triangle_reflection_complex/billiard_words.hs

import Data.List
import Data.Ord
import Text.Printf
import System.Environment

main = do
  argv <- getArgs
  listWordsUpToLength $ read $ argv !! 0

listWordsUpToLength :: Int -> IO ()
listWordsUpToLength n = do
  putStr $ unlines [printf "%s %d/%d %f"
                             w
                             (p `div` gcd p q)
                             (q `div` gcd p q)
                             (atan (sqrt 3 / (2*q_/p_ + 1))) |
                      ((p,q),w) <- wordlist (n `div` 2, n `div` 2),
                      let p_ = fromIntegral p :: Double,
                      let q_ = fromIntegral q :: Double,
                      length w <= n,
                      let x = 2*q + p,
                      let y = 2*p + q]

--                             (sqrt 3 / 2 * fromIntegral p / (fromIntegral q + fromIntegral p / 2) :: Double) |
--                             (slopeWord "bca" (orthogonalSlope (p,q))) |

wordlist :: (Int,Int) -> [((Int,Int),String)]
wordlist (pmax,qmax) = nub $
                     sortBy (comparing sl)
                     [((p `div` gcd p q, q `div` gcd p q), slopeWord "bca" (p,q)) |
                      p <- [0..pmax],
                      q <- [0..qmax],
                      q /= 0]   -- use p /= 0 || q /= 0 for more symmetric output
    where
      sl ((p,q),_) = fromIntegral p / fromIntegral q

-- letters: reflection along e_1, reflection along e_2, other one; p,q >= 0
-- the "slope" (p,q) means the Euclidean vector q*e_1 + p*e_2, where e_1,e_2 are at a 60 degree angle
-- in Euclidean coordinates this is (q + p/2, sqrt(3)/2 * p)
slopeWord :: [Char] -> (Int,Int) -> String
slopeWord [x,y,z] (p,q)
    | p > q = slopeWord [y,x,z] (q,p)
    | otherwise = concat $ map word $ zipWith step list (tail list)
    where
      p_ = p `div` gcd p q
      q_ = q `div` gcd p q
      xmax = if (p_-q_) `mod` 3 == 0 then q_ else 3*q_ :: Int
      list = [(x,(x*p) `div` q) | x <- [0..xmax]]
      step (x1,y1) (x2,y2) = ((x1-y1) `mod` 3, y2-y1)
      word (0,0) = [z,x]
      word (1,0) = [y,z]
      word (2,0) = [x,y]
      word (0,1) = [z,y,x,y]
      word (1,1) = [y,x,z,x]
      word (2,1) = [x,z,y,z]

-- assuming p, q >= 0
orthogonalSlope :: (Int, Int) -> (Int, Int)
orthogonalSlope (p,q)
    | p > q = (p-q, p+2*q)
    | p < q = (q+2*p, q-p)
    | otherwise = (1,0)