symbolic or mathematical, depending on the
purposes of the specific application. Modeling can be utilised
for analysing, evaluating, forecasting and simulating complex
systems to support decision-making. From the perspective of
spatial science, modeling must take both the spatial and
temporal dimensions.
Model can be represented as “a schematic representation of
reality, developed with the goal of understanding and
explaining it”. Spatial interactions can also be expressed as an
influence of a location on another, without being explicitly
embodied in the form of a measurable exchange or flow.
Spatial dynamics are easy to implement when compared to
that of temporal dynamics since the change in time should be
also be taken into account while modeling. Many techniques
were currently used to model spatial and temporal growth
especially in the field of crystal growth. 'Crystal' comes from
a Greek word meaning clear ice. In the late sixteenth century,
Crystal growth has been widely studied for many years.
Andreas Libavius, made the theory, which said, "Mineral salts
could be identified by studying the shapes of the crystal
grains." In 1669, Nicholas Steno observed that corresponding
angles in two crystals of the same material were always the
same. The first synthetic gemstones were made in the mid-
1800s, and methods for making high-quality crystals of
various materials have been developed over the course of the
past century. Since the mid-1970s such crystals have been
crucial to the semiconductor industry. Systematic studies of
the symmetries of crystals with flat facets began in the 1700s,
and the relationship to internal structure was confirmed by X-
ray crystallography in the 1920s.
Crystals form whenever a solid is formed from fluid. Crystals
form from vapors, solutions or molten materials, and are built
from repeating units. Crystals grow from the outside. Crystal
formation is called crystallization. Crystallization means
"become crystals". At a microscopic level, crystals consist of
regular arrays of atoms laid out much like the cells in a
cellular automaton. Crystals always start from a seed such as
a grain of dust and then progressively adding more atoms to
their surface. Some of the examples are: snowflakes
formation from water vapor, rocks like felsite, and most non-
living substances.
Snow crystals have a rich diversity of forms with striking
hexagonal symmetry. The two-dimensional types include
dendrite, stellar, sector and plate forms. Physical studies have
shown that the particular form of a snow crystal is dependent
upon the temperature and saturation in the growth
environment. Snow crystals exhibit remarkable, intriguing six

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fold symmetry while displaying a wide diversity of forms.
Kepler suggested in 1611 that the symmetry of snow crystals
was related to the hexagonal packing of spheres [1], which is
remarkable foresight given the modern understanding of the
hexagonal molecular packing of ice crystals[2].
The formation of complex structures during solidification
often results from a subtle interplay of nonequilibrium,
nonlinear processes, for which seemingly small changes in
molecular dynamics at the nano scale can produce profound
morphological changes at all scales. One popular example of
this phenomenon is the formation of snow crystals, which are
ice crystals that grow from water vapor in an inert
background gas. Although this is a relatively simple,
monomolecular system, snow crystals display a remarkable
variety of columnar and plate-like forms, and much of the
phenomenology of their growth remains poorly understood,
even at a qualitative level [3]. The crystal growth and
the cellular Automata have the same features.
2. SNOWFLAKE CRYSTALS
A crystal is a material in which the molecules are all lined up
in a specific way called the "crystal lattice". The water
molecules in ice form a hexagonal (six-sided) lattice, and all
snow crystals have six sides. The crystals form when water
vapor condenses directly into ice around tiny bits of dust that
have been carried up into the atmosphere. The beautiful
patterns form as the crystal grows, dressing up the dirt to look
real purrrdy!. Snow is not frozen rain. Sometimes raindrops
freeze as they fall, but this is called "sleet". Sleet particles are
just frozen water and don't have any of the patterns found in
snow crystals.
