TITLE: Examining The Factors That Effect Random Events Through
Coin Flips And Jingle Cups
STUDENT RESEARCHERS: Jon Bram, Abigail Neely, Nicole Warner
SCHOOL ADDRESS: Belmont High School
221 Concord Ave.
Belmont, MA 02178
GRADE: 11/12
TEACHER: Paul Martenis- plmartenis@aol.com
I. STATEMENT OF PURPOSE AND HYPOTHESIS:
We wanted to explore the theory of randomness using a coin
flip. We wanted to see if we could control factors that effect
the coin flip to increase the probability of the coin landing
heads up. We were looking to see what factors had what effect
on the outcome. Our hypothesis stated that if one controlled
enough factors they could make the outcome of a coin flip more
predictable (the probability of getting heads up would
increase).
II. METHODOLOGY:
To test our hypothesis we flipped coins, looking at different
factors, and seeing the factors' effects on the outcome of the
flip.
1. We used the normal hand flip, as our control, knowing that
the probability of landing head up was roughly even for both
heads and tails. Flipping it with the thumb and letting it
land on the ground, we took the results once it was at rest.
We repeated this 200 times.
2. We flipped coins with our hand and eliminated the bounce.
We eliminated the bounce by flipping the coins onto a rug to
cushion their fall. We did this 200 times.
3. We flipped coins using a mechanical catapult-type device.
We did this onto a hard surface bringing the end of the
catapult back to the same place each time. We repeated this
200 times.
4. We used the catapult and eliminated the bounce by letting it
land on a soft rug. We also only counted the coins that landed
in a one square foot area next to the catapult. The reason
that we did this was that we wanted to keep the force by which
the coin was propelled constant. By only recording the flips
that landed in a particular area, we kept the force constant in
our data. We did this time consuming process several times but
only were able to count 98.
5. We used an electromagnet and eliminated the bounce. We
created the magnet with a six volt battery connected to wire
coiled around an iron nail. We put a washer on the magnet and
broke the circuit so that the washer would fall the same way
every time. Because of time constraints we stopped after we
got 30 "heads" in a row.
To evaluate number one and four we used a video. We slowed
down the video and counted the flips in the air of the trials
with the mechanical device and several trials of the hand flip.
We also watched bounces and counted the number of flips in each
of those.
III. ANALYSIS OF DATA:
Trial type #heads #tails #other %heads
1. hand 105 95 52.5%
2. hand 97 103 48.5%
(no bounce)
3. mechanical 102 97 1 51.0%
(start same)
4. mechanical 88 10 89.8%
(start same,
no bounce,
certain area)
5. electromagnet 30 0 100.0%
(with washer,
from same height,
start same)
1. bounce-we found that when a flip starts random the bounce
has no effect on the probability of heads turning up. However,
we found that when the flip starts controlled, but is allowed
to bounce it becomes random.
2. force-we thought that the mechanism (either hand, catapult,
or electromagnet) was the most important part of the flip.
When we combined this with the elimination of the bounce the
probability of getting heads jumps from 50% to 90%. And when
we added in the reliance of the electromagnet to let the washer
go the same way every time the probability of getting "heads"
jumped to 100%. The reason for this is that with the hand the
force varies with magnitude and direction.
3. air resistance-because we got consistent results with the
electromagnet we felt that the effect of air resistance was
negligible.
In our use of the slow motion video we found that the number of
times the coin flipped in the air for the mechanical flip was
constant, while the number of flips in the air for the hand
flip varied.
IV. SUMMARY AND CONCLUSION:
We found that when you control factors of a coin flip the
result becomes more predictable. The more factors we
controlled the higher the probability of getting a head; since
the probability increased our prediction of getting a head
became more accurate. Therefore we confirmed our hypothesis,
the coin flip became more predictable. We believe that by
controlling factors that contribute to the randomness of an
event one can make that event more predictable.
V. APPLICATION:
This knowledge, that by controlling factors an event can become
more predictable, can help in predicting the outcome of events.
