The National Student Research Center

E-Journal of Student Research: Math

Volume 4, Number 2, July, 1997

John I. Swang, Ph.D.

Founder/Director

National Student Research Center

2024 Livingston Street

Mandeville, Louisiana 70448

U.S.A.

E-Mail: nsrcmms@communique.net

http://youth.net/nsrc/nsrc.html

1. When Is A Fractal a Fractal? A Study Of ZnSO4 Aggregates At Different Molarity
2. s Euler's Formula Accurate?
3. The Fractal Dimension of Populations in the Program "Anthill"
4. Does Pi Always Equal 3.14?
5. The Effect of Various Anode Shapes on Fractal Growth in an Electrodeposition Cell
6. Probability Theory

TITLE:  When Is A Fractal a Fractal? A Study Of ZnSO4
        Aggregates At Different Molarity

STUDENT RESEARCHERS:  Daniel Brecher, Lelia Evans, and Philip
                      Ording
SCHOOL ADDRESS: Belmont High School
                  221 Concord Ave.
                  Belmont, MA 02178
GRADE:  12
TEACHER:  Paul Hickman - phickman@copernicus.bbn.com


I.  STATEMENT OF PURPOSE AND HYPOTHESIS:

The experiment with copper aggregate was fresh in our minds and 
we wanted to find out what other kinds of fractal shapes we 
could create by changing the solution or concentration.  
Originally, we wanted to compare the copper aggregates to the 
zinc aggregates at different molarities, but we realized that 
the scope of this investigation was too large.  So we settled 
on just using ZnSO4.  Our hypothesis states that as the 
molarity of ZnSO4 solution changes, the fractal dimension of 
deposition fractals will change in an inverse relationship. 

II.  METHODOLOGY:

We mixed solutions of ZnSO4 at different molarities; 0.1 M, 0.2 
M, and 0.4 M.  Originally we created a 0.05 M solution.  
However, after using the 0.2 M and 0.1 M to grow aggregates we 
noticed a trend.  We knew that the 0.05 M would most likely not 
be a fractal [see scan of aggregates; as molarity increased the 
branches fill more area--the 0.05M would be a blackened 
circle.]  So, we decided to go the other direction and make a 
0.4 M solution.

We used scotch tape to cover the bottom hole of the cell.  This 
technique, although excellent for scanning, produced 
troublesome bubbles.  It took at least three trials, when 
making a single cell, to remove the bubbles from the anode.

We grew two aggregates of each molarity simultaneously by 
connecting them in parallel.  To eliminate any variables, we 
made the anodes the same size and position and applied the same 
voltage to the cell.  We grew each aggregate for 15 minutes and 
then disconnected the power source to stop the growth. 

Next, we removed the scotch tape from the bottom of the cell 
and scanned the aggregates into the computer.  So as not to 
scan the anode wire, we placed a piece of white paper, with a 
slit for the anode, over the cell.  Then we saved the scanned 
images as MacPaint files.  Using MacPaint, we made some 
corrections on the aggregates: we touched-up stray points which 
appeared only after scanning and we removed the blobs at the 
end of the 0.4 M aggregates (these blobs were composed of 
copper which plated from the copper cathode ring as the 
branches of the zinc got too close to the ring).

We measured the fractal dimensions of the aggregates using the 
box and circle methods and compared these results.

We analyzed our results by putting the values we found for both 
the box and circle methods on a single graph, showing an 
inverse relationship. 

III.  ANALYSIS OF DATA:

          .1a M     .1b M     .2a M    .2b M    .4a M     .4b m
Box      1.934     1.938     1.771    1.750    1.386     1.422
Circle   1.993 *   1.974 *   1.806    1.839    1.566     1.636

* When we measured the aggregate by the fast circle method, the 
lowest radius was pulling the log graph to a too-steep 
position, making the fractal dimension greater than two, which 
is not possible. (.1a = 2.183,  .1b = 2.069).  To solve this 
problem, we eliminated the first data point which was a hole in 
the fractal, where the cathode was located. 

IV.  SUMMARY AND CONCLUSION:

As we discussed above, our data shows that there is an inverse 
relationship between the solution molarity and the fractal 
dimension of the aggregates formed by these solutions.  This 
conclusion contradicts our prediction that a higher molarity 
would produce a denser, higher-fractal-dimension aggregate.  