Fig- 1 Basic Shapes of Snowflakes
A snow crystal is a single crystal of ice, but a snowflake can
be an individual crystal, or as many as 200 stuck together to
form the big "puff-balls" that often fall when the temperature
is just below freezing. Snowflakes can be big clusters of snow
crystals, or just one individual and unique crystal. No two
snow crystals are exactly alike (which is rather surprising
given the amount of snow that falls), but their forms usually
fall into several basic types and categories. Ice crystal is
material in which molecules are arranged as a hexagonal
lattice (crystal lattice). Snowflakes patterns arise as snow
crystal grow their Growth depends on temperature and
humidity
Snow can be further classified into six basic patterns called:
Needles, columns, plates, columns capped with plates,
dendrites, and stars. Each type is the result of different
atmospheric and temperature conditions within the cloud.
Snowflake is one of the well-known examples of crystal
formation. Snowflakes are collections of snow crystals,
loosely bound together into a puff-ball. These can grow to
large sizes, up to about 10 cm across in some cases, when the
snow is especially wet and sticky. Thus, snow crystals are
individual, single ice crystals, often with six-fold symmetrical
shapes.
These grow directly from condensing water vapor in the air,
usually around a nucleus of dust or some other foreign
material. Typical sizes range from microscopic to at most a
few millimeters in diameter.
Fig- 2 Crystalline Structure of Ice
The structure of a given snowflake is determined by the
temperature and humidity of the environment in which it
grows, and the length of the time it spends there. Figure 3
shows some forms of snowflakes that are often seen in nature.
Fig -3 1) Simple sectored plate; 2) Dendritic
sectored plate; 3) Fern-like stellar dendrite

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2.1 Six-fold symmetry of snow crystals
When water freezes into ice, the water molecules stack
together to form a regular crystalline lattice and the ice lattice
has six-fold symmetry. It is this hexagonal crystal symmetry
that ultimately determines the symmetry of snow crystals.
2.2 Complex symmetrical shapes of snow crystals
The snow crystals form complex shapes because of their
simple six-fold symmetry and also because they are both
complex and symmetric, and it is this combination that gives
them their special beauty The growth usually begins with a
dust particle, which absorbs some water molecules that form a
nucleus for the ice crystal.
Typically snow crystals need some kind of surface on which
to get started. Faceting then causes the newborn crystal to
quickly grow into a tiny hexagonal prism. As the crystal
grows larger, the corners often sprout tiny arms, since they
stick out a bit further into the supersaturated air and thus grow
a bit faster. The crystal growth rate strongly depends on the
temperature. If there are variations in the temperature, the
snow crystal encounters different growth conditions, growing
into an intricate shape. Thus, we see such a rich diversity in
the shapes of snow crystals in nature. When a snow crystal
grows from air supersaturated with water vapor, there are two
dominant mechanisms that govern the growth rate. The first is
diffusion -- the way water molecules must diffuse through the
air to reach the crystal surface. The second involves the
surface physics of ice -- the efficiency with which water
molecules attach themselves to the ice crystal lattice.
Faceting is the one which operate at the molecular scale to
produce the crystal lattice, can control the shape of a snow
crystal some ten million times larger. Facets appear on many
growing crystals because some surfaces grow much more
slowly than others. If we imagine
Fig- 4 Crystal faceting
beginning with a small round ice crystal, then mostly we
would find that the surface was quite rough on a molecular
scale, with lots of dangling chemical bonds. Water molecules
from the air can readily attach to these rough surfaces, which
thus grow relatively quickly. The facet planes are special,
however, in that they tend to be smoother on a molecular
scale, with fewer dangling bonds. Water molecules cannot so
easily attach to these smooth surfaces, and hence the facet
surfaces advance more slowly. After all the rough surfaces
have grown out, what remains are the slow-moving facet
surfaces.
3. CELLULAR AUTOMATA CONCEPT
A cellular automaton is an array of identically programmed
automata, or "cells", which interact with one another. The
arrays usually form either a 1-dimensional string of cells, a 2-
D grid, or a 3-D solid. Most often the cells are arranged as a
simple rectangular grid, but other arrangements, such as a
honeycomb, are sometimes used.[4] The essential features of
a cellular automaton
("CA" for short) are:
 Its State - a variable that takes a separate value for
each cell. The state can be either a number or
a property.