With the analysis of data, to see the effect certain factors
have on an event, people will be able to better understand the
outcomes of many events. The application of our findings is
much more theoretical than practical. It is impossible to
think that in a world such as ours one would be able to
eliminate all of the factors that make an event random.
Hopefully our experiment will help us to better explain and
understand random events when we apply our method of breaking
down all of the factors that effect the event.
TITLE: Is There A Connection Between The Fractal Dimension Of
A Rorschach Inkblot And The Usual Clinical Response?
STUDENT RESEARCHERS: Daniel Brosgol, Karen Eagar, Ealaf Majeed
SCHOOL ADDRESS: Belmont High School
221 Concord Ave.
Belmont, MA 02178
GRADE: 12
TEACHER: Paul Hickman- namkcihp@aol.com
I. STATEMENT OF PURPOSE AND HYPOTHESIS:
We hypothesized that the inkblots with more common aggressive
responses, as had been found by psychologists, would have
higher fractal dimensions and the blots with more "placid"
responses would have lower fractal dimensions.
II. METHODOLOGY:
1. Find a set of the Rorschach inkblots with the common
responses included.
2. Remove the colored inkblots (the colors in these blots
heavily influenced the response to the blot to a point where
making them colorless would change the initial response).
3. Scan the blots onto a computer and clean up the images.
4. Make outlines of the images or, if the outlines are the only
images that you have available, fill in the blots.
5. Measure the fractal dimension of the blots on Fractal
Dimension 5.1 using both the box and the fast circle method.
6. Remove inappropriate data points from the graphs.
7. Compare the fractal dimensions of the blots to see if there
are any patterns.
We want to make it clear that there are only 10 blots in all
from the Rorschach set, 4 of which are colored. We felt that
the colors would have too much of an effect on the response to
the blot. This would lessen the effect of the FD (jaggedness)
on the response. We got our data concerning the responses to
each blot from the book which had the blots in it. The book
used information that has been gathered by many psychologists
over the years.
III. ANALYSIS OF DATA:
Fractal Dimension Measurements
FD Full FD Full FD FD
Blot Usual Response Blot Blot Coast Coast
Box Circle Box Circle
1. lions, pigs, and bears 1.83 1.75 1.41 1.57
2. bear, gorilla 1.88 1.90 1.37 1.44
3. animal hide or boat 1.84 1.73 1.36 1.40
4. 2 female figures 1.68 1.71 1.32 1.46
5. bat or butterfly 1.66 1.86 1.39 1.42
6. mask, jack-o'-lantern 1.71 1.85 1.15 1.45
The data we obtained for the coast line was not as helpful for
our hypothesis because the measurements were so similar. We
based our conclusion on the data for the filled inkblots
measured with the fast circle method, (the box method was too
erratic to conclude anything from it). From our useful data, we
found that images with a negative interpretation had the
highest fractal dimensions, unlike the more passive responses
which had the lower FD's.
IV. SUMMARY AND CONCLUSION:
We have accepted our hypothesis, that there is a correlation
between the fractal dimension and the interpretation of
Rorschach inkblots. The correlation is that the higher the
fractal dimension the more violent the response is. One
example is no. 2 which has a high fractal dimension and a
negative response ( bear and gorilla). The bear and gorilla
are big animals usually feared because of their size and their
power, making them a violent response. An image that has a
lower fractal dimension is no. 4. The interpretation of it is
two female figures facing each other. The females have a lower
fractal dimension, so they are responded to more passively.
These were the two blots with the most differing FD's.
V. APPLICATION:
We believe that someone could use this information in
advertising because they could subconsciously elicit a certain
response to a product that they were trying to sell. Someone
could use this for future psychoanalytical research by helping
to understand why certain shapes give different responses.
Psychoanalysts would then be able to understand the minds of
their patients better. This could also help those people
taking the tests because they would know what the psychologists
were looking for in terms of their answers.
TITLE: The Fractal Dimensions of Sponges: Is There A
Connection?