One observation of the aggregate growth helped us understand 
the inverse relationship between fractal dimension and 
molarity: the higher molarity aggregates grew faster than the 
lower molarity aggregates.  Aggregate growth is a process by 
which positively charged zinc ions in solution migrate towards 
the negative electrode and plate out.  Since the voltage was 
constant, the ions, whether in high or low concentration, in 
every cell move at the same rate towards the cathode.  The 
higher molarity solution contains more zinc ions in solution.  
In this case, the aggregate will grow faster because there are 
more ions present.  Aggregates which are formed at faster rate 
are more likely to form a node, a bump resulting from 
inconsistencies in plating ions.  

A node which protrudes from an aggregate has more edges than 
the rest of the aggregate, and therefore the node will collect 
more plating ions and grow at a faster rate.  High molarity 
solutions grow more nodes, and these nodes grow into branches.  
In the lowest molarity solution, 0.1 M, there are fewer ions.  
The aggregate will grow slowly and therefore fewer nodes will 
form.  The plating occurs equally around the cathode causing a 
nearly perfect, dense circle.  This accounts for the increasing 
fractal character as the molarity increases. 

V.  APPLICATION:

Many patterns of growth existing in nature have useful 
functions.  Crystals of silicon are sliced for microchip  and 
other electronic devices.  Plating is used in the production of 
jewelry and electronic parts.  We hope that our findings will 
contribute to a better understanding of the factors which alter 
and shape patterns of growth. 



TITLE:  Is Euler's Formula Accurate?

STUDENT RESEARCHER:  Paul Brand
SCHOOL:  Mandeville Middle School
         Mandeville, Louisiana
GRADE:  6
TEACHER:  John I. Swang, Ph.D.


I.  STATEMENT OF PURPOSE AND HYPOTHESIS:

I would like to conduct a research project to see if Euler's 
formula is correct.  Euler's formula is a mathematical formula 
that states that the number of edges on a polyhedron is equal 
to the number of faces plus the number of vertices minus two 
(E=F+V-2).  My hypothesis states that Euler's formula will be 
accurate with all of the polyhedrons that I test.

II.  METHODOLOGY:

First, I wrote my statement of purpose and hypothesis.  Then I 
conducted a review of literature on Leonard Euler, Euler's 
formula, and polyhedrons.

The manipulated variables in my experiment were the number of 
faces, vertices, and edges on the different polyhedrons I 
gathered.  The responding variables in my experiment were the 
answer to Euler's formula generated.  The variables held 
constant were Euler's formula.

I gathered 10 different polyhedrons.  The I counted the number 
of edges, vertices, and faces and then recorded them on my data 
collection sheet.  Then I used Euler's formula to compute the 
number of edges.  Then I compared the number of edges I counted 
and the number I computed with Euler's fomula.  Then I wrote my 
summary and conclusion where I accepted or rejected my 
hypothesis.  Then I applied my findings and published my 
research.

III.  ANALYSIS OF DATA:

For polyhedron  1, E=F+V-2 was 32.  The actual number of edges 
was 32.  For polyhedron  2, E=F+V-2 was 12.  The actual number 
of edges was 12.  For polyhedron  3, E=F+V-2 was 12.  The 
actual number of edges was 12.  For polyhedron  4, E=F+V-2 was 
15.  The actual number of edges was 15.  For polyhedron  5, 
E=F+V-2 was 12.  The actual number of edges was 12.  For 
polyhedron  6, E=F+V-2 was  9.  The actual number of edges was 
9.  For polyhedron  7, E=F+V-2 was 26.  The actual number of 
edges was 26.  For polyhedron  8, E=F+V-2 was 12.  The actual 
number of edges was 12.  For polyhedron  9, E=F+V-2 was 11.  
The actual number of edges was 11.  For polyhedron 10, E=F+V-2 
was 11.  The actual number of edges was 11.  Euler's formula 
was correct ten out of the ten times I tested it.

IV.  SUMMARY AND CONCLUSION:

In my research, I found out that Euler's formula worked on 10 
out of the 10 polyhedrons I tested.  Therefore, I accept my 
hypothesis which stated that Euler's formula would be accurate.

