Insideout Dragon

Attributed to Roger Bagula

Graphics By Paul Bourke
July 2003

This fractal is calculated in a similar way to the Mandelbrot, that is, a series is calculated and coloured according to how quickly it escapes to infinity. The main difference is that the series in this case is somewhat more complicated. For a coordinate (x₀,y₀) one forms a complex number z₀ = 1 / (x₀ + i y₀), this becomes the first term of the series.

The series is given by z_n+1 = z_n² + f(|z_n|) + i g(|z_n|)
where
f(r) = r (1 + 2 r + r²) (r² - 1) / (1 + r³)²
and g(r) = r (1 - 2 r + r²) (r² - 1) / (1 + r³)²

and where |z| denotes the modulus of the complex number z.

Variations
Change denominator from (1 + r³)² to (1 + r + r⁷)