STUDENT RESEARCHERS: Noah Dean Bullock, Maura Nolan Henry,
Joshua Charles, Asa Nissenbaum
SCHOOL ADDRESS: Belmont High School
221 Concord Ave.
Belmont, MA 02178
GRADE: 12
TEACHER: Paul Hickman- namkcihp@aol.com
I. STATEMENT OF PURPOSE AND HYPOTHESIS:
We wanted to find out more about sponges and their fractal
dimensions. We were particularly interested in the possibility
of a connection between the 3D fractal dimension of a sponge
and its 2D fractal dimension. Our hypothesis states that there
exists a mathematical connection between the 3D fractal
dimension of a sponge and the 2D FD of that sponge.
II. METHODOLOGY:
Materials:
3 large masses of different sponges (cubes would be helpful)
a band saw
goggles
a balance
felt tip marker
ruler
a scanner
Fractal Dimension 5.1
Graphical Analysis
Procedure:
Be sure to dry out all of the sponges before beginning.
1. The first cube, or mass of sponge is your first iteration.
Mass this sponge and record your data.
2. Using the felt tip marker, mark lines at half of each
dimension of the original (height, width, length). Cut the cube
using the bandsaw, making sure not to forget your goggles,
along the measured lines. The resulting cube is your second
iteration.
3. Mass this cube and record your data.
4. Repeat steps 2 and 3 two more times, finishing with four
individual iterations.
5. Create a table of scale size versus mass for the four
iterations.
6. Graph the natural log of the scale size versus the mass
using Graphical Analysis. This gives you your 3D fractal
dimension.
7. Take the sponges to the scanner and scan two surfaces of
each of them. (This ensures accuracy.)
8. Using Fractal Dimension 5.1, measure each of the scans'
fractal dimensions. Use the fast circle method and place the
center of the circle at different locations on the scan. This
will allow for results that better represent the whole scan.
9. The slopes of the trendlines of the graphs of your results
reveals the 2D fractal dimensions of your scans.
One of the variables that can be controlled is the moisture
within the sponge. Sponges are sold wet, and so to ensure
accuracy of mass, you must dry them out ahead of time. Another
variable is that of the accuracy of the sponge cut. Some of
the cuts may be less than accurate; this can be attributed to
human error.
III. ANALYSIS OF DATA:
Our first sponge had a 3D fractal dimension of 2.88 and a 2D
fractal dimension of 1.802. Our second sponge had a 3D fractal
dimension of 2.98 and a 2D fractal dimension of 1.922. Our
third sponge had a 3D fractal dimension of 2.83 and a 2D
fractal dimension of 1.883.
The first set of data and the third set of data are from two 3M
sponges. The second set of data is from one E-Z One sponge.
IV. SUMMARY AND CONCLUSION:
Our data do not support our hypothesis. We did not prove that
there is any mathematical connection between the 2D fractal
dimension of a sponge face and the sponge's 3D fractal
dimension. The data we obtained shows neither a connection
between the two brands of sponges nor one between the two 3M
sponges. That is not to say, though, that one does not exist.
With more data from both other brands of sponges and more 3M
sponges, a connection is possible.
V. APPLICATION:
Companies which create sponges could better ascertain the most
absorbent sponges by a combination of their material and their
fractal dimensions. But our research extends far beyond the
humble walls of the modern kitchen. If a connection between 2
and 3 fractal dimensions is established, material engineers
would be able to examine the porosity of ceramic building
materials based on a side and thus be able to determine their
strengths.
TITLE: The Effect Of Different Formation Methods On Fractal
Dimension Of Squish Fractals
STUDENT RESEARCHERS: Mary Tsien, Meaghan Brusch, and Steve
Martin
SCHOOL ADDRESS: Belmont High School
221 Concord Ave.
Belmont, MA 02178
GRADE: 11/12
TEACHER: Paul Martenis- plmartenis@aol.com
I. STATEMENT OF PURPOSE AND HYPOTHESIS:
The purpose of our project is to find out if the fractal
dimension of the squish fractals using different substances is
affected by the way in which the fractal is formed. We will
use four different methods: control, twist, one-corner, and
one-side. Our hypothesis states that the fractal dimension
will be the same for both the control and the twist method, but
will increase when the one-corner and one-side method are used
due to the higher peaks which we think are formed by
concentration on one side. Our general hypothesis is that the
fractal dimension will be affected by the different methods
that are used to form the fractal.
