SINGAPORE MATHEMATICS PROJECT FESTIVAL 2003
Project title:
Using Cellular Automata and Fractal Complexity to
Analyse Natural Phenomena
Team members:
Ch’ng Hui Yen, Melissa
Hu Yi Lan
Sia Jia Min, Charmaine
School:
Raffles Girls’ School (Secondary)
Teacher-in-charge:
Mrs Sia Wai Leng
Category:
Senior Section
1
CONTENTS
Summary
3
Introduction
5
Objectives
6
Procedure
6
Sierpinski’s Triangle
7
Oil Spill
8
Forest Fire
13
Wind in Forest Fire
17
Oil Spill Experiments
19
Modelling Forest Fire
20
Conclusion
22
References
23
Acknowledgements
23
2
SUMMARY
Cellular automata are discrete dynamical systems whose behaviour is completely
specified in terms of a local relation. One typically consists of a set of cells, which are
assigned a state value, namely shaded or unshaded. Each cell then computes its new
state by assessing its initial state, as well as those of its neighbours, according to a set
of given rules.
Cellular automata can be used to simulate natural phenomena in which direct contact
is between one unit and another is required to affect the second unit. Such phenomena
include the burn patterns of forest fires and the percolation pattern of oil spill seeping
into the earth.
It must be taken into account that oil spills are limited by the initial volume of oil that
was spilled, while forest fires are not limited to the initial number of burning trees.
Instead, forest fires are able to spread freely with the presence of fuel.
In this project, we discovered some of the patterns that dictate cellular automata and
fractal geometry, studied the applications of fractal geometry in modelling natural
phenomena, and compared predictions with experimental results using the knowledge
gained.
Firstly, we drew cellular automata using simple functions that were probability-free to
obtain a better grasp of fractal geometry. Sierpinski’s triangle, a famous fractal, is an
example of this type of cellular automata. Secondly, randomness was then brought
into the picture to increase the complexity of cellular automata. Next, the external
perimeter was measured using a box count technique. Their complexity was then
measured using log-log graphs, taking the gradient of the line of best fit as the
complexity. Fourthly, satellite pictures of forest fires were modelled using a Java
Applet to show that it is possible to obtain fairly accurate values of the probability of
forest fire spreading and wind direction within a short time.
An analysis of each oil spill and forest fire diagram is given to see how it is affected
by the probability. The complexity of both models increase from probability 0 to 0.5,
then decrease as the probability increases from 0.5 to 1.
Wind is an added complication to forest fires, as it can either increase or decrease the
probability of the fire extending in a particular direction. Increase in probability of the
fire spreading can transform a small fire into a potentially severe fire. Changes in
wind direction will alter the main direction of fire spread at intervals, thus leading to
immensely complicated fires.
Finally, the oil spill experiment and simulation proves that it is feasible to extract soil
samples from the site of spill and carry out experiments with them in order to
calculate the likely extent of percolation by the oil spill. Small-scale spills could be
created in various soil samples, and then matched with core samples taken from the
site of real spills to match the probability of oil percolation, and thus obtain an
estimate of how far the spill would percolate.
3
It is possible to compare the burn patterns of real fires with that of the models by
flying over the site of a burning/burnt forest and taking aerial photos of the site.
Models that are extremely similar to the burnt pattern could then be generated to
obtain information about the probability of the fire spreading and the wind direction.
This would then yield information about the potential severity of the fire, and
potential locations to locate fire fighters.
From this investigation, we see that fractal geometry plays a large part in predicting
the percolation of oil spills and spread of forest fires.
4
INTRODUCTION
Cellular automata are discrete dynamical systems whose behaviour is completely
specified in terms of a local relation. One typically consists of a set of cells, which are
assigned a state value, namely shaded or unshaded. Each cell then computes its new
state by assessing its initial state, as well as those of its neighbours, according to a set
of given rules.
Cellular automata can be used to simulate natural phenomena in which direct contact
is between one unit and another is required to affect the second unit. Such phenomena
include the burn patterns of forest fires and the percolation pattern of oil spill seeping
into the earth.