V.  APPLICATION:

I will apply my findings by telling students when they are 
working with a complex polyhedron that they can find the edges 
by adding the faces and vertices and subtracting 2.



TITLE:  The Fractal Dimension of Populations in the Program
        "Anthill"

STUDENT RESEARCHERS: Chris Baird, Julia Stoyanovich 
SCHOOL ADDRESS: Belmont High School
                  221 Concord Ave.
                  Belmont, MA 02178
GRADE:  11/12
TEACHER:  Paul Hickman - phickman@copernicus.bbn.com


I.  STATEMENT OF PURPOSE AND HYPOTHESIS:

We want to find out whether there is a connection between the 
initial parameters of population growth, such as time and size 
of population, and the fractal dimension or jaggedness of the 
pattern of sites visited by the population.   Our hypothesis is 
that there is a certain connection, but we do not know what 
kind of correlation there will be. 

II.  METHODOLOGY:

We carried out a research project using the Beta version of 
"Anthill," a two-dimensional random walk program by Paul 
Trunfio from Boston University.

1.  To acquire data, we ran Anthill many times, changing the 
corresponding variable(s).  After a certain amount of time we 
paused the growth and copied the image, using the program 
Capture 4.0 by Yves Lempereur, and saved them as a Macpaint 
files.  From there we used the program Fractal Dimension 5.1 by 
Brandon Volbright to measure the fractal dimension of the 
pictures.  We used both options of the program: box and circle 
method, and averaged them.  Than we compared the dimensions of 
these growths for one variable and plotted the values in the 
program Graphical Analysis 1.3.1.1 by Dave Vernier and Todd 
Bates.  

2.  We ran the program several times with constant parameters 
to make sure that the pictures are different each time, with 
their fractal dimension the same, showing the program is based 
on randomness.  

3.  We ran the program, changing only the number of ants.  We 
used 1, 2, 3, 4, 10, 25, 50, 75, 100, 150 for the number of 
ants. We organized sample images of population growth and their 
values into groups with 1, 2, 3, 4 in one group, and 1, 50, 
100, 150 in another. Since the fractal dimension for the low 
values of number of ants showed little change, we found it 
necessary to also investigate high values.

4.  The final step of our research was to observe the 
correlation between the time the program is run and the fractal 
dimension of the resulting pattern.  We ran Anthill with the 
same set of parameters (1 ant, no deaths or births, equal 
probabilities of direction) for 20, 30, 40, 60, 90, and 120 
seconds.  We organized a sampling of these images with 30, 60, 
90, and 120 seconds.

5.  To check the accuracy of our results, we picked a value for 
the variable we were testing and applied it to the relationship 
we had found. The fractal dimension value from this function 
was a prediction which we then compared to actual results to 
check the validity of the relationship. 

III.  ANALYSIS OF DATA:

1.  For the constant variables we did find different images 
that have similar fractal dimensions, showing the program's 
true randomness.

2.  When we first investigated the variable of number of ants 
we used 1, 2, 3, and 4 ants. We found that the fractal 
dimension did not change significantly.

3.  We thought that even though we had acquired almost constant 
results for fractal dimension of changing number of ants, there 
still might be an increasing relationship.  Therefore we tried 
much higher values of the number of ants: 10, 25, 50, 75, 100, 
and 150. We found that the fractal dimension did change, it 
proved to be an increasing non-linear function.  Our prediction 
made by applying this function was close to the measured value, 
showing the accuracy of our result.

4.  When we ran the program, changing only the time, and 
leaving all the other parameters constant, we discovered a 
correlation between the time and the fractal dimension of the 
resulting pattern.  The relation was a linear increasing 
function.  Predicting a value for fractal dimension through 
this linear function, we came up with accurate results, showing 
the validity of this relationship.

IV.  SUMMARY AND CONCLUSION:

1.  The Fractal Dimension does depend on both number of ants 
and time as we thought.

2.  The Fractal Dimension is an increasing function of the 
number of ants.  The function is not linear.  The more the 
ants, the greater the jaggedness.

