Quaternion Julia setsBA Quaternion Julia set is an extension to the normal type of JuliaAset which requires four variables to be set to describe the imageCinstead of two. With X and Y set to the location of the point beingBplotted the formula below is iterated until the sum of x,y,
and
becomes greater than 4.
x = x
y = 2xy + B
= 2x
= 2x
+ DCConstants A to D can be set in Alter Variables in the Fractal menu.?The demonstration image and the formula come from an Article by?Dr Ian Entwistle published in Fractal Report. See help for more&details and example vaules for A to D.U
Quaternion Julia setsfffffff
fffffff
? A , CReal
B , CImag - 1
C , CImag - 2
D , CImag - 3
Quaternion Julia setsV
Iterated Function System -
Iterated Function System>The IFS fractal is based on a series of transformations taking the form:
NewX = A*OldX + B*OldY + E
NewY = C*OldX + D*OldY + F@Each transformation has a probability which is used to determine@which one is applied next (e.g. 0
5 causes the transformation to@be used half the time). See the help on the IFS fractal for fullAdetails of the effects that can be created and what the different
numbers mean.U
Christmas treeU
The file
, which contains details of0the IFS fractals, cannot be found in the current"directory. What do you want to do?
IFS fractals.U
Edit/choose @I@F@S pattern
Iterated function system (IFS)
@5Integer arithmetic will not be available because some6of the numbers involved in the current fractal are too3large. Floating point arithmetic has been selected.U
Select option
New fractal
Rename fractal
Delete fractal Exit menu
?Select an option or press ESCAPE to return to selection list...
You cannot delete the last
fractal!(This option creates a new IFS fractal...
Please select
Copy current fractal
Create blank fractal
Do not create fractal<Do you want the new fractal to be a copy of the current one?
3You may have a maximum of 150 IFS fractals at once.6Since you currently have 150 it is not possible to add/any more without creating a new file. See help.U
Select FractalMEnter name to choose or use cursor keys to select. Press ~\~ or ~/~ for menu.
<
Enter new transformation line#
5Use the following keys: Return - Edit current value1 Escape - Finish editing6 D - Delete current line4 I - Insert blank line9 R - Replicate current lineU
Edit IFS dataBUse ~cursor keys~ to select box and then ~Return~ to edit value...: n A B C D E F PU
Enter new transformation lineU
Enter new transformation lineU
)You must have at least one transformation&available to be able to draw an image.
Delete line?
No=Do you really want to delete the current line of information?U
Memory full: you may only have up to 150 lines in each fractal.
v RSP
?&The sum of the probabilities in not 1.
What do you want to do?
Please select:
Scale probabilities to 1
Continue editing data;The sum of the probabilities must be 1 for all IFS fractals
v RSP
v RSP
.Some lines have a probability of 0, this means*they will never be called when drawing the fractal. What do you want to do?
Please select:
Remove these lines
Keep these lines
Continue editing dataJLines with probability=0 will never be executed, do you want to keep them?U
BUse ~cursor keys~ to select box and then ~Return~ to edit value...U
3Use ~cursor keys~ then select ~DONE~ or ~FORGET~...
Alter IFS starting location
'Please alter the options as required...
Default min. X value
Default max. X value
Default min. Y value
Default max. Y value
Done
Forget
7The initial location for the current fractal will cause9points to be plotted outside the screen area. Do you want.CAL to automatically set the initial location?
Please select:
Set location automatically
Enter new location
Do not change locationFPlease select what you would like CAL to do to the initial location...
v RSP
v RSP
v RSP
v RSP
Graphically
As numbers Last menu
GDo you want to type in numbers or draw the transformations graphically?U
#Plot colour from transformation no.
Increment colour when plotting
IFS menu
Edit @I@F@S parameters
Set initial location
Default pos. is current pos.
Choose from list
Open another @I@F@S datafile Last menu
=Please select which IFS-related option you wish to perform...U
>The IFS data used to draw this fractal is not stored in memory?and there are already 150 fractals in memory. To be able to useAthis fractal you must delete an existing IFS fractal from memory.
The IFS formula
has been created1since this was used to produce the image which is
being loaded.U
Function used:U
!Not enough memory for IFS fractal
data tables. These will not be
available.U
CAL.IFC
Iterated function system (IFS)U
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v RSP
v RSP
v RSP
v RSP
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Forget changes?
6Do you want to forget the changes made to the fractal?U
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Symmetrical Attractors7These fractals are drawn by using the variables X and Y2as co-ordinates for a point on the screen and then9repeatedly applying a series of transformations to derive7a new set of co-ordinates which are themselves plotted.
3A wide range of patterns can be drawn because it is8possible to alter many aspects of the formulae used. For6examples of patterns that may be drawn, and details of:the parameters that can be altered, select Alter Variables9from the Fractal menu and press F1 for help. This fractal:is based on an article in Fractal Report by Uwe Quasthoff.!See help for further information.U
Symmetrical attractors
,Press F1 for more details about the error...@The parameters that you have given in the Alter Variables optionAhave resulted in the calculation producing an arithmetic overflow8error. See help for examples of suitable numbers to use.U
Symmetrical attractorsU
, 9 F
@Please note that good images are only obtained in fast mode when=the number of iterations is set to around 150 or more. Either:increase the number of iterations or select accurate mode.U
Lyapunov spaceAThe Lyapunov images are drawn by going through each point in turn<and, starting with a seed value for X, applying the formula:
NewX = R*X*(1-X)@The value that R takes alternates between two possible values, A?and B, which correspond to the position of the point across andAdown the screen respectively. The Lyapunov exponent is calculated;from the values of X which are produced and this is used to?form the colour of the point. Points producing stable values of:X are coloured dark and those producing chaos are coloured>brightly if you are using the Lyapunov colour palette. You can8change the way the A and B values alternate by selecting'Lyapunov Details from the Fractal menu.U
Lyapunov space
No. initial iterations
Lyapunov detailsU
Set Lyapunov Details
;Please enter sequence required (press F1 for more details):
!The sequence must contain only As
and Bs, and must be at least 1
character long.U
Set lyapunov details
Set @A-@B pattern
Use accurate mode Last menu
ESee help for details of options. Note that accurate mode can be slow.
