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Information about “SCI-CALC AND MANDELBROT ZOOMLENS”
SCI-CALC is a scientific calculator with a simple, direct approach.
Various standard formulas are shown on the screen relating x and y.
Enter the numbers you want and the program calculates the results.
MANDELBROT ZOOM LENS lets you create, view, and magnify a Mandelbrot
fractal and demonstrate the unique nature of these complex mathematical
patterns which are found in nature. As a fractal image is magnified,
the smaller details will still look like the same fractal. The more a
fractal is magnified, the more hours it takes to fill in all the details
on the screen.
FILE1326.TXT
Disk No: 1326
Program Title: Mandelbrot Zoom Lens and SCI-CALC
PC-SIG version: 1
MANDELBROT ZOOM LENS lets you create, view and magnify a Mandelbrot
fractal and demonstrate the unique nature of these complex mathematical
patterns which are found in nature. As a fractal image is magnified,
the smaller details will still look like the same fractal. The more a
fractal is magnified, the more hours it takes to fill in all the details
on the screen.
SCI-CALC is a scientific calculator with a simple, direct approach.
Various standard formulas are shown on the screen relating x and y. You
simply enter the numbers you want and the program then calculates the
results.
Usage: Graphics/Entertainment, Calculator.
Special Requirements: CGA graphics for Mandelbrot Zoom Lens.
How to Start: Type GO (press enter).
Suggested Registration: $20.00 for Mandelbrot Zoom Lens; $4.00 for
Sci_Calc.
File Descriptions:
MANZOOM COM Main program.
MANZOOM PIC Picture file.
MANZOOM DOC Documentation.
SC BAT Batch file to start program.
SCI_CALC COM Main Program.
SCI_CALC DOC Manual.
SCI_CALC FMT Data file.
PC-SIG
1030D E Duane Avenue
Sunnyvale Ca. 94086
(408) 730-9291
(c) Copyright 1989 PC-SIG, Inc.
GO.TXT
╔═════════════════════════════════════════════════════════════════════════╗
║ <<<< Disk No 1326 Mandelbrot Zoom Lens, SCI-CALC >>>>> ║
╠═════════════════════════════════════════════════════════════════════════╣
║ To print the documentation, type: ║
║ ║
║ MANUAL (press enter) ║
║ ║
║ To start the Mandelbrot Zoom Lens, type: ║
║ ║
║ MANZOOM (press enter) ║
║ ║
║ To start the SCI-CALC program, type: ║
║ ║
║ SC (press enter) ║
╚═════════════════════════════════════════════════════════════════════════╝
MANZOOM.DOC
DOS FILE DOCUMENTATION
MANDELBROT ZOOM LENS
MANZOOM.COM Calculates graphics displays of the Mandelbrot fractal.
MANZOOM.DOC Documentation. Help file for MANZOOM.COM.
MANZOOM.PIC Picture data for MANZOOM.COM. Pre-calculated quick displays.
Requires IBM color graphics or compatible. Requires DOS 2.0 or greater.
Requires 64K RAM.
To start, enter MANZOOM in response to the DOS A> prompt.
MANZOOM.PIC must be on the default disk at run time. MANZOOM.DOC should be
on the default disk for the help features to work.
The Mandelbrot fractal is a mathematical object which provides
fascinating computer graphics displays. Use the MANDELBROT ZOOM LENS to
examine the beauty of the fractal. Move the image on the screen with the
cursor keys. Magnify the image for a new display by pressing the PgUp key.
There is an infinity of detail to be discovered in the Mandelbrot fractal.
But the more you magnify, the more hours it takes to fill in all the details
on the screen. At last! A computer solitaire program that gives your PC
something interesting and pretty to do when you aren't using it!
FRACTALS
The border of the Mandelbrot set is a fractal. A fractal
is a jagged edge. A map of a seacoast, for example, is a fractal. You can
magnify a fractal and it still looks like a fractal. A seacoast looks about
the same on a country map as on a city map. Another example is a view of the
horizon of a mountain range.