3.  The Fractal Dimension is an increasing function of time.  
The function is linear.  The longer the ants do their job, the 
greater the jaggedness.

V.  APPLICATION:

The results of our research can be used in predicting the 
behavior of some natural systems, such as forest fires, insect 
or animal population growth, the spread of disease, etc. The 
research can be continued by discovering the connection between 
the rest of parameters, and the final image.



TITLE:  Does Pi Always Equal 3.14?   

STUDENT RESEARCHERS:  Graham Rees and Colby Omner  
SCHOOL:  Mandeville Middle School
         Mandeville, Louisiana
GRADE:  6
TEACHER:  John I. Swang, Ph.D.


I.  STATEMENT OF PURPOSE AND HYPOTHESIS: 

We would like to do a mathematical research project to see if 
Pi is always equal to 3.14 no matter how big or small the 
circle is.  Our hypothesis states that Pi will always equal 
3.14 no matter what size the circle is.

V.  METHODOLOGY:

First, we wrote our statement of purpose and reviewed the 
literature on Pi, radius, circles, circumference, diameter, and 
ratio.  Second, we developed our hypothesis and a methodology 
to test our hypothesis.  Next, we listed our materials and made 
our observation and data collection form.  

Then we began our experiment by finding ten different sizes of 
circular objects that we were going to use.  They were a paper 
towel roll, a soft drink can top, a CD, a half dollar, a 
quarter, a clock, a Pog pad, a garbage can top, a Frisbee, and 
a Tupperware container.  Then we measured the diameter of the 
circles with a ruler.  Then we measured the circumference of 
the circles by putting a mark on the edge of the circular 
object.  Then we put another mark on the table where the mark 
on the circular object touched the table.  Next, we rolled the 
circular object until the mark on it touched the table again.  
Then we placed a mark on the table where the mark on circular 
object had touched the table again.  Then we measured the 
distance from the first mark on the table to the second mark on 
the table.  This measurement was the circumference.  Next, we 
divided the diameter of the circle into the circumference of 
the circle to find a value for Pi. 

Our manipulated variable was the size of the circles.  Our 
responding variable was the value of Pi.  Our controlled 
variables were the way we measured the circles' diameter and 
circumference and computed the value of Pi.                    

There were two sets of data, one from each student researcher.  
After we combined our data, we analyzed it.  Then we wrote our 
summary and conclusion where we accepted or rejected our 
hypothesis.  Finally, we applied our findings to the world 
outside of the classroom.    

III. ANALYSIS OF DATA:

We found out that a pog's diameter was 27 centimeters and its 
circumference was 86 centimeters.  When we divided them to see 
what their Pi equaled we got 3.19.

The CD's diameter was 11.9 centimeters and its circumference 
was 34.5 centimeters.  When we divided them to see what their 
Pi equaled we got 2.89.
 
The hair spray can top's diameter was 5.5 centimeters and its 
circumference was 19.6 centimeters.  When we divided them to 
see what their Pi equaled we got 3.38.

The silver dollar's diameter was 3.7 centimeters and its 
circumference was 11.5 centimeters.  When we divided them to 
see what their Pi equaled we got 3.11.

The cup's diameter was 8.8 centimeters and its circumference 
was 30 centimeters.  When we divided them to see what their Pi 
equaled we got 3.40.

The Frisbee's diameter was 22 centimeters and its circumference 
was 69 centimeters.  When we divided them to see what their Pi 
equaled we got 3.40.

The diameter of the Tupperware container was 15.4 centimeters 
and its circumference was 47.6 centimeters.  When we divided 
them to see what their Pi equaled we got 3.09.

The quarter's diameter was 2.7 centimeters and its 
circumference was 8.5 centimeters.  When we divided them to see 
what their Pi equaled we got 3.15.

The soft drink can top's diameter was 5.3 centimeters and its 
circumference was 20 centimeters.  When we divided them to see 
what their Pi equaled we got 3.77.

The diameter of the paper towel roll was 6.7 centimeters and 
its circumference was 21.5 centimeters.  When we divided them 
to see what their Pi equaled we got 3.19.

