Single Orbit Fractals are Tricky   Leave a comment


A Gumowski-Mira fractal at 1000x1000

A Gumowski-Mira fractal at 1000×1000

The fractal above was produced by plotting a single orbit. An orbit is a sequence of values: the results of each iteration of a single formula or a pair of formulae in turn. The values in the orbit can be iterpreted as a location in an image, the image can then be coloured based on the number of times each location has been visited by the orbit. The pictures produced are of limited resolution, most fractals can produced at much higher resolutions revealing finer details, this isn’t possible with single orbit fractals. The effect of increasing the resolution is to increase the number of locations in a picture that an orbit can visit. For example if the number of pixels per side is doubled then what used to a single location is now four locations, tripled and the original location becomes nine locations, so as the resolution is increased the plot density is decreased. If the orbit visits only one of the new locations the same colour will used but all the other new locations will not have been visited and will be black, if the the orbit visits a number of the new locations the number of visits to the old location will be spread among the new locations resulting in a different colour selection. Assuming that the orbit can visit all the new locations then one approach to counteract this problem is to increase the orbit length by whatever factor the resolution is increased by, 4 for doubling of dimensions, 9 for tripling of dimensions and so on.

The following two pictures shows the effect of increasing the length of the orbit in line with the increase in resolution for the example Gumowski-Mira fractal.

Resolution increased to 2000x2000

Resolution increased to 2000×2000

Resolution increased to 6000x6000

Resolution increased to 6000×6000

As can be seen for this fractal the image gets dimmer and dimmer. The only way increasing the length of the orbit could have maintained the image is if the greater number of locations used at higher resolutions were visited evenly, if only a few of the new locations were visited then black of the unvisited locations would have a greater overall effect resulting in a dimmer picture.

The parameters used for the example Gumowski-Mira fractal result in an orbit plot where all points in the orbit fall inside the picture’s boundaries, other parameters result in orbits that leave the boundaries of the picture and never come back, so increasing the orbit length is pointless.

Other single orbit fractals such as Chip (as named in FractInt) behave in interesting ways, for a period the orbit will visit locations confined to a small area and then will suddenly break out to a much larger area and will be confined to that area for a period and then again breakout and so on. The same limitations of increasing the orbit length in line with resolution applies to this single orbit fractal with the addition that the increased resolution picture will be different in appearance.

The following three pictures shows a Chip fractal at varying resolutions:

A Chip fractal at 1000x1000

A Chip fractal at 1000×1000

Resolution increased to 2000x2000

Resolution increased to 2000×2000

Resolution increased to 6000x6000

Resolution increased to 6000×6000

The problems associated with producing hihger resolution versions of single orbit fractals also applies to zooming in. When a small part of a single orbit fractal has been selected for zooming in not all of the points in the orbit can be plotted. To counteract this problem only the values in the orbit that can be plotted are counted as part of the orbit thus a much longer orbit is calculated, if the number of consecutive unplotted values reaches the required orbit length calculation is deemed to be complete because the orbit may never return to the area being plotted. Depending on how the orbit behaves this may or may not improve the resulting image and won’t in anycase produce an image that properly matches the appearance of the zoomed area.

In short single orbit fractals are interesting to play with but zooming in and the production of high resolution images doesn’t work properly.

A number of these single orbit fractals will be included with next release of Saturn and Titan, expansion using Titan will NOT be available.

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Posted 10 December 2013 by element90 in Fractal

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