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Fractals
& Strange Attractors - C++ Builder 6 Applications
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The Chaos Game - 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... |
Chaos Game "bent" algorithm ("+" to "-" switch) expanded to 1000 x 1000 pixels run to 4 billion iterations with modifications...
      
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Chaos Game sampling a range of distances expanded to 1000 x 1000 pixels with modifications...
    
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Chaos Game - 25 January 2012
The canonical algorithm has been changed so that the distance travelled can be selected as a multiplier between 0.01 and 2.00. The distance travelled may also be automatically varied between those points. The data generated creates patterns over a wide dynamic range of values, making visualization difficult. Several rendering schemes have been included with mixed results.
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Half Way There - 3D -int world[400][400][400] - January 2010
A 3D anaglyph requiring red/cyan glasses. The world is 400 cubed, a Boolean array, containing 64,000,000 cells and occupying 64,000,000 bytes of memory. In placing points (cities) on the map, use the TrackBar to give the location depth. A depth of 0 pops out at you; a depth of 400 is behind the plane of the display. The configuration of points (cities) can be written to and read from a text (.txt) file with the Open and Save buttons. When selecting the tweaked version of the code, please make sure to keep the distribution of cities relatively small. Otherwise the points may grow out-of-bounds and crash the application. At the end of each Run, the total number of cells occupied in the world is given, as well as the percentage. The Red/Cyan color separation can also be adjusted to control the perception of depth.
To create the 3D anaglyph, the three-dimensional world is read and the points (cities) projected onto a two-dimensional plane. The X and Y positions in the 3D world are mapped to their same positions on the 2D plane and colored RED. The X position in the 3D world is then shifted horizontally in proportion to its depth (the Z dimension) and then mapped, along with its unaltered Y position, on the 2D plane and colored CYAN. If the cell in the 2D plane was already RED, the position is then colored BLACK.
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Half Way There - 3D - int world[250][250][250] - January 2010
A 3D anaglyph requiring red/cyan glasses. In placing cities on the map, use the TrackBar to give the point depth. A depth of 0 pops out at you; a depth of 200 is in the plane of the display. When selecting the tweaked version, please make sure to keep the distribution of cities relatively small. Otherwise the points may grow out-of-bounds and crash the application. The world is 250 cubed, an Integer array, containing 15,625,000 cells. Interestingly, the RGB Color Cube is 256 cubed, containing 16,777,216 colors.
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Half Way There - Demonstration Version
- 2007
A somewhat bare-bones version used to demonstrate
the problem without necessarily revealing its solution.
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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
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Half Way There - April 2008 "B"
As far as we got on the first day of class. We'll
be enhancing this...
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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.
Playing with this, I did enter a configuration of dots which produced a fractal spiral pattern. However, I do not remember what that initial pattern was. Please let me know if you find it. |
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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.
 
 
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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.
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Intricate patterns in the cloud of points are more clearly resolved by increasing the number of iterations. Click on any image for the full scale screen capture.
In these images, the gray scale is applied only from the minimum number of hits per pixel to the maximum number of hits per pixel.
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Halfway There (Bent January 2009)
The original version ran only 1 million iterations. This version runs 10 million iterations by default although you can choose to run as many as 100 million or as few as the the original 1 million. Increasing the number of iterations makes the cloud of points develops the intricate patterns more clearly. Clicking on any image will call up a full scale image. In the image to the left the grayscale is applied four times, once to each 25% of the entire range from the minimum number of hits per pixel to the maximum number of hits per pixel.
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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.
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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...
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User Defined Polygons
A further modification allowing the user to define
the shape of the polygon.
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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.
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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.
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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.
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Langton's Ant:
Dave
Niebuhr's implementation of Langton's Ant... |
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Langton's Ant - Version
3
Dave
Niebuhr's implementation of Langton's Ant...
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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..." |
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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.
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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.
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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... |
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Lorenz Attractor -
3 views - Color and 3D Anaglyph
This application
shows all three 3-dimensional plane projections. The user may choose to color the points progressively from black,
through the cool and then warm colors, to white as the function is iterated, or choose to create a 3D anaglyph which requires red/cyan glasses to see. The red color is plotted as previously, but the cyan color is displaced to the right in proportion to the projection into the screen. The code was further modified to display white at any pixel where red was overlain by cyan or cyan was overlain by red. The image above is a composite of both visualization choices.
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Lorenz Attractor -
1 view - Color
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.
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Lorenz Attractor -
1 view - Color and 3D Anaglyph - Large
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 user may choose a color visualization to show
additional chaotic behavior that is obscured in a single color plot, or may choose a 3D anaglyph which requires red/cyan glasses to see. The red color is plotted as previously, but the cyan color is displaced to the right in proportion to the projection into the screen. The code was further modified to display white at any pixel where red was overlain by cyan or cyan was overlain by red. The image above is a composite of both visualization choices..
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OpenGL
Open Graphics Language |
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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 |
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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 |