The Mandelbrot fractal is neat because it is a mathematical object.
I wrote this program to graph the Mandelbrot fractal on your PC screen.
You saw an overview of the Mandelbrot set when this program first
started. If you wish, you can press A right now and see the overview again.
(Remember that F will get you back here.) The Mandelbrot set is the black
area. The fractal is the white border around the black area.
Another neat thing about Mandelbrot displays is the color around
the fractal. The colors are chosen mathematically. When a small section of
the white fractal is magnified, new colors are chosen. So go ahead and
magnify the fractal seacoast of the Mandelbrot ocean. Any magnification will
give a unique display of colors.
GENERAL INSTRUCTIONS
Press the A key to start the Automatic display. Watch the overview
of the Mandelbrot set get magnified three times. Then watch how the
Automatic Display moves around the fractal to give you a view of the whole
thing. I pre-calculated this display to get you started. This Automatic
display is loaded from disk into your computer's memory.
To stop the Automatic Display, press the RETURN key. Then use the
four cursor (arrow) keys to move the fractal around yourself.
Once you have the feel of it, pick a region which strikes your
fancy. Choose a region with some white near the center of the screen. Press
the PgUp key. The screen will go blank. The computer will start calculating
a new display. This is a 2X blow up of the center of the screen which you
chose. Dots of color will appear in a checkerboard pattern as the computer
calculates points (pixels) of the display. In a few minutes, you should have
a rough sketch of what the display will look like. The computer will
continue calculating more pixels for more resolution for a few hours.
Let the computer finish. Or adjust the position by moving the
image with a cursor key. Or press PgUp again. Whatever.
KEYSTROKE COMMANDS
This MANDELBROT ZOOM LENS has 16 commands. Just press one of the
16 command keys to issue a command. You can issue any command at any time by
pressing one of these 16 command keys.
All 16 command keys are listed with brief descriptions on two
menus. The 16 command keys are: Space, Return, 4 cursors, PgUp, A, D, F, G,
I, K, L, M, and Q. Both lower case and shift upper case work. Here are the
detailed descriptions:
Press the SPACE BAR for the main menu. (Actually, pressing any key
which is not a command will give you the main menu.)
Press the RETURN key (also called the ENTER key) to return to the
ZOOM LENS graphics displays. At low magnification, the RETURN key gives a
complete image in about one second, from computer memory. At high
magnifications, the RETURN key clears the screen and starts the computer
calculations which fill the screen with colored dots (pixels). You can tell
when the computer is calculating pixels, because pixel dots appear on the
screen in a checkerboard fashion. If you press any key while calculating a
display, the display is erased in preparation for the next display.
The four cursor (arrow) keys are for you to move the image on the
screen. Pressing one of these four keys causes the image to move in the
direction of the arrow. At low magnification, the moved image appears in
about one second, from computer memory. At high magnifications, a cursor key
clears the screen and starts the new calculations.
The PgUp key gives a 2X blow up. When you press PgUp, the center
of the screen display gets magnified. At low magnification, the blown up
image appears in about one second, from computer memory. At high
magnifications, PgUp clears the screen and starts the new calculations. At
maximum magnification, when the limits of this MANDELBROT ZOOM LENS are
reached, the PgUp key no longer magnifies the screen.
Press A for the Automatic Display. This program always starts with
the Automatic Display. This program can be restarted at any time by pressing
A. The Automatic Display can be stopped by pressing any key.
Press D for DOS file Documentation on the 3 MANDELBROT ZOOM LENS
files.
Press F for a one page introduction to Fractals.
Press G for a one page introduction to the use of this MANDELBROT
ZOOM LENS computer program.
Press I for the Information menu. All 16 keystroke commands are
listed either on the main menu or on the Information menu. The Information
menu lists the 5 commands which read from the documentation file MANZOOM.DOC
and display the contents on your screen.