 Its Neighborhood - the set of cells that it interacts
with. In a grid these are normally the cells
physically closest to the cell in question.
 Its Program - the set of rules that defined how its
state changes in response to its current state,
and that of its neighbors.
Cellular automata are discrete dynamical systems whose
behavior depends on local rules. Perhaps the most famous
cellular automata is The Game of Life described by John
Conway [5,6]. That automaton remains intriguing because its
complex behavior is capable of universal computation.
However, more serious applications are becoming common
because cellular automata allow parallel processing [7]. Real
valued cellular automata are also commonly used in
applications such as image processing [8]. Such automata
were used as a local model for snow crystal growth on a
hexagonal lattice [9]. A wide range of growth structures were
created, including stellar, dendrite, sector, and 2 plate forms
which includes the basic 2-dimensional types seen in physical
snowflakes. Of course, the growth exhibited, and was limited
to, having 6-fold symmetry as determined by the underlying
lattice.
In his cellular automaton (CA) model, each cell (or site) of a
planar lattice changes from empty to occupied as that location
turns to ice and remains occupied thereafter. A cell “freezes”
when it has one frozen neighbor, or when the number of
frozen neighbors equals some prescribed higher count. Cells
of the original 1984 model had six nearest neighbors,
reflecting the hexagonal molecular structure of ice and
observed symmetry of actual snow crystals
Cellular Automata models are used to simulate the process of
crystal growth and to generate various patterns that the
crystals create in nature. Crystals form whenever a solid is
formed from fluid. Crystals form from vapors, solutions or

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molten materials, and are built from repeating units. Crystals
grow from the outside. Crystal formation is called
crystallization. Crystallization means "become crystals". At a
microscopic level, crystals consist of regular arrays of atoms
laid out much like the cells in a cellular automaton. Crystals
always start from a seed such as a grain of dust and then
progressively adding more atoms to their surface. Some of the
examples are: snowflakes formation from water vapor, rocks
like felsite, and most non-living substances. There has been a
lot of research work done in the field of crystals, which make
use of cellular automata models for simulation purposes.
Cellular automata can be applied to emulate snowflakes. the
reason is that, it was not the case that every snowflake was
identical. However, if every snowflake had a random
structure, the information from each snowflake would be
meaningless. Therefore, there is a syntax within snowflakes,
several main types of structures, which are capable of
containing individual ariations.Packard discovered that
different weather conditions result in snowflakes taking on
different general aspects. One set of conditions yields
configurations that look like plates, another determines
snowflakes shaped like collections of rods, and another yields
dendritic stars. He wrote a cellular automaton simulation in
which the “off” cells, those with a value of “0”, represented
water vapour, and the “on” cells, those assigned a value of
“1”, represented ice, and appeared on the screen in colour.
The snowflake would grow outwards from its boundary. A
typical set of rules initiated a scan of a cell’s neighbourhood,
summed the values of the surrounding cells, and filled the
new cells with either ice or water vapour, depending on
whether the total was odd or even. The resulting artificial
snowflakes lacked the complexity of real snowflakes,
particularly those with structures based on patterns of needle-
like shapes but they did have plates and dendrites growing
from the corners of the plates, from which more dendrites
grew; they were easily identifiable as snowflakes.
4. TYPES OF SNOWFLAKE CRYSTAL
Fig- 5 Stellar Snowflake Crystal
Stellar snowflakes are thin crystals with six broad arms that
form a star like shape[10].
Fig- 6 Stellar Dendrites Snowflake Crystal
Dendritic means “tree like”. These types of snowflakes are
fairly large 2-4 mm and an be seen with the naked eye.
Fig- 7 Sectored plate Snowflake Crystal
Sectored snowflakes are hexagonal crystals divided into six
equal pieces.