Use fast modeMSee help for full details. Note the fast mode may slightly alter some images.U
Mode:+Fast - see Lyapunov Details in Fractal menu+Accurate - often much slower than fast modeU
Lyapunov spaceU
VVVVVVVV
fSfSfSfS
VVVVVVVV
fSfSfSfS
Ushiki's Phoenix>This fractal is drawn using an algorithm which is very similar=to that for the Mandelbrot set except that the product of the?imaginary part of C and the last value of Z is added to the new4value of Z. As in the Mandelbrot set, Z starts as 0.
< NewZReal = ZReal*ZReal-ZImag*ZImag + CReal + CImag*OldZReal* NewZImag = ZReal*ZImag*2 + CImag*OldZImag=The colour of the point is the number of times that the above@operation can be performed with |Z| remaining below 4. Note that?the computer needs to hold details about three values of Z: the@previous value, the current value and the new value. Based on an4article by Joyce Haslam published in Fractal Report.U
Choose @Julia @Set
Return to @PhoenixU
Ushiki's Phoenix
Choose @Julia @Setfffffff
Return to @Phoenix
Julia set real position
Julia set imaginary position
Julia set image selected
based on point at centre
of initial image.
Return to @PhoenixU
Main Ushiki's Phoenix option
selected: choose Draw to
calculate image on screen.fffffff
Choose @Julia @SetU
IUshiki's Phoenix option selected (use Fractal option to choose Julia set)CJulia set option selected (use Fractal option to choose main image)U
Ushiki's Phoenix
Sierpinski Gasket@The Sierpinski Gasket is formed from a large triangle with three=identical smaller triangles superimposed on it. Each of theseBcontains a further three triangles which in turn holds three more,?and so on ad infinitum. The pattern is created by iterating the?formula below, named after its discoverer, W. Sierpinski. It is.sometimes also called the Sierpinski triangle.
NewX = 2*X
NewY = 2*YBIf either NewX or NewY is >1 then 1 is subtracted from it. If bothCare greater than 1 then the subtraction only occurs for NewY. ThereAis an IFS version of this fractal available - select the Iterated:Function System fractal and then choose Sierpinski Gasket.U
Sierpinski gasket
fffffff
Sierpinski gasketU
$Solving Z
-1=0 using Newton's methodCThis fractal is produced by iteratively solving the above equation.BThe colours displayed on the screen can either be derived from the@number of iterations required to locate a solution, the solutionAfound, or a combination of both. n should be greater than 2, with?larger numbers offering more solutions, but requiring a greater@calculation time. CAL solves the equation using Newton's method,.whereby the following calculation is iterated:
(n-1) * Z
NewZ :=
n * Z^(n-1)AThe arithmetic is performed using complex numbers and the initial?value of Z is the co-ordinates of the point on the screen being"tested. See help for more details.U
Please note:'When the colour plotted depends on both+the solution found and iterations required,&it is best to use a specially designed,colour palette. Do you want CAL to construct
an example?
Example palette?
Construct example
Do not change palette
EDo you want CAL to construct a new palette for use with this fractal?U
Newton colour method
Iterations to find solution
No. of solution found
Do not change
.How should pixels in this fractal be coloured?U
Newton's method (solving Z
-1=0)
Power to use (2 upwards)
Max. capture distance Min. distance between attractors
Colouring method
Newton's method (solving Z
-1=0)U
v RSP
The Mandelbrot Set & Julia Sets@The Mandelbrot Set is drawn by taking a point, Z, and repeatedly@squaring it and adding another value, C, to give a new value for<Z. This process is repeated over and over again until either
|Z|>2 or it has been repeated
times. You can change@this maximum number of times by using the General Configurations
option in the options menu.
@Each point in the Mandelbrot set has an associated Julia set and6this can be viewed by choosing `Choose Julia Set' from:the fractal menu. For more details, choose this option and
press F1 for help.U
Choose @Julia @Set
Return to @M-@SetU
Mandelbrot/Julia Set
Choose @Julia @Set
Return to @M-@Set
Julia set real position
Julia set imaginary position
Julia set image selected
based on point at centre
of initial image.
Return to @M-@SetU
Mandelbrot set selected,
choose Draw to calculate
image on screen.
Choose @Julia @SetU
GMandelbrot set option selected (use Fractal option to choose Julia set)GJulia set option selected (use Fractal option to choose Mandelbrot set)U
Mandelbrot/Julia Set
F+F+F+F
FFF+F+F+F+FFU
The file
, which contains details of5the L-System fractals, cannot be found in the current"directory. What do you want to do?
