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Macintosh and Windows software to your host computer.
The NSF Applications of Advanced Technology project On Growth and Form: Learning Concepts of Probability and Fractals by ``Doing Science'' led to a number of prototype hands-on activities, laboratory experiments, and interactive visualization programs for the Macintosh platform have been developed to demonstrate how fundamentally random microscopic events can give rise to fractal macroscopic patterns.
A more developed set of applications and curriculum is available at Patterns in Nature.
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Macintosh and Windows software to your host computer.
For further information on the following simulations, please contact us. Feedback would be most appreciated.
The Random Walk program allows allows the student to flip a set of ten
coins a large number of times quickly and summarizes in the form of a
histogram the number of heads resulting from each trial. Another part
of the program models the direction of a one dimensional random walker
by the flip of a coin. Where the walker lands at the end of ten steps
is compiled in a histogram at the bottom of the screen.
One of the simplest and clearest demonstrations of probabilistic
phenomena is produced by dropping balls over an array of pins which form
Pascal's Triangle. By being forced to go either left or right, the
balls form a normal distribution in the catch bins at the bottom of the
array. Repeated trials demonstrate a number of phenomena: averaging;
statistical variation; etc. The combination of an actual physical model
with the computer simulation that can then run many trials rapidly
reinforces fundamental probabilistic concepts. In this way, students
are slowly exposed to the basic rules and statistical properties which
are necessary in order to explore diffusion processes in nature.
Diffusion in Gels experiment: Two rectangular slabs of gelatin,
agar, or some other gel are placed in contact. One is clear gel. The
other has concentrated food color added. Over a period of a few days
students watch and measure the diffusion front of color as it moves
through the clear gel into and through the clear gel.
Liesegang Rings experiment: A crystal of copper sulfate is
placed in the center of a gel containing dissolved potassium chromate.
As the copper sulfate diffuses into the gel, copper chromate crystals
precipitate out, creating a colored ring. Locally this depletes the
chromate below the precipitation threshold. The copper must diffuse
further to find chromate concentration above the precipitation
threshold. As this process repeats, the result is a set of colored
rings, developing in time, that illustrate diffusion.
The Diffusion Chamber experiment: Two different gases are
inserted at the opposite ends of a clear glass tube. (Typically these
gases are NH3 and HCl). The gases diffuse through the tube and when
they meet they form a white dust (NH4 Cl) part way along the tube.
Students are asked to predict mathematically where the "dust" will form
and how soon this will occur.
The Fractal Coastline program randomly generates coastlines
based on conditions set by the student. Student measures the fractal
dimension of the resulting coastline using the computer version of the
methods employed in the hands-on activity: "walking" with rulers of
different lengths or "covering" with grids of various sizes. The
computer records the data and allows students to create graphs based on
their data to find the dimension of their object.
Click here to see how one
teacher has used these materials in his class.
The Fractal Dimension program gives the student a variety of
objects to measure to determine their dimension. The program offers two
measuring methods: (1) the box method which covers the object with
different sized boxes and records the number of boxes it takes to cover
the object; and (2) the circle method which covers the object with
concentric circles and records how much of the object is within each
circle. The program graphs the results of its measurements and allows
the student to choose which data points they want the computer to graph.
Objects from the "real world" (including patterns grown in laboratory
experiments) can be scanned into the computer and imported into this
program for analysis.
The Aggregation Kit program automates the student hands-on
activity of pattern building. The student can generate computer
aggregates by setting a number of variables. The program allows the
student to split screens an observe two aggregates at once in order to
determine the effect that changing variables has on the aggregate that
grows. Aggregates generated by this program can be saved and
transferred into the Fractal Dimension program to be analyzed.
Experiments which demonstrate fractal growth phenomena include
Bacterial Colony Growth, Termite foraging,
Electrochemical Deposition, Viscous Fingering, and
Dielectric Breakdown patterns. The patterns for each of these
experiments can be digitized and imported into the Fractal
Dimension program for quantitative analysis.
Student laboratory experiments (i.e., Hele-Shaw Lichtenberg ) and real
images taken from nature (i.e., leaf patterns, photographs of lightning)
can be scanned into the computer or digitized through the use of a video
camera and frame grabber. The programs are rich enough that they can
also be used to create images from the numerical data of digital
elevation maps developed by the NATIONAL GEOLOGICAL SURVEY (i.e., river
patterns, canyon shapes, mountain ranges, and coastlines) as well as
from satellite data collected by NASA's MAGELLAN MISSION of the surfaces
of Venus.
