How to use it?
Just play around with it for awhile, then read the "tips" section, below.
What does it do?
The program plots several (rather complicated) functions that exhibit
chaos constrained to certain symmetries. It computes a new value
and then feeds the answer back into the input for another go
‘round. The values are complex numbers. Since complex
numbers are pairs of values that can be plotted on a plane, the effect
is that of a point jumping around the plane with each
iteration. Think of a pogo stick bouncing around in
the window, landing on the coordinates produced by the equation.
The computer counts how many times it has landed on each spot.
When it's done, the colors are determined by the count—purple
means lots of hits, down thru red, yellow, green, etc., until a blue
background means no hits. For those who might be interested, I've
listed the equations below. Math-phobics should avert their eyes
and skip to the "tips" section below :-)
(m, v, are parameters settable in the dialog box under theoption
menu. M should be an integer (vector), v should be 0 or .5)
True confessions here—I don't understand this hex quilt
mess. Field and Golubitsky (reference 1, below) say X,L,M,N are
vectors (complex numbers), where X is z and the others are
constants. R,F,L are linear transformations in the plane (2x2
matrices, I presume). If you're just dying to know more, dig up a
copy of the book and read the explanation on page 195. The slider
parameters lambda and omega are embedded in L, and beta and gamma are
the (x,y) of the vector a in the formula above.
Tips
There are many interesting
pictures lurking in this parameter space, but they aren't always easy
to find—it takes a bit of patience. Once you find something
interesting, it's best to mark the spot with the "Mark" button, and
then jiggle the sliders to look at the variations on the theme.
If you find yourself fuzzing out, hit the "To Mark" button to go back
to where you were before and start again. If you find the
gradations of the sliders too coarse or too narrow, you can change them
via the "option" menu.
The black spots you see when moving the controls are the first few hops
of the function, which gives a quick preview of the picture to aid in
finding the interesting ones. When the picture frame is outlined
in red in preview mode (while you're moving the sliders), the function
is hopping off to infinity and will not make a picture.
Clicking in the track of the slider will nudge the slider 1 pixel,
clicking and holding will cause sucessive previews until the mouse is
lifted. You can select the controls with the tab key. When
selected, the sliders and combo boxes respond to the arrow keys and the
buttons to the spacebar, enter, or return keys.
When you enlarge the window, the resolution of the image will
deteriorate. This is because the "pogo stick" now is bouncing over a
larger area, and will land on each of the pixels a fewer number of
times. Raising the number of iterations with the "Iter. Detail"
popup will make up for this. Those of you with fast CPUs can
raise the number of iterations of the highest setting through the
"Preferences" dialog. This will make for sharper (but slower)
pictures.
You can set the slider values in a dialog box. This is useful if
you want to make a picture found by someone else (or by yourself some
time ago). I've listed some parameters of interesting pictures
below to help you get started. The quilts have some additional
parameters settable only in the dialog box. It is suggested that "M" be
small integers, and "Shift" be either 0 or 0.5. Other values seem
to work OK though.
You can copy images to the clipboard. The image and the picture
parameters are both copied. Pasting into a paint or draw program
will give you a picture, while pasting into notepad or other text-only
program will give you the parameters of the picture. Copy/Paste
can be used as a primitive bookmark system. If there is a plain
text parameter list in the clipboard, the Paste menu is enabled, and
pasting the values will set the parameters and draw the picture.
You
can copy parameters for images you like, save them to a text file,
then
reproduce the picture by copying the text from the text file and
pasting it into the application. If you hold down the alt key
while copying a quilt image, a single cell will be copied which can be
used for desktop or web page wallpaper.