L-System fractals.U
)Unable to calculate pattern. Try reducing&the order selected - currently set at
L-System aborted. Select
Draw again if you do want
to produce the image.U
.Invalid transformation rule. Use Edit L-System1Transformations in the L-System Menu to alter it..There must be at least one transformation rule/specified. Use Edit L-System Transformations in
the L-System Menu to enter one.1The number of angles steps per revolution must be4greater than zero. Use Edit L-System Transformations)in the L-System Menu to change the angle.U
Scaling4Please wait, scaling image to current screen size...
@3333333
?3333333
v RSP
g<-u+
v RSP
Evaluating#Please wait, evaluating L-System...U
y<-u+
F
Angle:
A Amount turned through for every + or - is 360
/angle specified
Axiom:& Axiom is initial set of instructions
Rule
becomes@ Each rule is applied every iteration and any occurances of theF specified letter are replaced with the corresponding set of letters.C Move off bottom of list to add new transformation line, or deleteE contents of existing line to remove it. Press Return when finished.U
Edit L-System - 8Use ~cursor keys~ to select line and edit translation...U
Forget changes?
>Do you want to forget the changes made to the transformations?U
Select option
New fractal
Rename fractal
Delete fractal Exit menu
?Select an option or press ESCAPE to return to selection list...
You cannot delete the last
fractal!-This option creates a new L-System fractal...
Please select
Copy current fractal
Create blank fractal
Do not create fractal<Do you want the new fractal to be a copy of the current one?
8You may have a maximum of 150 L-System fractals at once.6Since you currently have 150 it is not possible to add/any more without creating a new file. See help.U
Select FractalMEnter name to choose or use cursor keys to select. Press ~\~ or ~/~ for menu.U
L-System menu
Edit @L-@System transformations
Choose fractal from list Open another @L-@System datafile Last menu
BPlease select which L-System related option you wish to perform...U
:The data used to draw this fractal is not stored in memory?and there are already 150 fractals in memory. To be able to use>this fractal you must delete an existing L-System from memory.
The L-System
has been created1since this was used to produce the image which is
being loaded.U
Function used:U
L-System -
L-System fractals<L-System images are produced using a form of turtle graphics>language where a set of instructions is built up involving the
operations:
, F - Draw forwards G - Move forwards( + - Turn right - - Turn left. | - 180
turn ! - Reverse + and -AFrom a simple instruction - the axiom - a set of rules is applied=which replace a single letter with a set of letters, allowingBelaborate patterns to be made by repeatedly applying simple rules.U
L-@System options
L-System
!Not enough memory for L-System to be available; remove TSRs etc...
and try again.
CAL.LSM
L-SystemU
!Not enough memory for L-System to be available; remove TSRs etc...
and try again.U
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Landscape FractalAThe landscape fractal is drawn by assigning a random value to the;corners of a 3x3 matrix. The colours of the other points in>this matrix are then found by averaging the points around them
and adding a random number.
AThis matrix is then broken down into four quarters, each of which?is itself a 3x3 matrix. This process of producing colour values;and then splitting the matrix up produces cloud-like images=if you use a blue-white type colour palette. It is especially?effective if you make the colours cycle by pressing < or > when
the image is on the screen.U
Landscape
Randomness
LandscapeU
- :
The Diffusion Image9This fractal is drawn by plotting an initial point in the9centre of the screen and then adding additional points to8it randomly. This is done by testing co-ordinates on the9screen until part of the image is found and then adding a:pixel to the side of the existing one. If the point tested2does not contain part of the image then one of its:neighbouring pixels is examined. The routine used is based.on a QuickBASIC program by Dr Gabriel Landini.U
Diffusion fractal
Colouring method
Diffusion colour method
Random colouring
Colour by position
Time-based colouring
Do not change
6How should pixels in the diffusion fractal be plotted?U
Diffusion fractalU
/ O f t
The Lorenz Attractor<The Lorenz Attractor is drawn by taking a starting point, in2this case (1,1,1) and applying the transformation:
NewX = X-(A*X*
T)+(A*Y*
T)" NewY = Y+(B*X*
T)-(Y*
T)-(Z*X*
NewZ = Z-(C*Z*
T)+(X*Y*
T):You can alter the values of A, B, C and
T using the Alter<variables option in the Fractal menu, but the default values9seem to give the most intersting pattern. The colour used<changes every few seconds to highlight recently drawn areas.=CAL uses the values of X and Z to form the location of a line
to plot on the screen.U
Lorenz attractor
T (time step between plots)
Lorenz attractorU
Logistic Equation9This fractal is drawn using a formula similar to that for
the bifurcation diagram:
Z = C * Z * (1-Z)8The initial value of Z can be set in the Alter Variables8screen, but must be non-zero because Z would then remain7zero as the formula is iterated. As with the Mandelbrot;set and Ushiki's phoenix, each point has an asociated Julia<image, which can be selected from the Fractal menu. See help,for details of suitable locations to select.U
Choose @Julia @Set
Return to @Log. @Eq.U
Logistic Equation
Choose @Julia @Set
Return to @Log. @Eq.
Julia set real position
Julia set imaginary position
Initial real value for Z
Initial imaginary value for Z
Julia set image selected
based on point at centre
of initial image.
Return to @Log. @Eq.U
Main Logistic Equation option
selected: choose Draw to
calculate image on screen.
