mathr / blog / #

I implemented yet another method to visualize the Mandelbrot Set in Haskell, using circles around points with radius their distance estimate divided by 4 (ensuring no points of the Mandelbrot Set are inside that circle), starting new circles where they intersect.

The algorithm in Haskell:

algorithm :: Integer -> Rectangle -> Set Circle -> Set Circle -> IO ()
algorithm count bounds active inactive = do
  when (((2 ^) . round . logBase 2 . fromIntegral $ count) == count) $ do
    writeFile (show count ++ ".svg") (toSVG bounds active inactive)
  case S.maxView active of  -- Ord Circle is by radius and then (arbitrarily) center
    Nothing -> do
      c <- randomBorderPoint bounds
      case distance . refine . vertex $ c of
        Nothing -> algorithm (count + 1) bounds active inactive
        Just d  -> algorithm (count + 1) bounds (S.singleton (Circle c (d/4))) inactive
    Just (me, rest) -> do
      let them = S.toList $ rest `S.union` inactive
          new = S.fromList
                  [ Circle q (d/4)
                  | q <- concatMap (intersect me) them
                  , q `inside` bounds
                  , all (q `outside`) (map shrinkSlightly them)
                  , let md = distance . (!!(2^(4::Int))) . iterate refine . vertex $ q
                  , isJust md  -- very highly likely, but better safe...
                  , let Just d = md
                  ]
      algorithm (count + 1) bounds (new `S.union` rest) (me `S.insert` inactive)

It's not the fastest algorithm in the world...