Parameters to try:
Icon 1
lambda alpha
beta gamma
omega N
-2.7
5.0
1.5
1.0
0.0 6.0
-2.08 1.0
-0.1 0.167
0.0 7.0
1.56 -1.0
0.1 -0.82
0.12 3.0
-1.806 1.806
0.0
1.0
0.0 5.0
1.56 -1.0
0.1 -0.82
0.0 3.0
-1.86
2.0
0.0
1.0
0.1 4.0
-2.34
2.0
0.2
0.1
0.0 5.0
2.6
-2.0
0.0
-0.5
0.0 5.0
-2.5
5.0
-1.9
1.0 0.188
5.0
-2.05 3.000
-16.79
1.0
0.0 9.0
-2.32 2.32
0.0
0.75
0.0 5.0
2.5
-2.5
0.0
0.9
0.0 3.0
1.455 -1.0
0.03 -0.8
0.0 3.0
1.5
-1.0
0.1
-0.8
0.0 2.0
1.5
-1.0
0.1 -0.805
0.0 3.0
-2.523 2.08
0.0
0.0 0.188
7.0
Icon 2
lambda alpha
beta gamma
omega
N P
-2.5
8.0
-0.7
1.0
-0.9
9.0 0.0
-2.38 10.000
-12.3
0.75 0.02
5.0 1.0
1.0
-2.1
0.0
1.0
1.0
3.0 1.0
-2.42 1.0
-0.04 0.14
0.088 6.0
0.0
1.455 -1.0
0.03 -0.8
-0.025 3.0
0.0
1.807 -1.125
-0.227 0.443
0.352 7.0
3.0
1.943 -1.5
0.818 -0.818
-0.352 4.0
3.0
1.5 -1.023
0.227 -0.784
0.057 4.0
2.0
1.5 -1.25
0.239 -0.886
0.068 6.0
5.0
2.011 -1.841
0.568 -0.852
-0.386 9.0
9.0
Square Quilts
lambda alpha
beta gamma
omega shift M
-0.59
0.2
0.1 -0.33
0.0
0.0 2.0
-0.59
0.2
0.1 -0.27
0.0
0.5 0.0
-0.2
-0.1
0.1 -0.25
0.0
0.0 0.0
0.25 -0.3
0.2
0.3
0.0
0.0 1.0
-0.28 0.25
0.05 -0.24
0.0
0.0 -1.0
-0.12 -0.36
0.18 -0.14
0.0
0.5 1.0
0.1
0.2
0.1
0.39
0.0
0.0 -1.0
-0.589 0.2
0.04 -0.2
0.0
0.5 0.0
-0.28 0.08
0.45 -0.05
0.0
0.5 0.0
-0.59
0.2
0.2
0.3
0.0
0.0 2.0
-0.28 0.25
0.05 -0.24
0.0
0.5 -1.0
-0.11 -0.26
0.19 -0.059
0.07
0.5 2.0
0.102 -0.273
0.034 -0.102
0.341 0.0
0.0
0.102 -0.273
0.034 -0.102
0.341 0.0
0.0
0.136 -0.273
0.034 -0.136
-0.239 0.0
0.0
Hex Quilts
lambda alpha
beta gamma
omega M
0.1 -0.076
-0.06
0.1
0.0 0.0
0.2
0.04
0.1
0.1
0.0 1.0
-0.105 -0.15
0.06 -0.03
0.0 0.0
0.02 -0.1
0.14 0.052
0.0 0.0
0.02 -0.1
0.14 0.052
0.04 0.0
0.1 -0.076
-0.06
0.1 0.101
0.0
0.17 -0.034
0.0
0.1 0.101
0.0
0.102 -0.136
0.091 0.0
0.136 0.0
0.0 -0.68
0.148 0.68
0.0 0.0
The dreaded License Agreement!!
The program is freeware; copy it all you want and send it to anybody
you think might like it. Please include this document along with
it.
For more information
Two books that I'm aware of discuss the math behind the pictures and show some pictures. Check out:
Symmetry in Chaos, by M.
Field, & M. Golubitsky, Oxford 1992 (many beautiful high res
pictures—this program is basically a port of the QBasic program
in the back of the book. Search Amazon, Addall.com)
Fearful Symmetry: Is God a Geometer?, by I. Stewart, & M. Golubitsky, Oxford (UK) Blackwell (USA) 1992 (Amazon, Addall.com)
The authors have a website also.
Warning!
I've wasted may hours drawing these pictures and you are at risk for the same affliction—they can be very addictive!
Have Fun!
Jim Burgess