See the ROUGH SURFACES MANUAL by
undergraduate David Futer.
The Lottery program confronts misconceptions about "winning
streaks" and the predictability of random events. This program flips
coins until it gets four heads in a row. When fours heads in a row
occur, the computer stops and allows the student to predict (bet) what
the result of the next flip will be. The computer keeps track of the
student's "score" by recording the number of times they correctly
predict the next flip.
Jump to:
Random Walks,
Anthill,
Deer,
Many Walkers,
Diffusion Chamber,
Coastline,
Fractal Dimension,
Aggregation Kit,
GasketMeister,
Surface Analyzer,
BlaZe,
Forest,
Hopfield,
Lottery,
Polymer,
DNA Walk, or
Chill Out.
Hands-on activities encourage students to predict the outcome of
flipping a set of ten coins 1000 times. How many times would all ten
coins come up heads? How many times would nine coins come up heads?
etc. Then students flip coins for themselves and average results over
the entire class. Students then are given a number line and a walker.
Students place the walker in the center of the number line and move it
left or right depending on the outcome of a coin flip. 
Random Walks
This program allows the student to experiment with multiple two
dimensional walkers. A large number of "ants" (walkers) start together
in the center of the computer screen. Each ant moves independently, in
a random walk, and the area covered by the ants grows. The student may
change the relative probability that each ant moves in each direction:
north, south, east, or west.
Anthill
The two dimensional random walkers in this program are deer. The deer
move around a field of grass that they eat. The lives and movements of
these deer are governed by a number of rules. How big are the deer?
How old are the deer when they reproduce? How many steps can a deer
take without food? Also, how long after it is eaten does the grass grow
back? All of these variables are controlled by the student. The
computer illustrates in the form of a chart the population of the deer
and the amount of grass in the field. The goal? To get a stable
population where neither the deer nor the grass overrun the field.
Deer
Many one dimensional random walkers are displayed in a vertical column.
Each walker moves independently and after each step the computer
displays a histogram of walker locations. The computer also displays
the average value of X, where X is the displacement of walkers away from
their initial position. Also shown is a graph of average X squared
vs. the number of steps; in the limit of a large number of walkers, this
graph approaches a straight line, a fundamental consequence of random
walks.
Many
Walkers
This program demonstrates how molecules of two different substances
starting at opposite ends of a tube diffuse along the tube in random
walks. A white precipitate forms where the substances meet each other.
The student controls the length of the tube and the step size of each
substance to see the consequences for the location and speed of
formation of the visible precipitate.
Diffusion Chamber
The Coastline activity allows students to measure a given coast
by "stepping it off" with calipers set to different lengths or by
"covering" the coastline with different sized grids. Does a caliper set
to five miles take twice as many steps along the coast as a caliper set
to ten miles? No. Why not? It has something to do with the dimension
of the coastline. Students collect data for various step sizes then
plot their results on a log-log graph, finding from the slope the
dimension of their coastline.
Fractal
Coastline
Students generate a "gasket" by folding pieces of paper and cutting out
certain sections. When they have completed their "gasket" the students
measure the dimension of their product by covering it with squares.
Students graph the results of their measurements and uncover what
exponent yields the straightest line plot. This exponent is the
dimension of their "gasket". Students are introduced to logarithms as a
tool to take the guesswork out of finding the dimension of an object.
In the previous activity, Fractional Dimension, students had to plug in
exponents until they got the straightest line. Use of a log-log plot
yields the dimension directly from the slope, eliminating number
plugging.
Fractal Dimension
The Pattern Building hands-on activity allows students to model
the movement of an ion in solution by charting the results of a two
dimensional random walk. An ion starts on the outer edge of a circle
and takes steps to complete a random walk. The direction of the steps
can be determined by rolling a tetrahedral die or by following entries
in a randomly generated table. If the ion reaches a structure
previously grown from the center, it sticks to it; if the ion exits the
circle before touching the center it is removed and replaced by a new
random walker. After many walkers stick to the growing structure, they
form a fractal pattern or "aggregate".