Choose @Julia @SetU
JLogistic Equation option selected (use Fractal option to choose Julia set)CJulia set option selected (use Fractal option to choose main image)U
Logistic Equation
The Hopalong Fractal9The Hopalong Fractal is drawn by taking a starting point,7in this case (0
1) and applying the transformation:
NewX = Y - Sign(X) *
|Q*X-R|
NewY = P - X:Sign(X) returns the value +1 if X is positive, -1 if it is8negative and 0 if X is 0. P, Q and R are constants which:control characteristics of the image. Certain values (e.g.:P=1.5, Q=0.5, R=-2.5) give rise to a stable image of three:rings whilst other combinations result in a chaotic image.U
Hopalong
HopalongU
. E \ q
The Henon Attractor<The Henon Attractor is drawn by taking an arbitrary starting&point and applying the transformation:
NewX = 1+Y-(A*X
NewY = B*X=You can alter the values of A and B using the Alter Variables?option in the Fractal menu, but the default values seem to give;the most interesting pattern. When a point (X,Y) is plotted<onto the screen the colour at that point moves progressively;down the colour scale that is displayed at the right of the:screen. If you define a colour scale that fades from black<to white you will see the points that the algorithm produces.most frequently as bright points in the curve.U
Henon attractor
Henon attractor
. E \ s
Gumowski and Mira Attractor8These fractals are drawn by taking starting values for X6and Y, along with two constants, A and B, and applying6a transformation to derive successive new values for X3and Y. A point is plotted at (X,Y) every iteration.;A selection of different feather-like patterns are possible8by changing A and B in the Alter Variables option in the
fractal menu.
:For examples of patterns that may be drawn, and details of:the parameters that can be altered, select Alter Variables9press F1 for help. This fractal is based on an article in,Fractal Report Issue 22 by Dr Ian Entwistle.U
Gumowski and Mira Attractor
b Initial X Initial Y
,Press F1 for more details about the error...@The parameters that you have given in the Alter Variables optionAhave resulted in the calculation producing an arithmetic overflow8error. See help for examples of suitable numbers to use.U
Gumowski and Mira AttractorU
The Gingerbread person?As with most of the fractals in CAL, this is drawn by iterating=a very simple formula and plotting the successive values of X
and Y that are calculated.
NewX = 1-Y+|X| NewY = XA|X| refers to the magnitude, or absolute value, of X, for example@|-4|=4. Initially variables X and Y are both set to around 1. If?left for long enough detail starts to emerge within the mass of
points that are plotted.U
Gingerbread person
Gingerbread person
User defined formulae -
User defined formulae?This option allows you to enter your own formulae into CAL. For:more information select Edit Formula from the Fractal menu<and press F1 for help. Formulae can be entered using a built<in text editor so that there is no need to leave CAL to make
alterations.
;CAL is now able to draw two types of user defined formulae:?those, like the Mandelbrot Set, where a formula is iterated for?each point on the screen and a colour plotted or those like the?Henon Attractor where single pixels are plotted which form into"shapes. See help for more details.U
Edit/choose formula
User defined formulaeU
+Not enough memory for User defined formulae#to be available; remove TSRs etc...
and try again.U
'Do these operations before calculation:*Initialise these variables for each point:/Repeat the following operations each iteration:&Until the following condition is true:#Colour the point by: (e.g. colr:=n)U
Edit formula3When finished editing press the escape (ESC) key...U
>H-<r
>H-<r
None
X-Axis
Y-Axis
X and Y Axes
Rotational U
Symmetry type
X-Axis
Y-Axis
X and Y Axes
Rotational
Do not change
HWhich type of symmetry does this fractal posesses - see help for detailsU
3Use ~cursor keys~ then select ~DONE~ or ~FORGET~...
Alter formula options
'Please alter the options as required...
Set symmetry type
Max. number stored as integer
Default min. X value
Default max. X value
Default min. Y value
Default max. Y value
Done
Forget
16384*The max. integer number must be 4, 16, 64,"256, 1024, 4096 or 16384 (press F1
for more informAtion)U
Select option
New fractal
Rename fractal
Delete fractal Exit menu
?Select an option or press ESCAPE to return to selection list...
You cannot delete the last
fractal!1This option creates a new User Defined fractal...
Please select
Copy current fractal
Create blank fractal
Do not create fractal<Do you want the new fractal to be a copy of the current one?*You may have a maximum of 150 user defined*fractals at once. Since you currently have)150 it is not possible to add any more...U
Select FractalMEnter name to choose or use cursor keys to select. Press ~\~ or ~/~ for menu.U
,Since this computer is fitted with a numeric,co-processor, the accurate mode may actually
be quicker than the fast mode.U
Accurate
Formula menu
Edit formula text Select options/position/symmetry
Default pos. is current pos.
Choose formula to use
Open another set of formulae
Write to @F@R@M file
Read from @F@R@M file
Alter precision (
) Last menu
APlease select which formula-related option you wish to perform...U
<4uh;
=The formula used to draw this fractal is not stored in memory?and there are already 150 fractals in memory. To be able to use=this fractal you must delete an existing fractal from memory.
The formula
has been created1since this was used to produce the image which is
being loaded.U
Formula used:U
EDrawing was aborted because an error occured... (F1 for more details)
Divide by zero error whilst
executing user defined formula
"Floating point arithmetic overflow%whilst executing user defined formula$Illegal value passed as index whilst
e.g. -6^2
4 is not allowedU
CAL.UDF
User defined formulaeU
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Mandelbrot set
z:=(0,0)
c:=(x,y)
z:=z*z+c
colr:=nU
The file
, which contains details of the5user defined fractals, cannot be found in the current"directory. What do you want to do?
user defined fractals.
F T ` y
What do you want to do?
