sqrt 2, visualized

This code, inspired by my earlier post, animates the digits of √2 in an unsuual way. Each digit advances the curve in the direction given by a numeric keypad, with 5 and 0 both considered (0,0). The thumbnail at left links to a 4000x4000 window on 9856041 digits of the curve, which originates at the image center.

The program makes for an interesting screensaver, but does it follow any pattern, or illustrate any special properties of √2 ? Though I'm no mathematician, from what I've read this seems unlikely. √2 is suspected (but not proved) to be a normal number. This would mean the digits of its expansion (respective to some number base) follow a uniform distribution; no digit would be more likely to appear than any other.

Nonetheless, I'd be curious to see the results of a really long run!

	import Control.Arrow ((***))
	import Control.Monad (join, liftM2, liftM3, when)
	import Data.Bits (shift)
	import Data.Maybe (fromJust)
	import Graphics.UI.SDL as SDL
	(xres, yres) = (1600, 900)
	main = withInit [InitVideo] $ do
	w <- setVideoMode xres yres 32 [NoFrame]
	s <- createRGBSurface [SWSurface] 1 1 32 0 0 0 0
	enableEvent SDLMouseMotion False
	setCaption "Sqrt 2 Curve" "Sqrt 2 Curve"
	run w s (xres `div` 2, yres `div` 2) dirs rgbs [1..]
	run w s p (f:fs) (c:cs) (n:ns) = do
	drawCell w s p c
	when (n `mod` 47 == 0) $ SDL.flip w
	e <- pollEvent
	case e of
	KeyUp (Keysym SDLK_ESCAPE _ _) -> save >> print n
	_ -> run w s (f p) fs cs ns
	where
	save = saveBMP w "out.bmp"
	drawCell w s (x,y) c = draw x y
	where
	rect x y = Just $ Rect x y 1 1
	draw x y = do fillRect s (rect 0 0) (Pixel c)
	blitSurface s (rect 0 0) w $ rect x y
	sqrt2 = let ns = f 2 0 in concatMap (show . (ns !!)) [0..]
	where
	f x r = d : f (100 * (x - (20 * r + d) * d)) (10 * r + d)
	where
	d = head (dropWhile p [0..]) - 1
	p n = (20 * r + n) * n < x
	dirs = map f . expand $ map numPad sqrt2
	where
	f (x,y) = (x+) *** (+y)
	rgbs = expand . cycle . map f $ liftM3 (,,) ns ns ns
	where
	ns = [71, 77.. 255]
	f (r,g,b) = shift r 16 + shift g 8 + b
	numPad = fromJust . (`lookup` pad)
	where
	pad = zip "0528639417" $ head ps : ps
	ps = (join $ liftM2 (,)) [0, 1, -1]
	expand (x:xs) = replicate 8 x ++ expand xs

view raw sqrt-two-visualized.hs hosted with ❤ by GitHub