Many factors influence the behaviour of a forest fire, among them being fuel type,
fuel moisture and wind:
 Fuel types determine how different terrains affect fire spread by comparing the
mass of fuel per unit area, live and dead, grouped by particle size classes.
 Fuel moisture has a tremendous impact on the intensity of a fire and the rate of its
spread. A fire model must account for the fuel moistures' affect on both fire
intensity and the heat necessary to bring the fuel ahead of a spreading fire up to
ignition temperature. These factors include: relative humidity, intensity of recent
rainfall and the types and developmental stage of various fuel types.
 Wind provides the most variable influence on fire behavior. The unpredictable
nature of wind derives from the large number of distinct classes of wind and from
its ability to change both horizontal and vertical direction.
In modelling forest fires, we took into account fuel types and fuel moisture to
generate a basic probability of fire spreading without the presence of wind. Wind was
later added to the model to see how its presence could dramatically alter fires.
It must be taken into account that oil spills are limited by the initial volume of oil that
was spilled, while forest fires are not limited to the initial number of burning trees.
Instead, forest fires are able to spread freely with the presence of fuel. Thus, besides
the fact that oil spills are one-dimensional while forest fires are two dimensional, oil
spills are also less likely to percolate out of control because of the limited volume of
oil spilt.
5
OBJECTIVES
1. Discover the patterns that dictate cellular automata and fractal geometry.
2. Study the applications of fractal geometry in modelling natural phenomena.
3. Compare predictions with experimental results using the knowledge gained.
PROCEDURE
1. Draw cellular automata using simple functions to obtain a better grasp of fractal
geometry, e.g. modelling Sierpinski’s triangle.
2. Introduce probability into simple cellular automata functions such as oil spills and
forest fires to increase the complexity of cellular automata diagrams.
3. Measure the complexity of the models by using log-log graphs and taking the
gradient of the line of best fit as the value of the complexity.
4. Add the factor of wind direction to the forest fire simulation to acquire more
realistic models.
5. Model diagrams based on satellite pictures of forest fires to obtain an approximate
gauge of probability of forest fire spreading and wind direction.
6
SIERPINSKI’S TRIANGLE
In one of the simplest forms of cellular automata, the state of a cell is determined only
by the state of the two neighbours located just above, one on the left and one on the
right. The rules for shading are given in the lookup table below. The resulting
diagram is Sierpinski’s triangle, a famous fractal.
7
OIL SPILL
The diagrams below represent the profile of the spill of oil or other liquid percolating
into the earth. The cells are arranged in a bricklike pattern to represent particles of
soil or other material in the ground; the cells represent the spaces around those
particles into which the oil can seep. Cells are considered neighbours if they are in
consecutive rows and share part of a face. For a predetermined probability that
models the density of the soil, the oil can flow downwards either to the left or right. If
the soil bed is porous enough, the oil will be able to percolate through the network of
cells through the soil bed. As the oil percolate through the soil bed, parts of the
percolation will end, and depending on the percolation probability, the oil may
eventually stop percolating altogether.
If a cell is shaded, it has a certain pre-determined probability of causing each of its
two neighbours in the next row to be shaded. A calculator with a random-number
function is used to generate digits. If the number generated by the calculator is lower
than the pre-determined probability, this would indicate percolation, while if the
number generated by the calculator is higher than the pre-determined probability, this
would indicate no percolation.
A limitation of modelling oil spills is that the density of the soil bed, i.e. the
probability of percolation remains constant throughout. More complex models could
include pockets of soil that are associated with higher or lower percolation
probabilities, however this would entail having knowledge of the soil bed.
Probability: 0.4
With a probability of 0.4, the oil spill stops very quickly. This indicates that the soil
is not very porous. It is very easy to deal with such a spill as only the top soil needs to
be cleared.
Probability: 0.45
With a probability of 0.45, the oil spill has a tendency to spread further down in
certain areas. However, once the percolation spreads out in several directions, the
different parts of the oil do not meet further down and thus the percolation still stops
rather quickly.