Append formula to end of file
Overwrite file
Do not export formula
The file
already exists - overwrite it?U
comment {M*****************************************************************************L The following fractals were exported from the CAL fractal drawing program:
K Further details about CAL and the features that it offers can be obtainedL from Tim Harris either at 5 Burnham Park Road, Peverell, Plymouth, England' or via EMail at tharris@uk.ac.plym.sc
real(pixel)
imag(pixel)
(x,y)
pixel
/2.302585093
sqrt(
The function * used in this formula may not be supported6in Fractint, or may be implemented in a different way.1This could cause slight differences to the image.U
colr:=n;This formula may appear differently in Fractint because the+ instruction, which changes how the colours,are used, has no counterpart in .FRM format.U
:=*This instruction is too complex to decode."Split it into several simple ones.U
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?The default formula in the User Defined Formula option containsAone or more errors and so cannot be compiled. Select Edit Formula0from the Fractal menu to correct these mistakes.U
?The default formula in the User Defined Formula option contains@operations which are not possible in the integer arithmetic modeBOnly floating point arithmetic will be available for this fractal.4This formula may only be used in floating point mode.because it involves operations which cannot be"performed with integer arithmetic.U
?The default formula in the User Defined Formula option contains@values which exceed the integer arithmetic limits. Only floating4point arithmetic will be available for this fractal.>The values used in this formula exceed the limit set in SELECT<OPTIONS. Integer arithmetic is not available until you alter7this limit or the formula. Floating point is available.U
=The default formula contains operations which may not work in<integer arithmetic mode depending on the exact numbers being<used. Floating point arithmetic will give the correct image.>Note that an inaccurate image may be produced because division<is not always possible in integer arithmetic mode. Using the2floating point option will give the correct image.U
This formula generates too
many variables/constants.
Out of memory - there may
only be 50 instructions
per formula.U
Syntax error
Incompatible operands
Invalid use of function
No such function
A comparison must take the form,source 1 operand source2, e.g. x<4, y=3 etc.U
!A constant may only appear on the
right hand side of a statementU
An operand may not exceed
25 characters in lengthU
A condition should include
only one comparisonU
"This variable has been used before
being assigned a valueU
Each statement must include
:= onceU
Only one operand should appear
on the left of a statementU
There are no statements
in the iterated loop!
!You must specify an end condition
for the iterated loop!
Missing open bracket
Missing close bracket
v RSP
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!*"@"h#~#
&$&1&>&K&X&e&r&
'''4'A'N'['h'u'
(((5(B(O(\(i(v(
Sorting fractals... U
(The default coordinates for this fractal%are now set to the coordinates of the
current image.U
v RSP
Rename fractal
What do you want to call the fractal?
Please enter the new name...
The name cannot
be blank
Please enter the name...
The name cannot
be blank
Delete fractal?
Do you really want to delete
.CAL requires a file to contain details of the
Please select:
Try another disc
Cancel
Create new file
Search other directories
Do not change fractal
Do not use @L-@System fractalU
+ ; K [ k {
v RSP
$The Flip-Mandelbrot Set & Julia SetsAThe Flip-Mandelbrot Set is a variation of the main Mandelbrot Set?whereby the real and imaginary parts of one of the intermediate;calculations are swapped over, of flipped, every iteration.
The formula used is:
Z = Flip(Z)^2 + Posn>Z starts off as 0 whilst Posn is set to the value of the point?being calculated where the X co-ordinate becomes the real value=and the Y co-ordinate the imaginary value. A Julia-set may be=produced for any point in the image. For more information see
the context sensitive help.U
Choose @Julia @Set
Return to @M-@SetU
Flip-Mandelbrot/Julia Set
Choose @Julia @Set
Return to @M-@Set
Julia set real position
Julia set imaginary position
Julia set image selected
based on point at centre
of initial image.
Return to @M-@SetU
Flip-Mandelbrot set chosen,
select Draw to calculate
image on screen.
Choose @Julia @SetU
LFlip-Mandelbrot set option selected (use Fractal option to choose Julia set)LJulia set option selected (use Fractal option to choose Flip-Mandelbrot set)U
Flip-Mandelbrot/Julia Set
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Chaotic dynamic systems>For each point in the image, X and Y are set to the horizontal>and vertical co-ordinates respectively. The following function&is then applied until X
>Threshold:
NewX = a + b*X + c*Y
NewY = d + e*X?Threshold and a-e can be varied from the Alter Variables optionAin the Fractal menu. Try using a=1, b=-2.4, c=-0.98, d=0, e=0
71.?Based on an article by John Topham published in Fractal Report.U
Chaotic dynamic systems
E Threshold
Chaotic dynamic systems
@fffffff
Composite Fractal Images?This fractal allows you to create composite images by combiningDsaved images on the screen. For example, the landscape fractal could@be displayed three-dimensionally across the bottom of the screen<with a selection of IFS trees `growing' on it. AlternativelyBseveral images of the Mandelbrot Set could be displayed as spheres
orbitting one another.
@If the images have been saved in different screen modes, or with=different colour palettes, CAL will select the closest colour<available from the current palette. The COMPOSIT.CGF paletteDincludes a wide range of colours and so is suitable for mode images.U
Composite fractal image
Composite images
Are you sure?
JClearing the screen will remove the current image, do you want to do this?U
The screen is already blank!
