Symicon Help
 Symmetry Help

Symmetry generates graphic images based on mathematical formulae.  There are four main types of images which can be generated:  Symmetric Icon, Square Quilt, Hexagonal Quilt and Fractal.   These images are generated by using  mathematical procedures to calculate a new point to visit, given the currently occupied point;  the points are then coloured according to how often they have been landed on.  By changing the parameters for the generating functions an infinite variety of forms can be produced (in theory!).

The mathematical theory and lots of beautiful examples can be found in

 Michael Field & Martin Golubitsky, SYMMETRY IN CHAOS, OUP 1992.

The generating procedure of the Symmetry program is based on the qbasic programs described in the book.

Symmetry application developed by Peter Lavers using BORLAND C++ 4.5.    COPYRIGHT

SYMMETRY is NO-NAG SHAREWARE

The Basic Idea

File Menu

Start

 Parameters

Display

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NO-NAG SHAREWARE

Please support the development of this program.

If you find SYMMETRY fun or useful,  send a development contribution
of AUS$5 or more to:

P.Lavers
 24 Grainger Circuit
 Melba
 ACT 2615
Australia.

Include your email address.  If I get enough support to cover the upgrade costs
I will port to Win95, and continue developing the program.  If not I will heed my
wife's exhortation: "Stop wasting time making pictures!" ("And get a job!")


COPYRIGHT

SYMMETRY program is SHAREWARE copyright © 1996 Peter Lavers.
Some components copyright © Borland International.

Non-commercial copying and distribution is welcomed.
Email: plavers@canberra.teknet.net.au

NO-NAG SHAREWARE:
If you find SYMMETRY fun or useful, please send a development contribution
of AUS$5 or more to
P.Lavers, 24 Grainger Circuit, Melba, ACT 2615, Australia.
Include your email address.  If I get enough support to cover the upgrade costs
I will port to Win95, and continue developing the program.  If not I will heed my
wife's exhortation: "Stop wasting time making pictures!" ("And get a job!")

If you use image(s) generated by this program in an application
or any form of document, for other than your personal use (including web pages),
 please include an acknowledgment that the image(s) were generated
using the program SYMMETRY developed by Peter Lavers.


The Basic Idea

The basic idea of SYMMETRY is that you do not do very much.

1.	Choose the type of icon (default is fractal):  Parameters|Set Icon

2.	Set the degree of symmetry and other parameter values (or stick with default values): Parameters|Set Parameters

3.	Have a look at the basic shape and check it is OK:  Start|Look

4.	Then generate the icon: Start|Go.
Switch off generation of the image to speed things up: Start|No Image.

5.	After a while have a look at the icon:  Display|Redisplay.
This stops the icon generation, so repeat step 4 to continue.

6.	Now really wait for a while and do step 5.
The fine structure of some of the figures requires tens of millions of iterations, which may take some hours (486dx66-time).

Variations on the theme

Copy the displayed image to a *.DIB file when it looks cool:  File|Save Image.

Try different colours (Display|Base colour and Display|Max colour).

Try the loopy colour option (Display|Loopy clr).  (Takes effect after you rechoose base or max colour.)

Change the size of the icon by selecting Display|Scale.  This may also help if you get the Exploding! error message and most of the shape does seem to be there.


File Menu

New  	Closes down the iterating loop (if running), and the generation of statistical data for colouration of image (if in train), saves the data and then resets so a new symmetric icon (and its data file) can be generated.  This takes time!
Note: The currently displayed image is not also cleared.  It will be when you select Background Blast

Open	Opens a data file for a symmetric icon (*.SMY) and resets so is ready to continue generating that icon.  The image is not automatically displayed - run Start|Go and then Display|Redisplay to see it.

Save Image	Saves the currently displayed image as a .DIB file which can be opened by a graphics program such as PAINTBRUSH.

Save Tile  	Quilt patterns only.  Saves a tile of the image - useful for generating HTML backgrounds, or wallpaper.

Exit	 Tidies up, saves data and shuts down the program.


Start

Initial Position	Sets the starting position for iteration (or the current position for iteration to be continued from).