II. METHODOLOGY:
We are going to test our hypothesis by using four different
methods in which to form these "squish" fractals: the control
method (pulling straight up), the twist method (twisting to the
right while pulling up), the one-corner method ( pulling up by
one corner), and the one-side method (pulling up from one
side). We will then test these methods by using the "squish"
method with two different substances, blue paint and Crest
toothpaste. We decided to use these substances because we
thought that they would form fractals that would stay for long
enough period of time to scan them. The materials that we will
need are: several plexi-glass plates, blue paint, toothpaste,
a scanner, a syringe, and some beakers. We will then begin our
experiment. First, we will make all of the blue paint fractals
in one day, to make sure that the paint doesn't change
consistency. We will then make the paint fractals by measuring
out 1.5 mL of the blue paint in the syringe and squeezing the
paint into the center of a plexi-glass plate. We then will
place another plexi-glass plate over the substance and we will
hold down the plexi-glass in the center for 25 seconds. We
then will pull the top glass off first straight up. We will
then repeat this control trial twice, to guarantee accuracy in
our results, and do this for each of the four methods.
Immediately, after we have made each fractal, we will scan it
in the scanner to make sure that we are able to measure its
dimension. We will do this before the paint dries, so that it
doesn't crack which would affect the fractal dimension. We
will do the same process for the toothpaste. We will then take
all of the scans that we have made from each substance, and
measure the fractal dimension using the computer program,
Fractal Dimensions 5.1. Then we will be able to compare and
discuss our results.
III. ANALYSIS OF DATA:
We found that the fractal dimension increased when using the
one-corner and one-side methods due to the higher, more
intricate peaks that are formed. The highest fractal dimension
was formed using the one-side method, with an average of 1.547
(paint) and 1.497 (paste), which happened in both substances.
The second highest fractal dimension was found when we used the
one-corner method in the blue paint (avg. 1.514). This is not
true for our toothpaste results, but we do not think that they
are accurate due to the fact that the peaks immediately fell
when we formed the one-corner toothpaste fractals, making the
dimension inaccurate. We therefore find the blue paint
fractals to be more accurate and will concur with the data we
collected from that. Unlike what we predicted, we found that
the lowest fractal dimension was made when we used the twist
method, average 1.300 (paint) and 1.384 (paste). Our control
group was the second to lowest and we used these results when
we compared the different methods.
IV. SUMMARY AND CONCLUSION:
We found that the fractal dimension of a squish fractal of a
certain substance is greatest when the fractal is formed using
the one-side method and is lowest when using the twist method.
The dimension is highest when formed with one-side because
there is less paint or toothpaste in one area, so there is more
force in one area, and the paint or toothpaste gets pulled into
more complex patterns, making the fractal dimension higher.
The dimension is lowest when formed by the twist method
because, when twisting, the complex patterns are smudged,
making the dimension lower. The one-corner method is the
second highest in fractal dimension, a conclusion we came to
with our paint, not toothpaste data. This is because there is
a greater amount of force in one area, but it is not as great
as in the one-side method, making the dimension higher than the
control group, but not as high as the one-side method.
From these results, we both accept and reject our hypothesis.
Our general hypothesis was correct, for the fractal dimension
was affected by the method used to form the fractal. However,
our specific hypothesis was not totally correct. We were
correct when saying that the dimension would be greater using
the one-side and one-corner method, but we were wrong when
saying that the dimension would be the same in the control and
twist method. When stating this, we had not taken into account
the "smudge" factor, and this is why the dimension of the twist
is lower than that of the control. Generally, we are quite
pleased with our hypothesis.