IV.  SUMMARY AND CONCLUSION: 

In summary, we got an average of 3.23 for our equivalent of Pi, 
but we performed our experiment very crudely.  When the 
scientists perform these experiments they use very precise 
instruments.  Therefore, we accept our hypothesis because if we 
did this experiment again and used very precise instruments we 
would probably come out to an average of 3.14 for Pi. 

V.  APPLICATION:

We can apply our findings to the world outside the classroom by 
telling students to listen to their teachers when they say Pi 
is always equal to 3.14.  We can also tell students that they 
would have to have advanced measuring tools to see if Pi is 
always equal to 3.14.  



TITLE:  The Effect of Various Anode Shapes on Fractal Growth in
        an Electrodeposition Cell

STUDENT RESEARCHERS:  Paris Gartaganis, Tom Glennon, Laureen
                      Laglagaron
SCHOOL ADDRESS: Belmont High School
                  221 Concord Ave.
                  Belmont, MA 02178
GRADE:  12
TEACHER:  Paul Hickman - phickman@copernicus.bbn.com


I.  STATEMENT OF PURPOSE AND HYPOTHESIS:

Electrodeposition produces fractal shapes. In our investigation 
we wanted to find out more about different anode shapes and the 
fractals they produced. This investigation dealt with 
symmetrical and asymmetrical shapes, the growth patterns 
produced and the ability to predict fractal shapes.  We also 
examined the difference in fractal dimension of each aggregate 
grown and its correlation with the anode shape.  We used five 
different anode shapes (A circle, a square, a triangle, a D-
shape, and a shamrock) with varying area but consistent 
perimeter.  The purpose of this investigation was to see if we 
are able to predict the aggregate growth, dependent on the 
anode shape, and to see what factors influence fractal growth.

We were interested in the effects of different shaped anodes on 
fractal growth.  The fractals in an electrodeposition cell are 
produced due to ions coming towards the center cathode.  Our 
hypothesis states that the aggregate would assume a pattern 
that was influenced by the shape of the anode, although not an 
exact replica of the shape. 

II.  METHODOLOGY:

We constructed five anode shapes made out of copper wire which 
all had the same perimeter of 24 cm.  These five shapes were 
square, triangle, circle, shamrock, and a D-shape.  We used an 
electrodeposition cell which is two glass plates held together 
with metal clamps, with a center hole for the wire, to conduct 
our investigation.  We then grew aggregates with a 0.2 M CuSO4 
solution with the same current of 10 volts for each.  After the 
aggregate finished growing we scanned it into the computer 
using the AppleScan program and touched up the aggregates using 
MacPaint.  Once we had our aggregates on MacPaint, we were able 
to measure the fractal dimension using the computer program 
called Fractal Dimension 5.1.

III.  ANALYSIS OF DATA:

The shapes of the aggregates were not in correlation with the 
shape of the anodes.  Since each anode perimeter was 24 cm, the 
area of each anode was different.  The area of the circle was 
38.4845 cm2, the area of the triangle was 27.71 cm2, the area 
of the D-shape was 25.13 cm2, the area of the square was 36 
cm2, and the area of the shamrock was 25.92 cm2.   The speed of 
the aggregate growth and the density of the aggregate were both 
affected by the area.   

Fractal Dimension:  Using the program Fractal Dimension 5.1 we 
measured the Fractal Dimension of each aggregate using the box 
method and the circle method.  The FD of each aggregate are as 
follows:


Circle:                                   Shamrock:
box= 1.587                                box=  1.616
circle= 1.650                             circle= 1.647

Triangle:                                 D-shape:
box=  1.649                               box= 1.616
circle= 1.650                             circle=  1.656

Square:
box= 1.650
circle= 1.659

IV.  SUMMARY AND CONCLUSION:

From the results that we obtained in our experiment, we found 
that it was nearly impossible to predict aggregate growth based 
on the shape of the anode. It is difficult to establish a 
concrete result that would provide an explanation for varying 
anodes because we did not notice a substantial difference 
between aggregates formed with different anode shapes.  We 
rejected our hypothesis that different anode shapes would 
affect fractal growth because we did not notice a significant 
difference between fractal growth shapes.

Once we analyzed our aggregates, it was easy to come up with 
reasons for our result, but they were not based on concrete 
evidence and could be influenced by our want to notice a 
difference between aggregate shapes.  To firmly establish any 
theory, we would have to repeat this experiment to ensure that 
the results were consistent and not limited to our first 
findings.