Look  	Does one iteration pass (10000 iterations) and displays the image.

Go/Stop	Starts/stops the iteration process.  After each 5 million iterations the data is saved to file.  So the kerchunks can give you an idea of speed.  The data is also saved when you stop.

No Image/Show Image	 Toggles generation of image or not.  Ongoing image generation slows up generation of the icon, so once you have an idea of the pattern select no image.

The cursor HOURGLASS flits every 10000 steps, so this gives an idea of speed.


Parameters

Set Icon Type 	Sets which generating function to use, which determines the basic type of pattern.

Set Parameters 	Sets the parameters which the generating function uses, and so determines the particular shape of the pattern.

These menu items can only be selected in FileNew mode.


Fractal Parameters

 denotes rotational and reflection symmetry.

 denotes rotational but not reflection symmetry.

Sample parameter values:

Degree of  symmetry	a11	a12	a21	a22	b1	b2
55	-0.4	0.75	0.2	-0.3	0	0.4
5	0.5	0	0	0.5	0.5	0
3	-0.1	0.35	0.2	0.5	0.5	0.4
5	0.4	-0.1	-0.31	0.45	0.01	0
See the book Symmetry in Chaos for more!

Contraction:

For the generated points to remain bounded in the case of a fractal the parameters must satisfy:

Put 	a1 = a11 * a11 +  a21* a21

a2 = a21 * a21  +  a22 * a22

Need	 a1 <= 1 	and	 a2 <= 1

and        a1 + a2   <=    1 + (a11 * a22    -    a12  *  a21) ^ 2

Otherwise the generating function is not a contraction.

If you get the `not a contraction warning adjust the parameter values to satisfy the above constraints.


Hex Quilt Parameters

The final parameter m should take integer values.

Sample parameter values:

lambda	alpha	beta	gamma	omega	m
0.2	0.04	0.1	0.1	0.0	1
-0.105	-0.15	0.06	-0.03	0.0	0
0.1	-0.076	-0.06	0.1	0.101	0
See the book Symmetry in Chaos for more!


Square quilt Parameters

The final parameter m should take integer values.

Period is the number of repetitions in the pattern.

Sample parameter values:

lambda	alpha	beta	gamma	omega	shift	m
-0.59	0.2	0.1	-0.27	0.0	zero	2
0.1	0.2	0.1	0.39	0.0	zero	-1
-0.11	-0.26	0.19	-0.059	0.07	half	2
See the book Symmetry in Chaos for more!


Icon Parameters

Sample parameter values:

Degree of symmetry	lambda	alpha	beta	gamma	omega
7	-2.08	1.0	-0.1	0.167	0.0	D
2	-2.18	10.0	-12.0	1.0	0.0	Z
16	2.39	-2.5	-0.1	0.9	-0.15	Z
See the book Symmetry in Chaos for more!


Display

Scale	Sets the size of the generated image (inversely: e.g. a value of 2 halves the size).  Only available for icon or fractal type of figure.

Background Blast	Sets the Background back to black.  (Select Redisplay to refresh the image. ) This is not done automatically because I found some of the juxtaposed images looked great, after generating some different icons (try a quilt and then a fractal).

Base Colour	Sets the colour assigned to least hit points and refreshes image.

Max Colour	Sets the colour assigned to most hit points and refreshes image.

Straight Clr/Loopy Clr	Toggles method for assigning colours.  Takes effect when base or max colour is reselected.  Loopy includes some chance elements.

Redisplay	Recalculates frequency data and refreshes the displayed image.  This command is only available when iteration is going.  Also it stops iteration (so the image is not dirtied).
This requires processing the data to obtain cumulative frequencies, and thence cutoff frequencies for colouring points.
BIG MEG WORK, be patient.


Online Maths Tutoring
Net Maths Tutoring

If you are seeking help with your learning of mathematics, consider a flexible and convenient Net tutoring service.  High-school to uni level.

Personal tutoring also available in the Belconnen region of the ACT (Australia).

Phone 	(06) 2584791

Email: plavers@canberra.teknet.net.au

http://www.teknet.net.au/~plavers/