Use Overlay Saved Image to
make a composite image.U
3333333
Which type of overlay? Rectangle Landscape
Sphere
Do not change
9How should this image be overlaid onto the composite one?U
Load in which way? Overwrite
In front
Behind
Do not change
0How should pixels from the new image be plotted?U
Method of plotting pixels
In frontU
Angle (see help)
Maximum height (see help)
High water mark (see help)
Show outline box (Y/N)
Horiz. rotation
5Set options below and then choose LOAD or FINISHED...
Choose image to load
Shape to load image into
Set position of image
Load FinishedU
/All of the information must be completed for it0to be possible to load an overlay. The images to+use must have been saved as CAL data files.&The angle used for drawing a landscape
must lie between -75
and 75
)The maximum height of a landscape must be
between 0 and 300%.'The rotation used to plot a sphere must
be between -360
and 360
.#You must reply Y or N as to whether)CAL should display a solid outline around
the landscape image./The high water mark must be between 0 and 100%,+this is the `water level' on the landscape.
See help for more details.U
5Use ~cursor keys~ then select ~LOAD~ or ~FINISHED~...
Load image as overlay
2This file could not be loaded, ensure it is a data,file and contains an image: it must not have'been saved as details or as a RAM dump.U
Rectangle
Composite images
Clear screen
Blank section of screen
Overlay saved image Last menu
MThese options let you overlay several images at once to create a composite...U
Composite fractal imageU
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Change screen
Edit from initial screen
Change current screen
Blank screen Last menu
EDo you want to edit the current screen, or return to the initial one?U
Life AlgorithmAThis algorithm, Conway's Life, is a computer model which displaysAsome of the properties of living organisms. The screen is treated?as a number of cells, which can either be alive or dead - shown?by coloured or blank pixels. Based on the current pattern a new2image is produced every iteration using the rules:
1 1. Cells die if they have less than 2 neighbours9 2. A cell does not change if it has exactly 2 neighboursA 3. Cells are born into an empty spaces with exactly 3 neighbours1 4. Cells in spaces with 4 or more neighbours dieASee help for more details, or load the example image LIFEXMPL.CGFBfor example patterns. Note that this must be loaded in the mode in
which it was saved.U
Status:
% livingU
Change screen
Life algorithm
DThe life algorithm requires a mode with at least four colours. Users>with a monochrome display should try using another mode since,Aalthough colours may not be visible, images can still be created.U
Life algorithmU
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The Bifurcation Diagram;The bifurcation diagram is drawn by taking a seed value for<each point across the screen (this can be set by the user in3the Alter Variables option in the Fractal menu) and'repeatedly applying the transformation:
" NewSeed = XPosition*Seed*(1-Seed)?XPosition will initially take values between 2
7 and 4, but youAmay zoom in/out to see other parts of the image. This calculationAis repeated a set number of times and the last few values of seedAare plotted on the screen. The image formed resembles a branching:tree with first 1 then 2 then 4 lines crossing the screen.>Eventually this order breaks down into chaos, although smaller*tree structures can still be seen in this.U
Bifurcation diagram
Initial value
Number of cycles
Points to plot per value
EDrawing was aborted because an error occured... (F1 for more details)3It was not possible to complete the drawing because2the values used in calculations grew too large for
the computer to handle.U
Bifurcation diagramU
points per second)U
v RP1
v RP1
Statistics;This screen shows information about the images available...
Fractal name:
Type of image:
X minimum:
X maximum:
Y minimum:
Y maximum:E This fractal does not allow zooming. It either creates the same typeG of image each time draw is selected, is a composite image, or includes+ its own facilities for editing the screen.
No. colours used:
No. points
No. iterations:
Dimensions / pixels:
Next image
Composite fractal image
Image loaded
Time taken:
(Image finished)
Time taken so far:
(Image still being drawn) minutes
minute
seconds'Calculated point-by-point across screen$Point-by-point according to forumala
Press any key...U
v RP1
1 6 ; J \ a f ~
o n f u s i o n
i g h t (v4
(C)1990-1993
F1 for help and how to
receive an updated copy.
Using enhanced 386 code
Using enhanced 386 and 387 code
Using enhanced 287 code
Using maths coprocessor6To use on a monochrome display reload by typing CAL /M=If CAL crashes in floating point mode reload by typing CAL /NU
K
If you have any suggestions and comments, or if you haveL
come across an algorithm for a formula or type of fractalN
not featured in CAL, then I can be contacted at the address*
on the title screen or
OFractal Report, which has provided the source for many of the fractals featuredNin CAL, is produced by Reeves Telecommunications Laboratories Ltd., West TowanPHouse, Porthtowan, Truro, Cornwall. Also, FRAC'Cetera provides information aboutLproducts that may be of interest to fractal enthusiasts along with practicalNdetails for experimenting with fractals. For more information on this write toOHi-Ho Enterprises, Le Mont Ardaine, Rue des Ardaines, St. Peters, Guernsey, CI,IUK. See the context sensitive help screens for details about the fractalsEavailable. Thanks to the following for their continued help with CAL:@ Dan Goldwater for supplying innumerable palettes and formulaeI John Bridges, on whose VGAKIT software the graphics routines are basedP Shankar Ramakrishnan and Jesse Jones for help with the complex maths routines9 Dave Stevens, Andy Jewell, Vaughan Bell, Timothy Evans*CAL is copyright 1990-1993 Timothy Harris.U
There are no variables to
alter with this fractal
type.
03Use ~cursor keys~ then select ~DONE~ or ~FORGET~...
Alter Variables
)Please alter the variables as required...