Press K for Keystroke Command details. That's what you are reading
right now.
Press L for Long Documentation. The Long Documentation tells you a
bit about complex numbers. There are several pages of details on this
MANDELBROT ZOOM LENS. I also have a much larger and elaborate program,
called MANDELBROT MICROSCOPE. The differences between this ZOOM LENS and the
MICROSCOPE are described when you press L.
Press M for a MANDELBROT MICROSCOPE advertisement and order form.
You can press M right now, if you like. Press K to get back here.
Press Q to Quit using the MANDELBROT ZOOM LENS and return to DOS,
or to your operating control system.
LONG DOCUMENTATION
Here are several pages of documentation. Press PgDn to turn the
pages. If you are tired of reading this stuff, press any other key to return
to the ZOOM LENS.
This program which you are running is called MANDELBROT ZOOM LENS.
It provides an introduction to fractals. It provides quick graphic displays
for a sample of the Mandelbrot fractal. I wrote this program with an
emphasis on simple user interface. This is a public domain program. Please
feel free to make copies for your friends and acquaintances. MANDELBROT ZOOM
LENS consists of three DOS files: MANZOOM.COM, MANZOOM.DOC, AND MANZOOM.PIC.
I have another program, called MANDELBROT MICROSCOPE. The MICRO-
SCOPE is intended for more extensive exploration of the Mandelbrot fractal.
The MICROSCOPE works the same as the ZOOM LENS, with command keys and menus.
But there are more menus and a lot more command keys. To be better than the
ZOOM LENS, the MICROSCOPE requires some knowledge of fractals, so plenty of
printed documentation is included. The microscope is too large and
specialized for public domain distribution. Press M for an order form. You
can press M right now if you wish. Press L to get back here to read more
pages of this long documentation.
CALCULATION DETAILS
The Mandelbrot graph-display is calculated one point at a time.
Each point is called a pixel. We use 64,000 pixels on your PC screen. There
are 200 horizontal lines of pixels, 320 pixels in each line.
Each pixel on the screen is treated like a number on a graph sheet.
The PC squares the number, then adds the number to the square, then squares
the sum, then adds the original number again, then squares the sum again,
then adds the original number again... on and on.
It sounds like the answer should get bigger and bigger, right? Not
necessarily! We're talking here about complex numbers. Like a graph, complex
numbers are a pair, x and y. The y number is imaginary. An imaginary number
squared is negative. Here is how you square a complex number:
(x + iy)*(x + iy) = x*x - y*y + 2i*x*y
* means multiply i means imaginary
The "Mandelbrot iteration" is just such a series of successive
multiplication and addition starting with one complex number.
Some of the pixels for the display have x,y pairs which never get
large with the Mandelbrot iteration multiplication and addition. These are
complex numbers which belong to the Mandelbrot set. The computer program
leaves such pixels black.
Some of the pixels have x,y pairs which get large with the
Mandelbrot iteration. The computer chooses colors for such pixels according
to how quickly they get large.
The pixels near the border are the exciting ones. Those pixels
near the edge of the Mandelbrot set get unpredictably larger and smaller with
iteration. More about them later.
The Mandelbrot displays are beautiful and detailed for purely
mathematical reasons, and the math is simply addition and multiplication!
There is a lot more documentation with the MANDELBROT MICROSCOPE.
If you are interested in learning what a complex number is, a tutorial on
imaginary and complex numbers is included. If you already know about complex
numbers, there is interesting information on fractals for you. If you
already know about fractals, you may be interested in the bibliography of
books and articles on fractals which comes with MANDELBROT MICROSCOPE.
COLOR CHOICES
That first magenta circle which you see in the opening of my
Automatic Display contains the entire fractal. Don't go exploring to the
side; there's nothing there but the wild blue yonder. The magenta circle is
the circle of radius 2 centered on the origin of the complex plane. Any point
in the complex plane beyond this radius of 2 quickly gets very large with
Mandelbrot iteration multiplication and addition.