Fig- 8 Fernlike Dendrite Snowflake Crystal
These are the largest crystals up to 5mm, with lots of side fern
like side branches
5. CELLULAR AUTOMATA APPLIED TO
CRYSTAL GROWTH
CA’s do quite easily reproduce the basic feature of the overall
behavior that occurs in real snowflakes. Snowflakes can be
modeled with the help of cellular automata to produce simple
faceted forms, needle-like forms, tree-like or dendritic forms,
as well as rounded forms.
One of the ways to create real snowflake patterns is to use
two-dimensional cellular automata displaying 3-state seven
sum totalistic rules on a hexagonal grid. Each cell has three

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states representing growth as a result of different temperature
conditions. The state of a cell is updated based on the sum of
its six neighbors and its current state.
Crystal growth is an excellent example of a physical process
that is microscopically very simple, but that displays a
beautiful variety of macroscopic forms. Many local features
are predicted from continuum theory, but global features may
be analytically inaccessible. For this reason, computer
simulation of idealized models for growth processes has
become an indispensable tool in studying solidification.
Packard presents a new class of models that represent
solidification by sites on a lattice changing from zero to one
according to a local deterministic rule. The strategy is to
begin with very simple models that contain very few
elements, and then to add physical elements gradually, with
the goal of finding those aspects responsible for particular
features of growth.
5.1 Packard’s Cellular Automata
The simplest deterministic lattice model for solidification is a
2D CA with two states per site to denote presence or absence
of solid, and a nearest neighbor transition rule. Packard
considers rules, which have the property that a site value of
one remains one (no melting or sublimation)[11]. The rules
also depend on neighboring site values only through their
sum:
= f( ) With =
The domain of f ranges from zero to number of neighbors; f
takes on values of one to zero. These rules display four types
of behavior for growth from small seeds:
i. No growth at all. This happens for the rule that
maps all values of σ to zero.
ii. Growth into a plate structure with the shape of the
plate reflecting the lattice structure
iii. Growth of dendrite structure, with side branches
growing along lattice directions; this
iv. type of rule is obtained by adding growth inhibitions
to the previous rule. Physically growth inhibition occurs
because of the combined effects of surface tension and
radiation of heat of solidification.
v. Growth of an amorphous, asymptotically circular
form. This form is obtained by adding even more growth
inhibition.
The two ingredients missing from the cellular automaton
model are:
i)Flow of heat
This may be modeled with the addition of a continuous
variable at each lattice site to represent temperature.
ii)Effect of solidification on the temperature field
When solid is added to a growing seed, latent heat of
solidification must be radiated away. This is modeled by
causing an increment in the temperature field.
An extremely simple prototype for planar growth was
proposed by Packard [12]. In his cellular automaton (CA)
model, each cell (or site) of a planar lattice changes from
empty to occupied as that location turns to ice and remains
occupied thereafter. A cell “freezes” when it has one frozen
neighbor, or when the number of frozen neighbors equals
some prescribed higher count. Cells of the original 1984
model had six nearest neighbors, reflecting the hexagonal
molecular structure of ice and observed symmetry of actual
snow crystals. Subsequent studies [13,14] include simulations
on the two-dimensional integers Z2 with 4 and 8 nearest
neighbours, the so-called von Neumann and Moore
neighborhoods, respectively. As they grow, hexagonal
Packard Snowflakes develop intricate patterns reminscient of
real snowflakes .Indeed, Wolfram and others have argued
(e.g., in [15,16], and [8]) that the similarity demonstrates the
ability of simple local interactions to capture essential
features of complex natural processes. Recent advances in our
understanding of real snow crystal growth, however, make it
clear that the Packard rules evolve in a very different manner
than do the sectored plates they resemble at certain stages of
development. More realistic lattice algorithms, based on
physical principles, are currently being developed by Gravner
and Griffeath [17, 18]; Our Packard Snowflakes evolve on the
two dimensional integer lattice, so the crystal of frozen sites
belongs to the state space
A = {finite subsets A ⊂ Z2}.