Done
Forget
.The fractal you selected supports only integer*calculations, so the integer mode has been
automatically selected./The fractal you selected supports only floating2point calculations, so the floating point mode has
been automatically selected.1Integer arithmetic mode has been selected because6no floating point co-processor is available. This will$help speed-up the drawing of images.'You have no floating point co-processor(so calculation will be very slow because#integer arithmetic is not selected.U
There are no fractals from
which to choose!
KUse ~cursor keys~ and ~RETURN~ to select or enter first few letters of name
Select FractalU
Set Iterations
@ Please enter the maximum number of iterations. Increasing this@ can give more detailed images but is generally slower. Typical" values are between 100 and 1500.
The number of iterations must
be between 2 and 65535.U
Select fractal
Iterations
Alter variables Main menu
Fractal
2Please select an option from the pull-down menu...U
UP DOWN !Arrow indicates selected fractal.( Arrow indicates selected fractal.&
Y<Iu$
1<Qu-
NWhat set of options do you wish to alter? Select ~MAIN MENU~ to alter nothing.
Options
Display
S@V@G@A card type
Colours
Palette mapping
Arithmetic type
Optimisation
General options
Write options to disk Main menu
9Do you want to zoom in, zoom out, or go to the main menu?
Zoom
In Zoom
Centre image
To absolute position
Set by centre / magnification
Reset Main Menu
There is no image loaded
into which to zoom.
You cannot zoom into
this fractal.U
There are no fractal definitions
available!
There is no algorithm available
to do the calculation!
@3The image is too small to be drawn. Select a larger8size in the General Options screen from the Options menu/or a higher resolution from the Display screen.
Life algorithmU
You have not yet started
drawing an image.
There are no fractal definitions
available!
There is no algorithm available
to do the calculation!
The image is already
finished, or it is no longer
possible to continue.U
The colours can only be
altered in 256 and 16 colour
VGA modes.U
'There is no image loaded at the moment,'either select to draw one, or load one.
'There is no image loaded at the moment,!into which to zoom. Either select$Draw or load a previously saved one.U
! 0 ; i x
How many colours?
256-colour @M@A@P file
16-colour @M@A@P file Last menu
4Do you want to create a 16 or a 256-colour MAP file?
Unable to export this file
as a .MAP paletteU
Load selected file?
Do you want to load
FRCTL
.CGFU
-This option saves the current file to disc...
All data
Image
Palette
Details only
Export palette
Write @P@C@X file Last Menu
(What pieces of data do you want to save?V
There is no image
to save!
Overwrite?
No'The file already exists - overwrite it?
4Please select which disc operation you want to do... Load/Save
Save Catalogue
Batch mode Main Menu
!The burst must be between 0
1 and
6000 minutes (100 hours).
+The maximum time for drawing an image which(will not normally finish must be between$1 and 9999 minutes (about 150 hours)U
+The maximum time for drawing an image which(will not normally finish must be greater than the time of a single burst!U
5Use ~cursor keys~ then select ~START~ or ~ABANDON~...
Batch mode options
@4Use return to toggle options, or enter new number...
Redraw at current resolution
Redraw without block removal" Use floating point whenever poss.
Burst mode for all images+ Time spent on each drawing burst / minutes. Maximum time spent on one non-finishing image
Start
Abandon
What do you want to do?
Continue with other images
Exit batch mode
HDo you want to ommit only this image, or stop the batch mode altogether?U
-----U
Files processed :
(times given as Min:Seconds)7Filename Total image time Result of drawingU
5Use cursor keys to move up/down and return to exit...
Please wait...+Scanning directory for incomplete images...
Searching for files...
*.CGF
Loading...
9There are no files on the disk to continue in batch mode!.No .CGF files could be found! It is necessary ,to start drawing images and save them before
batch mode can be used.
Completed and saved
Continued and saved
Graphics mode unavailable
Disk error
Not a CAL file
Fractal unaviailable
User aborted
User terminated batch mode
Image already finished
No image data stored
Aborted due to arithmetic error8Details of some files not recorded due to lack of memory
Results of batch-Press F1 for help on the different results...U
-This version of CAL does not support the type-of fractal that you have just selected, so it,will not be possible to zoom into the image.%Image was saved in a different screen
mode from the current one
what do you want to do?
Please select:
Maintain current mode
Select saved mode6How do you want the computer to try to load the image?
Convert to current mode
Abandon load(
Image was saved as RAM
dump so cannot be
converted.
The screen mode used
is not available with
your graphics card.
Load aborted.This image was saved using the save as details)option: to re-create the original picture(return to the main menu and choose Draw.U
v RP
v Y[+
%==As
v RP
v Y[+
=<@v#
Unable to open output file, ensure that directory exists and
that disk is not full...U
C ] b z
"*"9"M"W"e"
% - P x
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Life algorithmU
#E(W({(
v Y[;
%Storing configuration file on disk...U
/How do you want the colour palette to be used?
Which palette style?
Repeat
Palette
Linea
Near detail
Far detail
Alternating Last menu
3Use ~cursor keys~ then select ~DONE~ or ~FORGET~...
User Defined Mode
2Please set the details of the user defined mode...
Screen width
Screen height
Mode number
Done
5Which general type of graphics adaptor are you using?
Which adaptor?
16 colour @V@G@A
256 colour @V@G@A and @S@V@G@A Last Menu
Which mode?
A - @Green, red, yellow
B - @Cyan, magenta, white
C - @Monochrome Last menu"Which CGA mode do you want to use?
