Drawing spirals in cubes
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module Main where
import Control.Monad (forM_)
import Data.List (intercalate)
import Debug.Trace (trace)
import Math.NumberTheory.Roots (integerSquareRoot, integerCubeRoot)
type Cell = Int -- ^ The index of a cell in the cube
type Peel = Int -- ^ A peel of our cube
type Ring = Int -- ^ A ring of a 2D spiral
type Length = Int -- ^ The length of an line segment
type X = Int
type Y = Int
type Z = Int
type Point2D = (X, Y)
type Point3D = (X, Y, Z)
area :: Int -> Int
area a = a * a
cube :: Int -> Int
cube a = a * a * a
ring :: Cell -> Ring
ring o = integerSquareRoot o `div` 2
peel :: Cell -> Peel
peel c = integerCubeRoot c `div` 2
edge :: Peel -> Length
edge l = 2 * l + 2
atZ :: Z -> Point2D -> Point3D
atZ z (x, y) = (x, y, z)
reflectY :: Point2D -> Point2D
reflectY (x, y) = (x, -y - 1)
reflectX :: Point2D -> Point2D
reflectX (x, y) = (-x - 1, y)
reflectX3 :: Point3D -> Point3D
reflectX3 (x, y, z) = (-x - 1, y, z)
reflectZ3 :: Point3D -> Point3D
reflectZ3 (x, y, z) = (x, y, -z - 1)
rotate :: Length -> Point2D -> Point2D
rotate l (x, y) = (-y - 1, -x - 1)
spirals :: Length -> Cell -> Point3D
spirals l c = atZ h $ case h `mod` 4 of
0 -> growingSpiral l o
1 -> rotate l $ shrinkingSpiral l o
2 -> reflectX . reflectY $ growingSpiral l o
3 -> reflectX . reflectY . rotate l $ shrinkingSpiral l o
where h = c `div` area l
o = c - h * area l
z = h - l `div` 2 - 1
growingSpiral :: Length -> Cell -> Point2D
growingSpiral l o | o == 0 = (0, 0)
| ro < e - 1 = (r, ro - r) -- 64
| ro < 2 * e - 2 = (3 * r - ro, r)
| ro < 3 * e - 3 = (0 - r - 1, 5 * r - ro + 1)
| otherwise = (ro - 7 * r - 3, 0 - r - 1)
where r = ring o -- ^ the current spiral ring
ro = o - area (edge $ r - 1) -- ^ offset within this ring
e = edge r -- ^ edge of the this ring
shrinkingSpiral :: Length -> Cell -> Point2D
shrinkingSpiral l o = growingSpiral l (area l - o - 1)
peeledCubes :: Length -> Cell -> Point3D
peeledCubes _ c | o < a = reverseIfOdd
. atZ (e `div` 2 - 1) -- on top of the cube
$ growingSpiral e o
| o < a + (e - 2) * (e - 1) * 4
= reverseIfOdd
$ mantel p e (o - a)
| otherwise
= reverseIfOdd
. atZ (-e `div` 2) -- on the bottom of the cube
. reflectX
$ shrinkingSpiral e (o - a - (e - 2) * (e - 1) * 4)
where p = peel c -- ^ the current peel
o = c - cube (edge $ p - 1) -- ^ offset within the current peel
e = edge p -- ^ the length of the edge of the current peel
a = area e -- ^ the area of a side of the current peel
reverseIfOdd = if p `mod` 2 == 0 then id else (reflectX3 . reflectZ3)
mantel :: Peel -> Length -> Cell -> Point3D
mantel p e m | o <= r = (p - r + o, 0 - p - 1, p - r - 1)
| o <= r + e - 1 = (p, 0 - p - 1 + o - r, p - r - 1)
| o <= r + 2 * e - 2 = (p - o + r + e - 1, p, p - r - 1)
| o <= r + 3 * e - 3 = (0 - p - 1, p - o + r + 2 * e - 2, p - r - 1)
| otherwise = (0 - p - 1 + o - r - 3 * e + 3, 0 - p - 1, p - r - 1)
where r = m `div` (4 * e - 4) -- ^ revolutions since starting on the side
o = m - (4 * e - 4) * r -- ^ offset within this revolution
asMatrix :: Length -> [[[Cell]]]
asMatrix e = foldl (\c (i, x, y, z) -> replace c z $ replace (c!!z) y $ replace (c!!z!!y) x i)
(replicate e $ replicate e $ replicate e (-1))
[ (i, x + div e 2, y + div e 2, z + div e 2)
| i <- [0..cube e - 1]
, let (x, y, z) = peeledCubes e i
]
where replace :: [a] -> Int -> a -> [a]
replace l i e = take i l ++ [e] ++ drop (i+1) l
main :: IO ()
main = forM_ (reverse $ asMatrix 6) $ \plane -> do
forM_ (reverse plane) $ \row -> do
putStrLn $ intercalate ",\t" $ map show row
putStrLn ""