8
Probability: 0.5
With a probability of 0.5, the oil spill percolates deep below the ground surface,
branching many times on the way without the branches combining together again.
Notice that the oil spill has percolated much deeper into the soil as compared to
probability 0.45, despite the small increase in probability. It is around this probability
that oil spills begin to cause difficulty in clearing up.
The oil spill of probability 0.55 has
infiltrated the soil deepest but tends to
cluster more than at probability 0.5.
Probability: 0.55
9
Probability: 0.6
At probability 0.6, the oil spill pattern is very dense, with few unfilled cells within,
and has a regular shape. An oil spill of this probability would be difficult to clear if it
has had enough time to percolate deep into the soil.
10
INTERPRETING LOG-LOG GRAPHS
The measure of complexity used on the oil-spill model is based on the Hausdorff
dimension (D) of an object, which is also known as its complexity measure. For
irregular objects that have not been generated with precise steps, a technique known
as box count is used to collect data on the perimeter of the object to approximate the
relationship between the overall perimeter of the object and the size of the unit used to
determine that perimeter. In this process, an object’s size, in this case its perimeter, is
determined by the number of boxes on a grid containing it, using successively smaller
boxes.
A grid of squares one inch on a side is laid over the diagram of the oil spill/forest fire.
The number of squares that contain the external perimeter of the oil spill/forest fire is
counted. A new count is then done using a grid of squares one-half inch on a side.
This should yield a count that is slightly more than twice as much. Because of its
irregularity, doubling the number of box sides per inch will produce a count that is
more than twice the previous count. A smaller box size picks up more than twice as
many of the object’s nooks and crannies that would have been skipped by using a
bigger box. The gradient of the line of best fit should then be calculated to find the
complexity of the oil spill outline. The “rougher” the outline, the higher the gradient,
but it should not be very much higher than 1, while the “smoother” the outline, the
closer the value should be to one. Not that a perfectly smooth outline would be a
straight line or smooth curve with a complexity measure of exactly 1.
Plotting the data on graph paper using ordered pairs of the form (# of boxes/inch, # of
boxes containing the perimeter) yield a non-linear progression of points, which
D
P
suggests the graph of a power equation, similar to the Hausdorff equation, N    ,
 p
where D is the dimension of the object, P is the original length of the object, and there
are N objects that are similar to, but smaller than the original, with each of these
smaller objects having a length of p.
If P is 1 inch and p is the length of the successively smaller grid boxes, then the ratio
P
equals the number of box sides per inch and N corresponds to the number of grid
p
boxes that contain part of the perimeter of the object, yielding data points of the form
P
( , N).
p
D
P
P
By manipulating the equation N    , we obtain log N  D log   . This can be
 p
 p
expressed as a linear equation in the form Y = MX, where Y is log N, M is D and X is
P
log   . Plotting the logs of the box-count data would yield a straight line with
 p
gradient D, the complexity. Instead of taking the logs of the data, we instead plot the
data on log-log paper.
11
These are the results of the box count for the oil spill:
Probability
Number of boxes per inch
of spill
1
2
4
8
0.4
8
17
50
116
0.45
6
16
40
114
0.5
18
50
135
380
0.55
25
62
166
411
0.6
12
31
75
181
Gradient
12
229
186
586
> 600
275
1.30
1.41
1.42
1.37
1.32
From the log-log graph above, we see that the complexity of the oil spills, which are
measured by the gradients of their respective lines, increase from probability 0 and
reach a maximum at around probability 0.5. As illustrated by the models, at this stage,
the oil spills branch out more, resulting in longer, more complex perimeters.
The complexity then begins to decrease from probability 0.5 to 1. This is due to the
oil spill becoming denser and more clustered and the perimeter decreases.
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FOREST FIRE
A cell, which represents a tree, can be in three states – unburned, burning or dead. A
burning tree will only burn for one “turn”, after which is becomes dead. A dead tree
cannot reignite. A turn consists of examining burning trees to see whether they ignite
their neighbours. A systematic way of examining the cells would be to begin at the
top and then proceed clockwise around the perimeter of the fire, testing cells. Once
again, random numbers are generated to run the model with a pre-determined
probability.