Aggregation Kit
In the Chaos Game, students begin by placing three dots on a
piece of paper and numbering them 1, 2, and 3. They also place a fourth
dot, anywhere they choose inside the triangle drawn through the three
numbered dots. Now they choose a numbered tile from a bag. They place
a dot halfway between the fourth dot and the dot corresponding to the
number they chose. They return the tile to the bag, shake, and choose
again. This process is repeated a number of times. The class discusses
what would result if they continued this process. Will the triangle be
completely filled with dots?
The GasketMeister program automates the hands-on activity. The
student places the original three points and the computer rapidly
carries out the process of adding new dots. What results from the
process is not a solid triangle, but a triangle with empty spaces called
a Sierpinski Gasket. The Sierpinski Gasket is a deterministic fractal
that has infinite self-similarity. The student can experiment by
placing starting dots in different geometric configurations, and setting
the distance a new dot is placed from the starting dot, to see if a
Gasket always results.
GasketMeister
This program gives students a variety of options for analyzing the
roughness that is found in the world around us. A number of stored
images allows students to discover the roughness exponent of such things
as the "landscape" of a section of human DNA. Students may also choose
to scan into the program images created from laboratory experiments of
objects they find in the "real world".
Surface
Analyzer
Can computer games, similar to the ever-popular Nintendo, be used to
teach a concept? What format for the game is necessary to achieve this
goal? This ambitious project has seen many versions of a game whose
purpose is to teach about connected pathways, phase transitions, and
their applications to real-world phenomena. A forest fire model was
used to model how connectivity of pathways plays a crucial role in the
formation of many substances (such as gels). In the Blaze! game,
students must not only determine the critical concentration of trees
which maximizes their profit but they are given control over the
situation by flying a helicopter loaded with water to put out the
fire. Due to the limited supply of water, students soon learn that they
must drop their water at strategic places along pathways of burning
trees. These places are called ``red bonds'' in percolation theory---a
subject which students study more formally later.
Blaze!
A seed tree sits in the middle of a checkerboard. A pair of students
takes a handful of pennies and lays one in each of the four squares
adjacent to the dime, then looks to see if each penny laid down is a
head or tail. Each head represents a new tree that grows; each tail
represents a rock. In a second round, students lay pennies on the
spaces adjacent to each new tree. Each new head means a tree that
extends the new growth. Repeat the process again and again until the
board is filled or all trees are surrounded by rocks. Now remove all
``rocks,'' i.e., the pennies showing tails. The process described here
is called Leath percolation and resulting pattern of ``trees'' is a
random fractal. The computer screen shows a square lattice with a seed
tree in the center. As in the game, the student can place a tree or a
rock randomly with equal probability in the four spaces adjacent to the
seed tree, continuing the process until all trees are surrounded by
rocks.
Forest
This program begins by looking at a simple molecular model for a certain
class of disordered materials---spin glasses. We can think of the
molecules in spin glasses as little bar magnets whose spins are either
up or down. In trying to decide how these little magnets line up with
one anothers, it was discovered that the way these systems change the
lining up of magnets is quite similar to one important aspect of brain
functioning, the so-called associative memory activity. Knowing the
dynamics of spin glasses, one can make a program that actually learns a
number of patterns (based on connections between spins) and to correctly
classify a new pattern when it is loaded into it's "brain."
Hopfield
Students flip coins until they get three heads in a row. When they get
three heads in a row they predict the outcome of the next flip using one
of three alternative strategies: "On a Winning Streak," "Running Out of
Luck," or "Random." Students repeat this process a number of times.
After several trials, students compare the success of their strategy
with the alternative strategies employed by their classmates.
Lottery
A pattern of infinitely growing, self-avoiding, random
walks in two dimensions is created on the screen. This pattern models a
polymer chain in a good solvent. The program allows the student to
calculate the fractal dimension of the pattern. The second part of the
program simulates the dynamics of a polymer chain.
Polymer
This program is one of many that explain how a random walk is translated
into a natural phenomenon. Here a random walker generates a landscape.
The landscape created by the random walker is analyzed and compared to a
similar "landscape" derived from the profile of amino acids found in the
DNA of sample genes.
DNA WALK
The student can create and analyze a simulated aggregate, or choose to
analyze an aggregate from the computer's files. The student is able to
determine the effect energy has on the formation of the aggregate by
experimenting with the program's options. For example, the force lines
option draws force lines on the screen and allows the student to launch
a single particle which will join the aggregate. The path the particle
takes can be compared to the predicted lines of the force.
Chill Out!
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