A - 320x200 16 colour
B - 640x200 16 colour
C - 640x350 16 colour
D - 640x480 Monochrome"Which EGA mode do you want to use?
A - 640x200 16 colour
B - 640x350 16 colour
C - 640x480 16 colour
D - 800x600 16 colour
E - 1024x768 16 colour"Which VGA mode do you want to use?
A - 320x200 256 colour
B - 640x350 256 colour
C - 640x400 256 colour
D - 640x480 256 colour
E - 800x600 256 colour
F - 1024x768 256 colour
G - @User defined mode#Which SVGA mode do you want to use?+This screen mode is not available with your/graphics card. Ensure that you have the correct-card selected and you are using a valid mode.:The mode that you have selected only supports two colours.<Block removal will often downgrade the image quality - block6removal can be turned off in the Optimisations option.U
N3Use ~cursor keys~ then select ~DONE~ or ~FORGET~...
Alter Configuration.Please alter the configurations as required...
Automatic save filename
Size of fractal (1 to 100)
Root to use for log. palettes
Show colour palette scale?
Sound effects?
Done
Forget
$Warning: The save file name will not save files in the CAL directory:!Are you sure you want to do this?
Save in specified directory?
No9Do you want to save files in the directory you specified?
?)The size of the fractal must be 1 to 100,'this being the percentage of the screen
that is used"The root must be between 1 and 100
(1=Linear)
You must reply Y or N to
if you want the sound
effects or colour scale'The save filename should not contain an)extension: all CAL saved images will have
.CGF appended to the filename.U
v RP1
)The current fractal supports only integer4arithmetic - no floating point option is available.$Integer has therefore been selected.0The current fractal supports only floating point,arithmetic - no integer option is available.+Floating point has therefore been selected.LWhich arithmetic system should be used for calculations? (Integer is faster)
Which arithmetic type?
Floating point
Integer Last menu
'You have no floating point co-processor&so calculation will be very slow using
this option.U
Which graphics standard?
A - @Ahead(@A)
B - @Ahead(@B)
C - @A@T@I @V@G@A
D - @Chips and Technologies
E - @Compaq
F - @Everex
G - @Genoa
H - @N@C@R
I - @Oak Technologies
J - @Paradise
K - @Trident
L - @Tseng @E@T@4000
M - @Tseng @E@T@3000
N - @V@E@S@A
O - @Last menu
2Which type of super VGA card do you intend to use?U
:These options increase speed, but can decrease accuracy...#Alter which optimisation algorithm?
Large
Block removal
Small block removal
NON-BIOS display access Last menu
2 J Z _
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03Use ~cursor keys~ then select ~DONE~ or ~FORGET~...
Set to Absolute Position
5Please set the co-ordinates of the section to draw...
Left position
Right position
Bottom position
Top position
Done
Forget`
03Use ~cursor keys~ then select ~DONE~ or ~FORGET~...&Set Position by Centre / Magnification(Please enter centre and magnification... Centre X Centre Y
Magnification
Done
Forget0
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minutes
minute
seconds# spent drawing (image now complete)U
v RP1
v RSP
minutes
minute
seconds
So far ! spent drawing (image incomplete)U
v RP1
v RSP
CAL.CFG
.#No valid configurations file found;
constructing new one...
FRCTL,Unable to construct new configurations file.*Ensure that disc is write-enabled and that
it is not full. There are no fractal definitions
available!U
CAL.CFGU
( 3 c
File not found
Unable to find
the requested path
Too many open
files
File access
denied
Invalid file
handle
access code
Invalid drive
Cannot remover
current directory
Cannot rename
across drives
Disk read error
Disk write error
(disk may be full)
File not assigned
File not open for input
for output
Invalid numeric
format
Disk is write-protected
Unknown unit
Drive not ready
Unknown command
CRC error in data
Bad drive request
structure length
Disk seek error
Unknown media type
Sector not found
Hardware failureU
=<Avr
v RP
v Y[+
'==As"
v RP
v Y[+
Please insert the disc
for Drive
Change drive5Which drive do you want to use? (e.g. A, B, C etc...)
Please enter drive letter:
Unable to locate drive
: on this machine.U
}#v P
\-Please wait, examining directory structure...
Select FileXUse ~cursor keys~ / ~RETURN~ to select ~D~rive ~G~o to file De~l~ete ~R~ename ~N~ew nameYUse ~cursor keys~ / ~RETURN~ to select Current ~D~rive ~G~o to file De~l~ete ~R~ename
Delete
No2Do you definately want to delete it from the disc?
It is not possible to
delete directories from
within CAL.
Rename file
Enter the new name for
Please enter the new name...
Save to new file>This option lets you save the current file under a new name...
Go to file0This option moves directly to a specific file...
Please enter the name...
This file cannot be found.
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P /Confusion and Light (C)1990-1993 Timothy HarrisO
Fatal execution error
Error code no.:
Reported at address:
Press a key...U
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... U
Error
Note:
Press a key...U
Note:
Help!
Help :
CAL.HLP
.,,The CAL.HLP file, which contains the context2sensitive help, cannot be found. Try re-installing
CAL from the original archive.OUse ~Up~ and ~Down~ cursors to select option, ~Return~ to choose, ~ESC~ to exit
Help!
Last pageU
$Not enough memory for initialisation!remove TSRs etc... and try again.
P U
P !Confusion and Light (C)1990-1993 U
O Is all that we see or seem but a dream within a dream? - Edgar Allen Poe O All that we see or seem is but a dream within a dream - Edgar Allen Poe U
(.txt)