In fact, we happen to know that any complex number which reaches an
iteration magnitude of 2 will very quickly get indefinitely large.
(Magnitude, or radius, is equal to the square root of x squared plus y
squared.) This program counts the number of iterations needed to reach a
magnitude of 2. This program does that for each point, or pixel on your PC
screen. So each pixel ends up with an iteration number.
To get a display, we need to assign colors to the numbers. It's
just like painting by number, but the computer does it. The assignment of
colors is not unique. I chose the colors for MANDELBROT ZOOM LENS. It is an
art to choose colors for Mandelbrot displays. Choose different colors and
you get dramatically different effects. MANDELBROT MICROSCOPE has keystroke
commands which let you pick the color assignments.
THE MANDELBROT FRACTAL IS INFINITE
The infinite action is at the fractal edge. That's the border
between the black Mandelbrot set and the colored region around it.
Mathematicians have proven that as you get closer to the fractal, the
iteration numbers get larger and larger. To get closer, this program does a
magnification. Magnification is done by spreading a smaller range of complex
numbers across the space of your PC screen. Each time the Mandelbrot fractal
is magnified, a new graphic display is produced. That goes on to an infinity
of magnifications.
Of course, this program is not infinite. I programmed the PgUp key
to do 2X magnifications. This ZOOM LENS PgUp key is not programmed to go to
magnifications for which I have not assigned color numbers. The MANDELBROT
MICROSCOPE lets you choose the color numbers. So the MANDELBROT MICROSCOPE
goes on to much greater magnification. You can even manually type in the
screen coordinates and magnification when using the MICROSCOPE.
This ZOOM LENS and the MICROSCOPE use Turbo Pascal 5 byte floating
point numbers. The MICROSCOPE also has an optional mode which uses 16 byte
fixed point numbers for very high magnification.
HOBBY PROGRAMMING
Most of the fun of fractals is in writing your own programs. There
is enough information in this Long Documentation for you to write your own
Mandelbrot Display program. The fastest way to get started is to glance at
the source code of a fractal generating program. I wrote a simple BASIC
program just for this. MANDEL1.BAS is included as a DOS file on the floppy
diskettes which you get with MANDELBROT MICROSCOPE. This BASIC program does
nice displays of the Mandelbrot fractal on IBM PC's. But the purpose of the
program is to provide a starting point for hobby programmers. The program
has only about 50 lines of BASIC statements, but there are over 100 lines of
comments and documentation in case you get stuck.
The source code for MANDELBROT MICROSCOPE is also included as a DOS
file. The source code is Turbo Pascal. There are useful procedures for
display of fractals. I invite you to use these procedures in your own
programs. Press M for ordering details.
The documentation for MANDELBROT MICROSCOPE includes references to
literature on other fractals for computer hobby programming.
SAVING DATA
The MANDELBROT MICROSCOPE saves the iteration numbers which are
calculated. There are 64,000 two byte numbers, so 128,000 bytes are required
just for this "Mandel Map". All told, the MICROSCOPE requires at lease 200K
free RAM, so do not order it if you have less than 256K.
The MANDELBROT MICROSCOPE allows you to save partially calculated
Mandelbrot displays as DOS files. A screen copy, calculation parameters, and
current position is also stored in the same file. Saving and restoring is
done with menu driven command keys, just like the commands in this ZOOM LENS.
Included with MANDELBROT MICROSCOPE are some pretty pre-calculated
Mandelbrot displays. I calculated these as samples. They come as DOS files,
ready to be restored on your computer.
The MANDELBROT MICROSCOPE moves and blows up the Mandel Map when
you press the cursor keys and the PgUp key. So calculations do not need to
start from scratch. Saved DOS file displays provide a jumping off point to
quickly get going in a particular region of the Mandelbrot fractal.