In a real forest, the trees have neighbours in all directions, and the trees are not
uniformly distributed. There are also other factors like wind and slope that will vary
the direction and speed of spread of the fire. During the spread of the fire, the fire may
be stopped by obstacles like rivers or it may be put out by a fire brigade. Thus, it may
be possible to stop potentially severe fires very early.
The fire dies out very quickly, without the need of obstacles
to control it.
Probability: 0.4
The fire dies relatively quickly without
branching out much, without a need for
obstacles to control it.
Probability: 0.45
13
Probability: 0.5
The fire branches out in most directions in a non-uniform way with large patches of
unburnt forest interspersed with burnt forest. Notice that the fire has become much
larger than the fire with probability 0.45, despite the small increase in probability.
There is now a need for obstacles to control the fire. Otherwise, there exists the
chance that it will burn until all the fuel available has been used up.
Probability: 0.55
At probability 0.55, the fire spreads everywhere and dies out slowly. The patches of
untouched forest are much smaller now and the perimeter had decreased.
14
Probability: 0.6
At 0.6 probability, the fire does not die out without control or dies out extremely
slowly. As the fire is very dense, it spreads out more evenly. Notice that the unburnt
spaces are not significant, and thus similar measures can be used to control a fire that
has probability 0.6 and above.
15
These are the results of the box count for the forest fire:
Probability
Number of boxes per inch
of spread
1
2
4
8
0.4
4
9
22
48
0.45
9
18
40
86
0.5
24
60
159
377
0.55
28
66
155
312
0.6
30
64
143
315
Gradient
12
76
129
> 600
459
486
1.18
1.07
1.33
1.13
1.14
From the log-log graph above, we see that the highest complexity occurs at
probability 0.5, similar to the oil spill. Unlike the oil spills, the complexity of the
forest fires undulate, which may be due to experimental error. Since both are
examples of fractal geometry, the forest fires should exhibit the same characteristics
as the oil spills, i.e. the complexity increases until probability 0.5, then decreases.
16
WIND
Wind is an added complication to forest fires, as it can either increase or decrease the
probability of the fire extending in a particular direction. Increase in probability of the
fire spreading can transform a small fire into a potentially severe fire. Wind can also
change direction, thus leading to immensely complicated fires.
Starting with a basic probability, this was altered for all four directions to give four
separate probabilities of the fire spreading in each direction. Below are diagrams
showing the spread of fire in the presence of wind. The square marked in red is the
initial burning tree.
0.45
0.1
0.7
0.35
Basic probability: 0.4 Wind change: 0.35
17
Clearly, the fire spreads intensely in the direction of wind. A fire with a basic
probability of 0.3 or 0.4, which would ordinarily have died out without the presence
of wind, is able to spread freely with the aid of strong wind, as the wind may increase
the probability of spread in a certain direction to 0.7. Thus, it is important to have
information about the direction and the strength of the wind when trying to control a
fire.
0.7
0.1
0.7
0.1
Basic probability: 0.3 Wind change: 0.6
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OIL SPILL EXPERIMENTS
Procedure:
1. Cut different materials to dimensions of 11 cm x 6.5 cm x 0.5 cm.
2. Using a dropper, place 5 cm3 of potassium manganate (VII) on the top and allow
the liquid to infiltrate the material for 5 minutes.
3. Use a Java Applet to model the results to find an approximate gauge of the
porosity of the sample.
Both glossy magazine paper and sponge results are of low complexity and have
percolated very deep. Two models, one of probability 0.65 and another of probability
0.63 fit the two materials respectively. This experiment shows that it is possible to
model real oil spills by taking samples of the soil, carrying out experiments and
finally generating similar models to find the porosity of the soil sample.
Glossy magazine paper
Sponge
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FOREST FIRE MODELLING
Below is a satellite image of an actual forest fire in Los Alamos, Mexico.
Using a Java Applet, we generated a model similar to the photographed fire to obtain
estimates of the probability of fire spreading and wind direction.
20
Below is a satellite image of an actual forest fire.