One hypothesis that we considered was in the difference of 
'branching' between the aggregates and the density of this 
branching.  Our theory was that aggregates formed with an anode 
which had a small area were less branched, possibly because the 
aggregate could not grow to its full potential without hitting 
the anode.  However, this theory was limited only to our 
findings and must be validated through numerous experiments.

We also confirmed our assumption that the strength of the field 
did have an effect on aggregate growth.  As the strength of the 
field increased, the rate of growth of the aggregate also 
increased.  We were surprised that the difference in anode 
shape did not produce a significant, visual change in 
aggregates, but we realized that because aggregates are formed 
so randomly, it was difficult to constrict this randomness to 
particular shapes.

V.  APPLICATION:

Since our hypothesis was not validated in our findings, we have 
developed procedures which might help other researchers 
investigate the topic of fractal growth.  Other researchers who 
investigate the topic of fractal growth might try to limit 
their research by using a consistent area as opposed to a 
consistent perimeter, varying the solution used, or using 
larger anodes.  In order to apply our results, other 
investigators might want to verify our results by repeating our 
experiment numerous times.



TITLE:  Probability Theory 

STUDENT RESEARCHER:  Gordon Spring
SCHOOL:  Mandeville Middle School
         Mandeville, Louisiana
GRADE:  6
TEACHER:  John I. Swang, Ph.D.


I.  STATEMENT OF PURPOSE AND HYPOTHESIS:

I would like to do a mathematical research project on 
probability theory.  I want to know if it is true.  My 
hypothesis states that any number on a die will come up 
seventy-five times when the die is thrown four hundred and 
fifty times as probability theory predicts. 

II.  METHODOLOGY:

The first thing I did was decide on a topic and then I wrote a 
statement of purpose.  Next, I collected and reviewed the 
literature on luck, chance, fortune, and probability.   
Following that I constructed a hypothesis and developed a 
methodology to test my hypothesis.  After that I compiled a 
list of materials.  I then performed my experiment and wrote an 
analysis of data, summary and conclusions, and application.

Then I got a cup and a die, placed the die in the cup, and 
rolled out the die.  I did this four hundred and fifty times, 
three sets of one hundred and fifty.  Each time I recorded the 
outcome of the roll of the die on a piece of paper.

While doing this procedure, I rolled the die the exact same way 
each time by shaking the cup three times before rolling the die 
and letting it bounce off a wall.

III.  ANALYSIS OF DATA:

Out of three trials, the numbers five and six were rolled most 
frequently, followed by one, two, four, and three in that 
order.  One came up 27 times in the first trial, 25 in the 
second, and 26 in the third.  Two was rolled 22, 29, and 25 
times.  Three came up 18, 27, and 12 times.  Four's outcome was 
21, 21, and 26.  Five was rolled 29, 26, and 30 times, while 
six was rolled 33, 22, and 30 times. 

IV.  SUMMARY AND CONCLUSION:

The die was rolled 450 times.  With all three trials combined, 
one was rolled 78 times.  Probability theory predicted that the 
number one would be rolled 75 times.  Two was rolled 76 times.  
Probability theory predicted that the number two would be 
rolled 75 times.  Three was rolled 57 times.  Probability 
theory predicts that the number three would be rolled 75 times.  
Four was rolled 68 times.  Probability theory predicts that the 
number four would be rolled 75 times.  Five was rolled 85 
times.  Probability theory predicts that the number five would 
be rolled 75 times.  Six was rolled 85 times also.  Probability 
theory predicts that the number six would be rolled 75 times.  

Because of the small number of times I performed my experiment, 
I rejected my hypothesis which states that any number on a die 
will come up seventy-five times when a die is thrown four 
hundred and fifty times as probability theory predicts.  I 
believe that if I performed this experiment many more times I 
would be able to accept my hypothesis because the observed 
values would better approximate those predicted by probability 
theory.   

V.  APPLICATION:

I will apply my findings to the world outside the classroom by 
using my new knowledge about probability theory to increase my 
chances of winning a game with dice.

© 1997 John I. Swang, Ph.D.