SPEED
Mandelbrot displays take time. Most of the time is spent doing
iterations in the Mandelbrot set, where the magnitude never gets large so a
color is never chosen. It is necessary to set a maximum iteration number,
and quit when the maximum is reached, leaving the corresponding pixel black.
MANDELBROT ZOOM LENS postpones hang ups in black regions by
skipping pixels in a checkerboard fashion. That way, you quickly get a
sketch of the overall display without immediately getting hung up in a black
region. The ZOOM LENS colors every 32nd pixel in every 32nd row, then goes
back and does every 16th pixel and row, then every 8th, then 4th, then every
other one, then does every pixel. Then the ZOOM LENS increases the maximum
iteration number and checks every pixel again to look for more details at the
fractal edge.
MANDELBROT MICROSCOPE saves time with the Mandel Map of iteration
numbers. The same skipping is followed, but previous calculated iteration
numbers are saved, so time is not spent on recalculations. Also, the
MICROSCOPE keeps track of the location of the fractal, so pixels deep in the
black Mandelbrot set are not evaluated at all. The algorithm for keeping
track of the edge is an interesting puzzle. A solution is in my source code.
The best way to speed up the calculations for a fractal display is
to control it yourself. The MANDELBROT MICROSCOPE has an optional manual
mode. You watch the display develop and modify the calculation parameters to
get just enough resolution. It's fun. With keystroke commands, you change
the number of bytes accuracy. You change the maximum number of iterations.
You change the skipping distance between pixels. With practice, you can run
the MICROSCOPE about 100 times faster than this ZOOM LENS. Of course, with
the MICROSCOPE, you will be lured deeper and deeper into the fractal, where
calculations get longer and longer.
On the other hand, long calculations have a mystique of their own.
One motive for deep searches into the Mandelbrot fractal is to give your PC
something to do at night, to keep it from getting bored! I like to let my PC
run all night, then save the result in the morning for further calculations
on another night. Somehow, a graphic display which took my PC 10 nights to
complete seems more beautiful to me than a display which only took an hour or
so.
1/13/88
Pete Gwozdz
SCI_CALC.DOC
JOY Scientific Calculator (c) 1987 by Bill Joyner. All rights reserved.
If you find this calculator useful in some way, please send $4 to Bill
Joyner, 603-C First Street, Oceanside, CA 92054. Although no warranties of
any kind are made for this product, every effort has been made to ensure
the accuracy of the calculations. Feel free to pass the SCI_CALC.ARC file
on to others in any manner whatsoever.
The only thing to be careful of is not using the "Enter" key when enter-
ing values into X and Y. After typing numbers into the X or Y field use
the arrow keys to move to the next field. The "Enter" key is reserved for
1) placing a result into the MEMory field when the cursor is located on a
result, or 2) placing the MEMory field value into the X or Y field when
the cursor is located on the X or Y field. This is why, when you first
start the Scientific Calculator, if you hit "Enter" after typing a value
for X a "0" comes up, since "0" is in the MEMory field upon start-up.
All calculations are made as numbers are typed into the X or Y field. Note
that if a field has a value of 1.7976931349E+308 the field is maxed-out and
should not be considered accurate.
Directory of PC-SIG Library Disk #1326
Volume in drive A has no label
Directory of A:\
FILE1326 TXT 1262 2-22-89 12:15p
GO BAT 38 10-19-87 3:56p
GO TXT 1156 2-22-89 12:16p
MANUAL BAT 147 1-19-89 9:21a
MANZOOM COM 24049 1-13-88 9:05p
MANZOOM DOC 19238 1-13-88 8:59p
MANZOOM PIC 27691 1-13-88 9:08p
SC BAT 31 10-18-87 6:27p
SCI_CALC COM 79583 10-18-87 6:25p
SCI_CALC DOC 1200 10-18-87 6:27p
SCI_CALC FMT 1479 10-18-87 6:22p
11 file(s) 155874 bytes
1024 bytes free