V. APPLICATION:
We think that our research on fractal dimension will help the
world by bringing science to children of all ages who might
not get interested in science normally. We think that with
this research, children's toys and posters can be made of these
interesting pictures formed by different processes. With this
game, children could make their own fractals using our four
different methods and compare them to what they resemble in
nature. This game would get children interested, at an earlier
age, in learning about physics.
TITLE: Termites: The Fractal Dimensions Of Their Random Paths
STUDENT RESEARCHERS: Brian Deese, David Millar, and Lindsay
Miller
SCHOOL ADDRESS: Belmont High School
221 Concord Ave.
Belmont, MA 02178
GRADE: 12
TEACHER: Paul Hickman- namkcihp@aol.com
I. STATEMENT OF PURPOSE AND HYPOTHESIS:
We wanted to find out more about termites and the fractal
dimension of their paths. Does the fractal dimension of
termite colonies increase over time as we expect? Would the
termite colonies resemble the aggregates and other fractal
objects that we have studied as we expect? Would the results
be linear, we didn't think so. Our hypothesis was that the
paths that termites take, in fact, make fractal objects and
that we can trace this fractal growth over time. We also
hypothesized that the growth would resemble that of the Cu
aggregate grown in class.
II. METHODOLOGY:
We first acquired a movie of a growing termite colony and a
termite computer program from B.U. We captured pictures every
15 frames and placed them in many different graphics programs.
Because of the great difficulty in getting a clean, clear
picture, we resorted to a different option. We printed images
of the movie out with a contrast setting of +50 in Adobe
Photoshop. We then outlined the path of the termites in black,
traced the paths onto tracing paper and scanned them back into
the computer. We were able to measure the fractal dimension of
these images in Fractal Dimension 5.1 and, using the fast
circle method each time, we established specific fractal
dimensions at each time interval. We then plotted the fractal
dimension with respect to time, and compared the visual result
of the termite paths to our Cu aggregate.
III. ANALYSIS OF DATA:
We took pictures of the termite movie at 15 frame intervals.
We hypothesized that the fractal dimension would grow over time
and our results are as follows:
# of frames Fractal Dimension
15 1.259
30 1.314
45 1.458
60 1.487
75 1.548
90 1.597
105 1.613
120 1.597
From the data above, we concluded that the fractal dimension of
paths of termites does in fact increase over time. The graph
leveled off at the end because our movie screen captured only a
set portion of the termite's burrowing. We hypothesize that
the fractal dimension would increase more if we were to see
more of the picture.
In regards to the termites similarity to our man-made
aggregates, we found an interesting comparison. The final
fractal dimension of both were similar, but this is only due to
the fact that we stopped the growth of both at a time when
their fractal dimensions coincided. There was, however, an
interesting visual similarity in the growth of both the
termites and the aggregate. In the initial growth of both when
looking at them up close, they looked surprisingly similar.
IV. SUMMARY AND CONCLUSION:
We found out three major things in this project: Termite paths
are fractal objects, the fractal dimension of their paths
increases over time as we expected in our hypothesis, and
projects in science often become a process in how you can
accomplish your task with the resources available.
V. APPLICATION:
We have learned many important things in this project. For
research in the future, our findings can help in different
ways. First off, from the process that we had to go through,
we would suggest to researchers in the future to pick a limited
and achievable goal. In terms of our specific research, our
project could be a spring-board for further research. From our
findings, an interesting investigation would be to actually
grow a termite colony in two or three dimensions in order to
watch, as well as chart, the growth of the colony. Although
the growth of termite colonies is not the scientific gateway to
the 21st century, knowing how these household pests' paths'
grow could be helpful in stopping them from damaging property.
