Robert P. Munafo, 2011 Feb 14.
The commonly-seen views of the Mandelbrot Set have been challenged by arguments based on techniques like error-term analysis (see Propagation of uncertainty). They show that if you want to get an accurate view of a lemniscate ZN you need to use at least N digits in all your calculations. The result is that most views of the Mandelbrot Set that people see on computers are (in theory) completely inaccurate or even "fictitious".
However, except for certain specific purposes (notably using iteration to trace an external ray), the problem is much smaller. The overwhelming amount of experimental evidence shows that ordinary Mandelbrot images (plotting e.g. the dwell for each point on a pixel grid) are indistinguishable from equivalent results produced by exact calculation. The images look the same to the human eye regardless of how many digits you use, as long as the number of digits is sufficient to distinguish the coordinates of the parameter value C.
This is because the roundoff errors added by each step in the iteration tend to cancel out as they would if randomly distributed, rather than systematically biased in one certain direction. See Systematic error, "Systematic versus random error".
revisions: 19990202 oldest on record; 20110214 expand discussion of systematic vs. random and link to external ray
From the Mandelbrot Set Glossary and Encyclopedia, by Robert Munafo, (c) 1987-2018. Mu-ency index
This page was written in the "embarrassingly readable" markup language RHTF, and was last updated on 2018 Feb 04. s.11