Using a Java Applet, we generated a model similar to the photographed fire to obtain
estimates of the probability of fire spreading and wind direction.
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CONCLUSION
In this project, we have used fractal geometry to investigate how the probability of oil
spills and of forest fires affects both the size as well as the complexity of the oil spills
and forest fires. We see that both share similar characteristics. We then added in the
element of wind to forest fires to see how they would affect the direction of spread
and strength of a fire. Actual forest fires were then modelled using a Java Applet to
show how it is possible to obtain estimates of the probability of fire and wind
direction while the fire was still burning.
Data relating to porosity of the soil could be collected at the site of real spills by
obtaining soil samples and carrying out experiments with them. Small-scale spills
could be created in various soil samples, and then matched with core samples taken
from the site of real spills to match the probability of oil percolation, and thus obtain
an estimate of how far the spill would percolate.
Percolation patterns can reveal how strong preventive measures need to be to avoid a
spill. Knowing the potential percolation in a certain area would give rise to
appropriate recommendations of how far structures located in the ground (e.g. wells)
should be located from a site that may be prone to spills, so as to prevent pollution. It
would also facilitate recommendations as to what type of materials should be used to
contain waste.
For example, with the large number of oil refineries in Pulau Bukom, just south of
Singapore, care must be taken in the planning of the storage of oil. Concrete barriers
should be placed around the sides to ensure that oil percolation is contained in case of
a spill. Also, suitable ground should be chosen such that oil will not percolate too far
into the ground, which might potentially cause harm to the environment.
It is possible to compare the burn patterns of real fires with that of the models by
flying over the site of a burning/burnt forest and taking aerial photos of the site.
Models that are extremely similar to the burnt pattern could then be generated to
obtain information about the probability of the fire spreading and the wind direction.
This would then yield information about the potential severity of the fire, and
potential locations to locate fire fighters. This method is being used in Canada to
contain forest fires.
Both sets of models have critical probabilities that produce the most complex
perimeters. When the complexity of a burn pattern is close to 1, this indicates that the
fire is either not severe or very severe. Fires with low probabilities stop quickly and
have a smooth outline. Fires with high probabilities spread in an almost circular
pattern (in the absence of wind), and also have smooth outlines. The complexity of a
burn pattern is greatest around a probability of 0.5. Thus, it is possible to use both the
complexity of the burn area as well as the size of the burn area to obtain the
probability of the fire spreading.
With the presence of wind, the fire spreads intensely. A fire that would ordinarily
have died out without wind is able to continue spreading far in the presence of wind.
Thus, it is important to have information about the strength of the wind when trying to
control a fire.
22
REFERENCES
 Biehl, L. Charles. Forest Fires, Oil Spills, and Fractal Geometry: An Investigation
in Two Parts. Part 1: Cellular Automata and Modeling Natural Phenomena. The
Mathematics Teacher, Vol. 91 No. 8. Reston, Va.: National Council of Teachers
of Mathematics, 1998.
 Biehl, L. Charles. Forest Fires, Oil Spills, and Fractal Geometry: An Investigation
in Two Parts. Part 2: Using Fractal Complexity to Analyse Mathematical Models.
The Mathematics Teacher, Vol. 92 No. 2. Reston, Va.: National Council of
Teachers of Mathematics, 1999.
 Biehl, L. Charles; Java Simulations of Forest Fires and Oil Spills:
(http://dimacs.rutgers.edu/~biehl/camenu.html)
 Fire
Propagation
Simulation:
Background
&
(http://www.npac.syr.edu/REU/reu94/mveach/section3_2.html)
Description:
 Fire Scar Type (http://www.fire.uni-freiburg.de/photos/satex/burnscar.htm)
ACKNOWLEDGEMENTS
We would like to thank the following people for their support and encouragement:
 Our teacher-in-charge, Mrs Sia Wai Leng, for her guidance,
 Mr Kenneth Lui, for offering his comments to improve the project,
 Mr Dominic Ch’ng, for the preparation of experimental materials,
 And all others who have helped us in one way or another.
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