An orbit plotted fractal based on one of the formulae developed by Clifford A. Pickover commonly called Pickover Popcorn. Available as a print, poster or card from Redbubble and Artflakes.

## Archive for the ‘**orbit fractal**’ Tag

## Daily Fractal No. 86 Leave a comment

## Fractal No. 3 Leave a comment

Available as a print, poster or card from Red Bubble.

## Orbit Fractal Square Root Colouring Leave a comment

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.

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.

## Orbit Fractals Leave a comment

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):

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: