Archive for the ‘Titan’ Tag

The Lyapunov fractal available in Saturn & Titan is different to all the other fractals as its algorithm isn’t based on the Mandelbrot or Julia algorithms. I use the name Lyapunov for these fractals as it the name used when I first came across them in the September 1991 edition of Scientific American. The formula used to generate these fractals is named after the Russian mathematician Aleksandr M. Lyapunov, this formula was used by Mario Markus of the Max Plank Institute for Nutrition to model the breakdown of carbohydrates using enzymes and he produces the first pictures, so the fractal should more poperly be called the Markus-Lyapunov fractal. For more information see Introduction to Markus-Lyapunov Fractals.

Unlike all the other fractals implemented in Saturn & Titan colouring is limited to “multi-colour” and “single colour”. The colouring can be be applied separately to positive Lyapunov exponent values and negative Lyapunov exponent values. The best results are obtained by using multi-colour for both positive and negative values with different colour maps (or gradients).

The pictures alter depending on the “sequence string” which in Saturn consists of As and Bs, the simplest sequence is AB, the sequence BA produces the same picture flipped about the diagonal. There are two values which control the quality of the pictures and they are the number of settling cycles and the number of calculating cycles, a cycle is the calculation of the sequence, so if there are 100 settling cycles and 400 calculating cycles and the sequence is AB there are 200 and 800 calculations, the longer the sequence the greater the number of calculations for a give pair of settling and calculating cycles.

There is a distinct shape that crops up frequently in Markus-Lyapunov fractals and that is the “swallow” for a reason that should be obvious, an example is shown below. This example was produced using Mars & Phobos which were fore runners of Saturn & Titan.

Red Swallow

There now follow some more Markus-Lyapunov fractals.

Lyapumov 25

Lyapunov 27

Lyapunov 34

Lyapunov 35

More examples can be found in my Deviant Art Gallery, use the search gallery box to find the Lyapunov fractals.

Posted 5 March 2012 by in Fractal, Software

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When plotting orbit fractals some means of colouring the points is required. One method of determining a colour is to use the “square root” method, the colour is determined as follows:

`value = trunc(n*(sqrt(hits)/sqrt(max_hits)))`

Where

• sqrt is the square root function.
• hits is the number of times a point has been visited by an orbit.
• max_hits is the maximum number of visits a point in the picture has been visited by an orbit.
• n is an arbitrary scaling factor.
• trunc strips the fraction part of the number leaving an integer.

The value obtained from this formula is used to select a colour from an array of colours, the index to the array is found by adding a user defined offset to the value and using the remainder of its division by the number of colours in the array.

The problem with this method is that as the fractal is explored the maximum number of hits in any given picture is likely to be different to any other given picture thus altering the colours. This problem also affects the expansion of the image by Titan, the seed file in the following examples has dimensions of 750×500 points, the expansions have 6000×4000 points and 12000×8000 points and have been scaled down to 1200×800 for this post using GIMP.

6000×4000

12000×8000

Clearly there has been a shift in the colours used and neither match the colours in the Saturn seed file. I don’t mind about the colour changes that occur while exploring orbit fractals but I do mind about the colour changes that occur as the result of expanding the image. I have an idea of how to solve the colour change due expansion and if it works it will be incorporated into version 2.0 of Saturn & Titan.

Posted 22 February 2012 by in Fractal, Software

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Not exactly, just three versions of the same Nova plotted used all orbits, escaped orbits and captive orbits. Currently Saturn doesn’t support rotation of Orbit Plotted fractals so I couldn’t produce a zoom of the major feature of Orbit Nova:

Orbit Nova

I’ve now added custom image sizes to Saturn allowing portrait style pictures in addition to the usual landscape style, so I zoomed in on the feature on the right saved a Saturn seed file (500×750 pixels) expanded it using Titan to 8000×12000 and finally rotating it using eye-of-gnome image viewer resulting in this:

Orbit Plotted Nova

Obviously I’ve also changed the colour map. Additionally there are two more versions one plotting escaped orbits and the second plotting captive orbits.

Escape Orbit Plotted Nova

Captive Orbit Plotted Nova

Posted 21 February 2012 by in Fractal, Software

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While experimenting with orbit plots of various fractals types I found that the expansion of the Magnet Model 2 fractal produced a blank rectangle. This has now been fixed, the package version has been increased to 1.1.1 as has the version of Titan, Saturn’s version remains at 1.1.

Posted 8 February 2012 by in Fractal, Software

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The only orbit plotted fractals (orbit fractals for short) I’ve dealt with so far are Pickover Popcorn fractals. For the next release of Saturn & Titan I decided to try plotting the orbits of a Mandelbrot instead of colouring it in the many ways that are available for escape time fractals. I modified the code that handled Pickover Popcorn to add the Orbit Mandelbrot fractal type.

This is what I got with accumulation colouring (i.e. colour based on the number of times a location is visited by an orbit):

Orbit Mandelbrot

As the fractal is essentially a Mandelbrot I considered how to add transforms to orbit fractal code, I realised that I’d modified the wrong code, the escape time fractal code already supports transforms so I added orbit plotting as an option to the escape fractals. The upshot of this is that all the escape fractals could be plotted as orbit fractals and the Pickover Popcorn fractal types are no longer required, this is because there are Pickover Popcorn escape time variants: PP Mandelbrot 4F, PP Mandelbrot 6F, PP Julia 4F & PP Julia 6F, the Pickover Popcorn 4F & Pickover Popcorn 6F fractals can be produced using PP Julia 4F & PP Julia 6F with the julia ‘seed’ value set to zero. I reduced the number of fractals types by adding function parameters and now the number is being further reduced by the introduction of the orbit plotting option.

Here are some examples of more orbit fractals:

New versions of Saturn & Titan are now available and can be downloaded from the Download page (a subpage of the Software page).

In terms of functionality not much has changed, parts of the user interface have changed and some of the fractal types have been generalised. See the Release Notes (subpage of the Software page).

The most important aspect of this release is that the software in now available for Linux AND Windows 7.

Posted 31 January 2012 by in Fractal, Software

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Progress towards Saturn and Titan Version 1.1 for Linux and Windows 71 comment

The last remaining feature changes to Saturn are complete. The editing of transforms in version 1.0 was rudimentary, I have added some functions to make the process easier.

Saturn v1.0 Transform Tab

The transforms tab allowed transforms to be added to a transform set (Complex Plane, Transforms A & Transforms B) or deleted from a set, once the transforms had been entered the order of the transforms could only be changed by deleting and re-adding transforms. If the transforms were not no longer required, each transform had to be deleted one by one.

Saturn v1.1 Transforms Tab

In Saturn version 1.1 The editing buttons have been moved from below to the right hand side of the transforms “tree”. Additional buttons have been added:

• Up -active when the cursor is on a transform that is not the first the transform in a set. When pressed the transform is moved above the previous transform.
• Down – active when the cursor is on a transform that is not the last transform in a set. When pressed the transform is move below the succeeding transform.
• Delete Set – active when the cursor is on a transform set name and there are transforms in the set. When pressed all the transforms for that set are deleted.
• Delete All – active when there is a total at least one transform in the all transform set combined. When pressed all the transforms are deleted.

Transform editing is disabled if the fractal type is Pickover Popcorn 4F, Pickover Popcorn 6F or Lyapunov.

So that is the last feature change completed which leaves a bit of testing, possible bug fixing, package preparation and documentation to complete by the 31st January 2012. I have successfully created my first Windows installation package so as planned version 1.1 will be available for Windows 7 as well as Linux.

Here is a picture of the Saturn installation window for Windows 7:

Saturn Installation window

The appearance of Titan has not changed for version 1.1, as with Saturn 15 Pickover Popcorn fractals have been replaced with 6 more generalised versions, it will however accept Saturn level 1 seed files (from Saturn Version 1.0.x) as well as the new level 2 seed files produced by Saturn version 1.1.

Finally, unlike version 1.0.x packages the version 1.1.x packages will not contain User Manuals, User Guide or example seed files these will be made available as separate downloads. The Windows package will contain both Saturn and Titan is considerably larger than the combined Linux packages as the Gtkmm libraries are distributed as part of the package.

Posted 24 January 2012 by in Fractal, Software

For version 1.1 of Saturn I’ve rearranged the layout of the Colour tab and the Fractal tab now supports non-complex and function parameters. There are a lot of changes to the Colour tab as it changes depending on the fractal type, the biggest changes are:

• The list of colour maps on the right hand side has been removed.
• When the Colour tab is selected, the list on the left is the list of colour maps instead of the list of fractal types
• Selection of the outer colour method is display at the top of the Colour tab for both inner and outer colouring.
• At the bottom of the Colour tab there are two rectangles, the upper rectangle shows the colours in the outer colour map and the lower shows colours in the inner colour map. The maps are redrawn as the colour offset is changed starting with the colour at the offset.

Here is the Colour tab (outer colour selection) from Saturn 1.0:

Here is the Colour tab (outer colouring) from Saturn 1.0:

The functions of the two previous tabs have been combined for Saturn 1.1:

The Colour tab for Lyapunov fractals from Saturn 1.0:

The Colour tab for Lyapunov fractals from Saturn 1.1:

The Colour tab for orbit fractals from Saturn 1.0:

The Colour tab for orbit fractals from Saturn 1.1:

The changes to Fractal tab are:

• Addition of non-complex parameters, each parameter is identified using a capital letter starting with “A”. The are a maximum of 6 non-complex parameters.
• Addition of function parameters, each parameter is selected using comboboxes and is identified using “fn” where n is a number in the range 1 to 6.
• Function parameters are displayed first followed by non-complex parameters and finally complex parameters.
• The Pickover Popcorn parameters h, a and b have been replaced with A to D for the new Pickover Popcorn 4F fractal and A to F for the new Pickover Popcorn 6F fractal. The five old Pickover Popcorn fractals have been removed.
• The displayed for fractals with function parameters changes as the function parameters are changed.

In addition to the five Pickover Popcorn fractals being replaced with 4F (4 function) and 6F (6 function) versions, the same has been done for the five PP Mandelbrot and five PP Julia fractals. The changes to the Fractal tab can be best illustrated by comparing the Fractal tab for PP Julia 5 from version 1.0 to the Fractal tab for PP Julia 6F from version 1.1.

The Fractal tab for PP Julia 5 from Saturn 1.0:

The Fractal tab for PP Julia 6F from Saturn 1.1:

There are still some user interfaces to be done affecting the Transform tab. The majority of the changes for versions 1.1 of Saturn and Titan are complete, before the new versions can be released I need to do the following:

• Complete Tranform Tab changes.
• Update the user manuals for version 1.1.
• Update the user guide for version 1.1.
• Create a Windows 7 installer.

I’m still on track for releasing the software at the end of the month, i.e. on or before the 31st January 2012.

Posted 19 January 2012 by in Fractal, Software

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The release files for Saturn and Titan are now hosted on ubuntu one instead of OpenDrive. The problems at OpenDrive, still have not been resolved after a week this does not engender confidence in the service, hence the migration to ubuntu one.

While testing version 1.1 I came across a bug in Titan Version 1.0.2: PP Julia 4 fractals were incorrectly expanded because the real part of the beta parameter used in the formula was mistakenly taken from the alpha parameter.

Titan Version 1.0.3 can be downloaded from the My Downloads page and the release notes are on the Release Notes page, although the above explanation tells you all you need to know.

Posted 18 January 2012 by in General, Software

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