Paul Bourke has some images and explanation of the thorn fractal. This implementation features over sampling, gamma correction, iterative improvement, and it's open source.

The Julia set version of this fractal is generated by computing the escape time for the functions

$$ x_{n+1} = \frac{x_n}{\cos(y_n)} + c_x $$

$$ y_{n+1} = \frac{y_n}{\sin(x_n)}+ c_y $$

I look forward to trying a Mandelbrot version of the fractal soon.