|
Fractals
& Strange Attractors - Borland C++ Builder 6 Applications
|
|
|
Langton's Ant - Version
3
Dave
Niebuhr's implementation of Langton's Ant...
|
|
Mandelbrot Set:
From Wikipedia,
the free encyclopedia: Work on the Mandelbrot set "coincided with
a huge increase in interest in complex dynamics, and the study of the
Mandelbrot set has been a centerpiece of this field ever since. It would
be futile to attempt to make a list of all the mathematicians who have
contributed to our understanding of this set since then..."
|
 |
New Mandelbrot Explorer
enhanced from Jeff Suer's
This
application, contributed by Jeff Suer, allows the user to explore the
Mandelbrot set by zooming in from a magnification of 1 to a magnification
of 230 or over one billion power (1,073,741,824). That is
20 times the magnification of an electron microscope.
Zoom-and-Center, either
in and out is accomplished by clicking the left or right mouse button
respectively. Magnification and doublings (powers of 2) are shown in
Edit boxes with each click. The user may choose from 5 different coloring
schemes including a "contour" map based on coloring odds and
evens either black or white.
|
|
|
Jeff Suer's Mandelbrot
Explorer
This
application, contributed by Jeff Suer, allows the user to explore the
Mandelbrot set by zooming in from a magnification of 1 to a magnification
of 230 or over one billion power (1,073,741,824). That is
20 times the magnification of an electron microscope.
|
|
The Lorenz Attractor:
From Wikipedia,
the free encyclopedia: The Lorenz attractor is a chaotic map, noted for
its butterfly shape. The map shows how the state of a dynamical system
(the three variables of a three-dimensional system) evolves over time
in a complex, non-repeating pattern, often described as beautiful. The
attractor itself, and the equations from which it is derived, were introduced
by Edward Lorenz in 1963, who derived it from the simplified equations
of convection rolls arising in the equations of the atmosphere...
Also
see Andrew Ho's pages on chaos
and the Lorenz
attractor...
|
Lorenz Attractor -
3 views
This application
shows all three 3-dimensional planes and colors the points from black,
through the cool and then warm colors, to white as the algorithm generates
them from the first to the last iteration. The added dimension of color
shows additional chaotic behavior that is obscured in a single color
plot.
|
|
|
Lorenz Attractor -
1 view
This application
shows only the YZ plane and colors the points from black, through the
cool and then warm colors, to white as the algorithm generates them
from the first to the last iteration. The added dimension of color shows
additional chaotic behavior that is obscured in a single color plot.
|
|
Half Way There
The simulations below are all variations
and enhancements on a simple theme. They represent an apocryphal story
which goes something like this:
Once upon a time there was an indecisive
nomadic family. They were always looking for a better life and set out
to make their fortune in surrounding cities. The user can left-click on
the screen to set the location of each of up to scores of cities, but
try beginning with just three. The user can then right-click on the screen
to set the location of that family's first camp. The family then follows
a compulsive and seemingly random behavior: It picks one city at random
and travels towards it. When it is half-way there, it tires and so stops
and camps for the night. In the morning the family awakes, and being indecisive,
selects another city as its goal. It travels half-way there and camps.
Compulsively, it keeps selecting a city at random, travelling half-way
there, and camping for the night.
The question is: "What sort of pattern
is generated by the campsites it has left?" Can you figure this out in
your head? Can you figure this out on paper? I have never met anyone who
could accurately predict what will happen. One's usual intuition goes
like this, "It's a random process, so the pattern should be random,
right?" But usually no one will say that because they suspect the
algorithm will produce counterintuitive results.
Try the white version below to understand
how the story fits the simulation.
Then try the enhanced versions to set your own city and camp locations
and explore the other variations...
|
|
|
Half Way There - Demonstration Version
- 2007
A somewhat bare-bones version used to demonstrate
the problem without necessarily revealing its solution.
|
Half Way There - April 2008
"C"
Some enhancements we may make on the second day
of class.
Full-Size Screen
Shots (the pattern clarifies after 1000 hits): cold-to-hot
/ hot-to-cold / faux-shadows
|
|
|
Half Way There - April 2008 "B"
As far as we got on the first day of class. We'll
be enhancing this...
|
|
|
Half Way There - January 2008
Getting a bit fancy... Introducing Buttons, TrackBars,
Edit, Panels, RadioGroups and PaintBox. Sleep has been changed to Maximum
Iterations. There is a choice of displaying the iterations or not and
a choice of tweak or not. The array may be rendered in three ways. The
image to the left shows both the normal (+) version and the tweaked (-)
version. The tweaked (-) version is run ten times (10x) more than the
normal version to provide comparable shading.
|
|
|
Half Way - Start - What we did in
class April 5, 2007
Not too fancy, but we added color visualization to
the rendering. For this image, a dot was placed at each vertex of a triangle
and one in the center. Later, three more dots were added to the center
of each of the three large inverted black triangles.
 
 
|
|
|
Half Way There - Sonified
A modification in which the RENDER function automatically
scales the color ramp to accommodate the full range of the data. The display
is also sonified. By right clicking on any row of the image, that row
will be scanned and the data values automatically scaled to accommodate
the full range of the MIDI keyboard. The speed of the scan, and thus the
speed of the sonification, are user adjustable. The cursor position and
the maximum value of the array (used for automatic scaling) are also shown.
|
|
|
Regular Polygons (Bent)
Here we bent the rules, following a suggestion in
class to change the "-"s in the algorithm to "+"s.
Additionally, we included the option of an extra point at the center of
the polygon, suggested by a "bug" in the "user defined
polygons" applications below. This now produces quite a variety of
fractal images.
|
|
|
A Simple Tale: "Half Way There."
A quick simulation you can write at a single sitting.
An indecisive family sets out every morning for a city which they believe
will make them rich. When they are half-way-there they camp for the night.
The next morning they change their minds about which city is the best
to travel to. They travel half-way there an camp. Next morning... The
user left-clicks the mouse to set the locations of the cities and then
right-clicks to set the location of the first camp. Pressing "RUN"
takes the family through 100,000 days and nights...
|
|
|
User Defined Polygons
A further modification allowing the user to define
the shape of the polygon.
|
|
|
Jeff's Polygon
A modification of the application above by Jeff Shih
who extended the process to polygons of many sides, by Dave Niebuhr who
helped debug it, and by Nick Gessler who added the coloring algorithms.
A slider bar allows you to enhance the detail of the algorithm.
|
|
|
Triangle - New
A demonstration of emergence from simple rules to complex
global behavior which produces Sierpinski's gasket. A simple rule repeated
thousands of times produces unexpected results. Modified from the application
above.
|
|
|
Triangle
A demonstration of emergence from simple rules to complex
global behavior which produces Sierpinski's gasket. A simple rule repeated
thousands of times produces unexpected results.
|
|
|
Open Graphic Language (OGL) - Globe
In this application, from
the C++ BUILDER 6 IDE, the objects rotatate automatically until any key
is pressed, whereupon the application will close.
Executable
|
|
|
Open Graphic Language (OGL) - Ring
In this application, from
the C++ BUILDER 6 IDE, the objects may be rotated by the arrow keys. The
red, gold and blue .bmp files are used to wrap the objects. Save them
in the same folder with the executable before running the application.
See what you can figure out about
the code. Try changing it! Replace the .bmp files with your favorite photos.
Here are a couple of different .bmp files: grid.bmp
rummy.bmp
|