TITLE: Heat Conduction And Its Relation To The Diffusion Of
Particles Through A Chamber
STUDENT RESEARCHERS: Paul Adler, Colleen Baker, and Nate
Holloway
SCHOOL ADDRESS: Belmont High School
221 Concord Ave.
Belmont, MA 02178
GRADE: 11/12
TEACHER: Paul Hickman- namkcihp@aol.com
I. STATEMENT OF PURPOSE AND HYPOTHESIS:
We wanted to compare the conduction of heat through a metal rod
with the behavior of gas particles diffusing in a diffusion
chamber as simulated by the program "diffusion Chamber." We
knew that certain aspects of the diffusing gas exhibit "random
walk" properties and wondered if the flow of heat through a
conductor would also have these characteristics.
Our hypothesis states that if heat was supplied to one end of a
metal rod, the flow of heat through the rod would not take an
amount of time linearly related to the distance it had to
travel. Rather, we thought that the heat would take an amount
of time to get to the end that was exponentially related to the
distance down the rod at which the temperature was measured.
II. METHODOLOGY:
Equipment needed-
1. Hot plate or other source of heat and insulating material
(we used Styrofoam, but expect that using a better insulator
like fiberglass would yield better results)
2. Thermometers (we used 4) [If electronic thermistor sensors
were used, we suspect that more numerous and better quality
data would be forthcoming]
3. Metal Rod (specifically, we used aluminum)
4. A lab stand to suspend the rod, sleeved in Styrofoam,
horizontally with its revealed end against the hot plate, stood
on its side.
5. A knife to cut the Styrofoam into blocks and poke holes in
which to insert the thermometers.
The variables in the experiment are:
1. The dimensions and material of the rod
2. The heat output of the hot plate
3. Ambient room temperature
4. Placement of the thermometers
5. Leakage of heat past the insulation
6. Convection of heat from the hot plate to the unshielded far
end of the rod
We were able to control the dimensions and material of the rod,
as well as the placement of the thermometers, but the other
factors were not well controlled.
The procedure used in the experiment is as follows:
1. Procure an insulator such as Styrofoam and poke long holes
through the foam to allow the metal rod to pass through as
snugly as possible.
2. Place the rod, with the insulation surrounding it, into the
clamp of the lab stand. Use a ruler to mark off equal lengths
on the insulator in order to locate the thermometers or
sensors.
3. Place a hot plate on its side at one end of the rod, and
adjust the clamp so that the rod-end makes a good contact with
the face of the hot plate.
4. Make small holes in the insulation at the marked intervals,
to allow a thermometer or sensor to protrude through the
insulation and contact the top of the rod (it is important to
be consistent in the side of the rod contacted, we chose the
top since it would radiate the most heat). Insert the
thermometers or sensors, and ensure that a good contact is made
(we checked the contact with our thermometers by twisting and
observing a loud grinding sound of glass on aluminum).
5. Turn on the hot plate, and take measurements of the
temperatures of each of the thermometers or sensors at regular
intervals until there is a noticeable flow of heat to the end
of the rod, where the last thermometer is placed.
III. ANALYSIS OF DATA:
The data indicates that although the overall trend of
temperature vs time for each distance is a linear relationship,
the relation of distance vs time taken to reach a certain
temperature is apparently also linear.
The data show that the temperature at the thermometer closer to
the heat source increased much quicker than the furthest
thermometer, however the relationship does not seem to be the
square of the distance. There are numerous problems with the
consistency of this data, though; for example, the fourth
thermometer, at 12 inches away from the heat source, seemed to
rise faster than the third, only 9 inches away from the heat
source. This may be due to the fact that the end of the rod
opposite to the hot plate is not insulated, so that the
increase in ambient temperature caused by the hot plate and the
convection currents it creates are heating the far end
indirectly. The thermometers used, too, although they were all
made by Gita in the same factory in China, showed some minor
discrepancy in their readings at room temperature.
