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? Please select: Try another disc Cancel ;CAL requires a file to contain details of the IFS fractals. Create new file Search other directories Do not use @I@F@S fractal Do not change fractal Edit @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 Sorting fractals... U Select Fractal UP DOWN %Arrow indicated selected IFS fractal., Arrow indicates selected IFS fractal.J~RETURN~ to select ~G~o to fractal ~N~ew IFS fractal ~R~ename De~l~ete& Delete fractal? Do you really want to delete You cannot delete the last fractal! New fractal(This option creates a new IFS fractal... Please enter the name... The name cannot be blank Please select Copy current fractal Create blank fractal Do not create fractalThe 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 J x .!6![!r" #B#j# &.&9&W& &$')'6';'R'W's' '%(V(~( (()G)L) *2*w* + +/+9+Z+i+}+ 0 0/0E0U0Z0i0y0~0 0R1b1g1 2!21262G2M2 5#535C5S5c5s5 5&6}6 7#7.7=7M7R7a7}7 8(888H8X8h8x8 9(9E9O9 :0:X: ;cL>n> ?/?k?z? @)@.@9@I@N@X@l@ A(A8AZAyA D4DMDRDaDpDuD E/E4ESEXEvE{E F(F3FDFVF^FxF H>HJHRHqHvH I#I-I5ITIYI N)N.N O.O=OLO[OjOyO PMQ_Q R'R8R S.SU V&WWWuW X#Xbrightly 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 Lyapunov spaceH 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 $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 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 for2 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 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 especially3effective if you make the colours cycle by pressing'< or > when the image is on the screen.U Landscape LandscapeU @ W n 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 " B Y g ~ The Lorenz Attractor1C1Z1_1z1 2-222X2b2 3$333H3 6'7A7P7f7v7{7 8 8u8 9$9)9D9T9Y9j9p9 <+<;E>U>Z>e>t> ?-?5?O?_?o? @/@?@O@_@|@ A A?AtA A+B\B D"D2DBDMD|D E.EJEXE F&FYFsF F#G2GBGGG H$H4HYHqH I)IBIVIaIjI J L,LFL\LlL|L M!M1MAMQMaMqM N)N2N=NNNjNoN O5P`R W(XfX [6[t[ ].]M^d^ _6_~_ aHa_ava a4b[b d)d9dDd e&e1eTeve f1f:f g!g;hShXhnh}h k1kKk]kwk l'l9lSlal~l 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 Images5This fractal allows you to create composite images by6combining saved images on the screen. For example, the8landscape fractal could be displayed three-dimensionally7across the bottom of the screen with a selection of IFS7trees `growing' on it. Alternatively, several images of:the Mandelbrot Set could be displayed as spheres orbitting one another.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 fffffff Which type of overlay? Rectangle Landscape Sphere Do not change9How should this image be overlaid onto the composite one?U Load in which way? Overwrite In front Behind Do not change0How 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 !#!6!T!}! "-"E"]"u" #5#M#e#}# %G&S& (d(m( ):)?)N)^)t)1*T* +'+6+E+T+c+r+ +",<,V, - -d-{- . ./.>.M.\.k.z. /2/y/ /80J0O0^0c0r0w0 2"21262E2P2_2d2s2x2 3)3.3A3K3a3 4=4S4 7 7/7:7]7l7{7 8@8O8^8i8 95:D:d:|: ;(;-;M;d; ;><^< @,@7@M@V@ B B/B>BMB\BkBzB C)C7CEC D*D/D The Bifurcation Diagram;The bifurcation diagram is drawn by taking a seed value forEventually 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 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. 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 ( - 2 A S X ] l } o n f u s i o n i g h t (v3 (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:I 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 + 4 F X ` h y 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! &\&k& '#'E'g' (.)3)D)I)Z)_)p)u) 3333333 fffffff ^??S? E)E>EREgE|EIFqF L2LKL P P?PWP minutes minute seconds# spent drawing (image now complete)U minutes minute seconds So far ! spent drawing (image incomplete)U 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 %T&%' +{,p- .1.8.G.X. /!/0/A/F/ 0@0O0z0 0U1i1 172N2o2 3%4g4t4 595F5S5`5m5z5 6%626?6L6Y6t6 7+7F7S7`7m7z7 808=8J8W8d8q8 9<9I9V9c9p9}9 :,:9:F:S:g:t: ;!;.;;;H;U;b;}; INSERT NUM CAPS U $ < u $@<@u Fatal menu system error #001U $LLLUL"M,M1MJM`MeMyM N N"N'N O*OTO[OeO O2P9PDPSPXP R'RCRNRXRqR U+V9V|V X Y.YVYeY{Y [([:[O[a[v[ ^4^H^M^R^Z^ _!_&_ Insufficient memory to run CALU