Elements of A will generally be denoted by u or v and
represented coordinate wise by (x, y).
The state of the crystal at time t is denoted
At = {occupied sites at time t}.
To specify whether or not a site is occupied, we write
At (u) = 1 if u ∈ At ,= 0
otherwise. Our analysis will focus on crystals started from a
singleton; i.e., we usually set A0 = {0}. A focus of our
analysis will be the final state
.
Simulations are as follows Temperature is set to a constant
high value when new solid is added. Hybrid of discrete and
continuum elements. Different parameters used
 diffusion rate
 latent heat added upon solidification
 local temperature threshold

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Fig- 10 simulated crystal
6. ADVANTAGES OF CELLULAR AUTOMATA
IN CRYSTAL GROWTH
The structure of cellular automata and crystal are one and the
same. They have the similar structure. Its very easy to
simulate the crystal growth using cellular automata. crystals
consist of regular arrays of atoms laid out much like the cells
in a cellular automaton. Crystals always start from a seed such
as a grain of dust and then progressively adding more atoms
to their surface. The crystal growth and the cellular Automata
have the same features.
CONCLUSIONS
Cellular automata (CA） modeling is one of the recent
advances in spatial–temporal modeling techniques in the field
of various growth dynamics.These models provide novel tools
that support for better understanding of the modeling process.
CA models have been successfully used to simulate different
growth behavior of crystals. It is very fascinating to see the
different intricate and complex forms that one sees during
crystal growth. In this paper, I have described the different
types of crystals, their formation and how to model using
cellular automata This paper shows some of the snowflake
patterns generated using packards CA model.
REFERENCES
[1]. J. Kepler ,The six sided snowflake (1611):
translation,Oxford University Press, Oxford (1966)
[2]. Libbrecth k,Rasmussen P,The Snowflake:winters
Beauty.stillwater: voyageur press 2003.
http:/www.snowcrystals.
[3] Physically Derived Rules for Simulating Faceted Crystal
Growth
using Cellular Automata," K. G. Libbrecht, arXiv:0807.2616
(2008).
[4]. Wolfram,A new kind of science,2002
[5]. E. Berlekamp, J. Conway, and R. Guy. Winning ways for
your mathematical plays.Academic Press, New York, 1982.
[6] Callahan P. Paul's page of Conway's life miscellany.
http://www.radicaleye.com/lifepage/lifepage.html..
[7]. Callahan P. Paul's page of Conway's life
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ge.html.
[8] Wolfram S. A new kind of science. Champaign: Wolfram
Media; 2002.
[9] Reiter CA. A local cellular model for snow crystal growth,
Chaos, Solutions and Fractals, to appear
[10]www.snowcrystals.com
[11]. Packard Snowflakes on the von Neumann
Neighborhood. Journal of Cellular Automata (2008). [pdf] C.
D. Brummitt, H. Delventhal, M. Retzlaff
[12]Packard N. Lattice models for solidification and
aggregation. Institute for Advanced Study Preprint. Reprinted
(1986). Theory and Application of Cellular Automata,
Wolfram S. (ed.).World Scientific, 305–310, 1984.
[13]. Wolfram S. A New Kind of Science.
Champaign:Wolfram Media. 2002.
[14] Packard N. and Wolfram S. Two dimensional cellular
automata. Journal of Statistical Physics 38 (1985), 901–946.
[15] Wolfram S. Computer software in science and
mathematics. Scientific American 251 (1984),188–203.
[16]Levy S. Artificial Life: The Quest for a New Creation.
New York City: Pantheon Books,1992.
[17] Gravner J. and Griffeath D. Modeling Snow Crystal
Growth II. To appear, 2007.
[18] Gravner J. and Griffeath D. Modeling Snow Crystal
Growth III. In preparation, 2007.

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