In relation to our hypothesis, the data does not support our
theory that the time required would vary in a square of
distance relationship, as the movement of particles in gas
diffusion chambers does. Rather, the time required seems to
follow a linear relationship from a cursory analysis. The
first, or 3-inch, thermometer took about three minutes to reach
the 26° C mark, whereas the second thermometer, at 6 inches,
took only five minutes to reach the same temperature. Our
hypothesis would predict that it should take four times as long
to reach the same temperature at twice the distance. This may
be because the phenomena of metal conduction is wholly
different from diffusion, or it may be due to the numerous
uncontrolled factors in the experiment such as the heat
supplied by the hot plate was not constant and there was a
noticeable leakage of heat around the insulation by convection
which heated the far end. Also, we were unable to control the
heat radiation that was emitted in large quantities from the
surface of the hot plate. It is more than likely that much
heat penetrated past the first 3 or so inches of insulation and
caused the second thermometer to rise in temperature more than
it should if all the heat were reaching it through simple
conduction.
IV. SUMMARY AND CONCLUSION:
Our results in this research were not conclusive. Based on one
trial with poorly controlled heat supply and heat leakage, we
found that the relationship between distance and time taken to
reach a temperature was a linear rather than an exponential
relationship. This would seem to contradict our hypothesis,
but we must remember that this was only one experiment, and it
was far from perfect. If we were to continue this direction of
research, we should redesign our experimental setup, look for
an alternate source of heat (a blowtorch, perhaps), and
concentrate a great deal of thought on the problem of limiting
heat leakage and convection.
V. APPLICATION:
It is difficult to find an application for inconclusive
results. However, if more conclusive experiments did vindicate
our hypothesis, we could recommend that campers worried about
burning their hands when toasting marshmallows on metal skewers
should purchase longer ones, since the time required to heat
their hands will increase rapidly with longer skewers. Cooks,
or those using frying pans, should avoid holding the handle
close to the pan.
TITLE: Bumble Ball- Random Walker Or Not?
STUDENT RESEARCHERS: Kevin P. Baratta, Deirdre A. Connolly,
Geetika Diddee
SCHOOL ADDRESS: Belmont High School
221 Concord Ave.
Belmont, MA 02178
GRADE: 11/12
TEACHER: Paul Martenis- plmartenis@aol.com
I. STATEMENT OF PURPOSE AND HYPOTHESIS:
We wanted to find out more about the path of the bumble ball.
We hypothesized that the bumble ball behaves as a random
walker.
II. METHODOLOGY:
We tested our hypothesis using the bumble ball, chalk, a stop
watch, a paper grid to collect data on, blue and red pens to
mark the path of the bumble ball, and paper labels A-I. We
composed a sixteen by sixteen square grid on the floor. We
traced the path of the bumble ball on the paper grid and
dropped labels marking the position every ten seconds. We ran
this process for ninety seconds per trial and performed as many
trials as we had time for. We ran the Many Walkers program for
the purpose of comparing the average X 2.
III. ANALYSIS OF DATA:
Our data indicated that the bumble ball was a random walker.
We recorded the location of the bumble ball every ten seconds
for ninety seconds. We analyzed this data in two ways. First,
we plotted the locations of the bumble ball for each trial at
every ten second interval on nine separate grids. We found
that the points were not distributed to any particular side and
there were very few if any repeating points. This indicated
that if the process was run more than once under identical
circumstances, the results would not be the same and thus
unpredictable. This means that the bumble ball is a random
walker.
We measured the distance from the starting point of the bumble
ball's path to each of its locations at the ten second
intervals. After collecting the distances for each location A-
I, over forty trials, we composed an Excel spread sheet to
find the overall average squared value at each distance. In
finding this value, we were able to compare our results to the
results of the many walker simulation. The graphs of the
bumble ball and many walkers are similar for the same number of
walkers and data points. Both sets of data points fall close
to a line, but not on it, due to the limited number of trials
we were able to perform.
IV. SUMMARY AND CONCLUSION:
Through the analysis of our data, we found that the bumble ball
is a random walker, which is in accordance with our hypothesis.
After forty trials of performing the experiment, we were unable
to predict the location of the bumble ball at any particular
time.
V. APPLICATION:
Our project gave us a "hands-on experience" in the world of
Randomness. The method we used can be applied to study other
mechanisms (toys) like the bumble ball.
TITLE: Comparison Of The Growth Patterns In Forest And Fire
And An Analysis Of The Programs' Representations Of
Real Forests
STUDENT RESEARCHERS: Boris Klimovitsky, Erik Landfried, Kaitlin
McGaw
SCHOOL ADDRESS: Belmont High School
221 Concord Ave.
Belmont, MA 02178
GRADE: 11/12
TEACHER: Paul Hickman- namkcihp@aol.com
I. STATEMENT OF PURPOSE AND HYPOTHESIS:
We want to explore the different ways of growing a forest,
using the programs Forest and Fire. We want to determine which
program more accurately portrays a true forest, as well as the
differences between the two methods. We think that Forest will
better represent an authentic forest, as it grows from the
center out with each seed's placement dependent on the growth
of the previous; whereas, Fire grows in rows, regardless of
previous seeding. We also believe that the two methods should
produce dissimilar fractal dimensions.
II. METHODOLOGY:
The first thing we did was explore the two programs
extensively. We attempted to look for the possible quirks of
each density. To compare the programs, we found the fractal
dimensions of each at the same density and compared the values.
The method to find each fractal dimension is as follows:
1) Create a forest picture by clicking on the appropriate
density.
2) Capture this image. (For Fire , simply select "Save
Picture." from File menu.
3) Open MacPaint, and paste the captured image into a file;
select "Invert" to switch coloring for proper scan.
4) Open Fractal Dimension 5.1 , and then open the MP file you
just created.
5) Measure the image using both the box and circle methods.
6) Look at the graph, and clean it up appropriately,
eliminating points which are not on the line.
7) The value for the slope of this line is the fractal
dimension.
8) Compare the fractal dimensions for both programs to see if
the method of growing the forest affects the fractal dimension.
III. ANALYSIS OF DATA:
The formation of forests is based on an intricate chain of
events, including wind, topography, climate, earth history, and
soil development. Real forests also start out from a central
seed and grow outward until they reach a stage called the
climax forest, which is the point at which little change
occurs. Climax forests have no new types of trees growing
there, and there is minimal change in the overall shape of the
forest. Climax forests perpetuate endlessly. ( Farb, Peter.
The Forest. New York: Time-Life Books, 1963)
IV. SUMMARY AND CONCLUSION:
In order to make a comparison between the two programs, we
sought to find the fractal dimension of the forests that each
created. We decided to look at the forests created at the
critical probability, when there is a 50% chance that the
forest will percolate. Percolation is the growth of a fractal
design from one side to the other. In our case, we examined
images where the forest reached two sides of the grid to which
it was limited.
We decided to examine Forest at .5 and .6 since we were unable
to look at its critical probability of .59. We looked at one
forest grown at each of the two probabilities. We found the
fractal dimensions to be for .5 and .6, 1.321 and 1.431,
respectively. For Fire, we found that the fractal dimension
was 1.668. For the three sample forests we measured, the shape
made by the Fire program had a higher fractal dimension.
Although we measured only three sample forests, we proved our
hypothesis that the different methods would produce dissimilar
fractal dimensions.
We also found that both programs have similarities with real
forests, yet the differences clearly outweigh these few
similarities. Forest, we believe, represents an individual
forest. It starts in the center and grows out from there until
something curbs its growth (amount of moisture, poor soil,
rock). It also realistically reaches a state of "climax
forest", where there are minimal future changes (on the program
the final forest is surrounded by rocks). The forest in Fire,
however, seems like it reaches beyond the grid endlessly. The
method of growth for Fire is inaccurate in that forests simply
do not grow from a single seed, independent from surroundings
and previous growth. Neither program accounts for the forces
of nature that affect a real forest, such as wind, topography,
climate, soil development, etc. We have decided that Forest
more accurately represents a true forest.
V. APPLICATION:
We think that the two programs are excellent for the unit on
fractals, but fail to accurately portray true forests for
students to examine. A better program, including a combination
of Forest as well as the natural factors which affect a forest,
would be a more useful tool for learning about forests and
natural fractals.
© 1997 John I. Swang, Ph.D.