X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=src%2FCube.hs;h=3a997a4ab25e917eccd7e9e18587589b7bc98a32;hb=279936ce0f59cbc2b5c1c3c748ae9fae03ee1146;hp=d741f490c6e7499a309b991c0e5d3ba0a9c6cd55;hpb=551cd4237e70d474ac00125644157f506c367ed5;p=spline3.git

diff --git a/src/Cube.hs b/src/Cube.hs
index d741f49..3a997a4 100644
--- a/src/Cube.hs
+++ b/src/Cube.hs
@@ -1,187 +1,1201 @@
-module Cube
+module Cube (
+  Cube(..),
+  cube_properties,
+  find_containing_tetrahedron,
+  tetrahedra,
+  tetrahedron
+  )
 where
 
-import Grid
+import Data.Maybe (fromJust)
+import qualified Data.Vector as V (
+  Vector,
+  findIndex,
+  map,
+  minimum,
+  singleton,
+  snoc,
+  unsafeIndex
+  )
+import Prelude hiding (LT)
+import Test.Framework (Test, testGroup)
+import Test.Framework.Providers.QuickCheck2 (testProperty)
+import Test.QuickCheck (Arbitrary(..), Gen, Positive(..), choose)
+
+import Cardinal
+import Comparisons ((~=), (~~=))
+import qualified Face (Face(Face, v0, v1, v2, v3))
+import FunctionValues
+import Misc (all_equal, disjoint)
 import Point
+import Tetrahedron (Tetrahedron(..), c, volume)
 import ThreeDimensional
 
-class Gridded a where
-    back :: a -> Cube
-    down :: a -> Cube
-    front :: a -> Cube
-    left :: a -> Cube
-    right :: a -> Cube
-    top :: a -> Cube
-
-
-data Cube = Cube { grid :: Grid,
+data Cube = Cube { h :: Double,
                    i :: Int,
                    j :: Int,
                    k :: Int,
-                   datum :: Double }
+                   fv :: FunctionValues,
+                   tetrahedra_volume :: Double }
             deriving (Eq)
 
 
+instance Arbitrary Cube where
+    arbitrary = do
+      (Positive h') <- arbitrary :: Gen (Positive Double)
+      i' <- choose (coordmin, coordmax)
+      j' <- choose (coordmin, coordmax)
+      k' <- choose (coordmin, coordmax)
+      fv' <- arbitrary :: Gen FunctionValues
+      (Positive tet_vol) <- arbitrary :: Gen (Positive Double)
+      return (Cube h' i' j' k' fv' tet_vol)
+        where
+          coordmin = -268435456 -- -(2^29 / 2)
+          coordmax = 268435456  -- +(2^29 / 2)
+
+
 instance Show Cube where
-    show c =
-        "Cube_" ++ (show (i c)) ++ "," ++ (show (j c)) ++ "," ++ (show (k c)) ++
-        " (Grid: " ++ (show (grid c)) ++ ")" ++
-        " (Center: " ++ (show (center c)) ++ ")" ++
-        " (xmin: " ++ (show (xmin c)) ++ ")" ++
-        " (xmax: " ++ (show (xmax c)) ++ ")" ++
-        " (ymin: " ++ (show (ymin c)) ++ ")" ++
-        " (ymax: " ++ (show (ymax c)) ++ ")" ++
-        " (zmin: " ++ (show (zmin c)) ++ ")" ++
-        " (zmax: " ++ (show (zmax c)) ++ ")" ++
-        " (datum: " ++ (show (datum c)) ++ ")\n\n"
-
-empty_cube :: Cube
-empty_cube = Cube empty_grid 0 0 0 0
-
--- TODO: The paper considers 'i' to be the front/back direction,
--- whereas I have it in the left/right direction.
-instance Gridded Cube where
-    back c = cube_at (grid c) ((i c) + 1) (j c) (k c)
-    down c = cube_at (grid c) (i c) (j c) ((k c) - 1)
-    front c = cube_at (grid c) ((i c) - 1) (j c) (k c)
-    left c = cube_at (grid c) (i c) ((j c) - 1) (k c)
-    right c = cube_at (grid c) (i c) ((j c) + 1) (k c)
-    top c = cube_at (grid c) (i c) (j c) ((k c) + 1)
+    show cube =
+        "Cube_" ++ subscript ++ "\n" ++
+        " h: " ++ (show (h cube)) ++ "\n" ++
+        " Center: " ++ (show (center cube)) ++ "\n" ++
+        " xmin: " ++ (show (xmin cube)) ++ "\n" ++
+        " xmax: " ++ (show (xmax cube)) ++ "\n" ++
+        " ymin: " ++ (show (ymin cube)) ++ "\n" ++
+        " ymax: " ++ (show (ymax cube)) ++ "\n" ++
+        " zmin: " ++ (show (zmin cube)) ++ "\n" ++
+        " zmax: " ++ (show (zmax cube)) ++ "\n" ++
+        " fv: " ++ (show (Cube.fv cube)) ++ "\n"
+        where
+          subscript =
+              (show (i cube)) ++ "," ++ (show (j cube)) ++ "," ++ (show (k cube))
+
 
 -- | The left-side boundary of the cube. See Sorokina and Zeilfelder,
 --   p. 76.
 xmin :: Cube -> Double
-xmin c = (2*i' - 1)*delta / 2
+xmin cube = (i' - 1/2)*delta
     where
-      i' = fromIntegral (i c) :: Double
-      delta = h (grid c)
+      i' = fromIntegral (i cube) :: Double
+      delta = h cube
 
 -- | The right-side boundary of the cube. See Sorokina and Zeilfelder,
 --   p. 76.
 xmax :: Cube -> Double
-xmax c = (2*i' + 1)*delta / 2
+xmax cube = (i' + 1/2)*delta
     where
-      i' = fromIntegral (i c) :: Double
-      delta = h (grid c)
+      i' = fromIntegral (i cube) :: Double
+      delta = h cube
 
 -- | The front boundary of the cube. See Sorokina and Zeilfelder,
 --   p. 76.
 ymin :: Cube -> Double
-ymin c = (2*j' - 1)*delta / 2
+ymin cube = (j' - 1/2)*delta
     where
-      j' = fromIntegral (j c) :: Double
-      delta = h (grid c)
+      j' = fromIntegral (j cube) :: Double
+      delta = h cube
 
 -- | The back boundary of the cube. See Sorokina and Zeilfelder,
 --   p. 76.
 ymax :: Cube -> Double
-ymax c = (2*j' + 1)*delta / 2
+ymax cube = (j' + 1/2)*delta
     where
-      j' = fromIntegral (j c) :: Double
-      delta = h (grid c)
+      j' = fromIntegral (j cube) :: Double
+      delta = h cube
 
 -- | The bottom boundary of the cube. See Sorokina and Zeilfelder,
 --   p. 76.
 zmin :: Cube -> Double
-zmin c = (2*k' - 1)*delta / 2
+zmin cube = (k' - 1/2)*delta
     where
-      k' = fromIntegral (k c) :: Double
-      delta = h (grid c)
+      k' = fromIntegral (k cube) :: Double
+      delta = h cube
 
 -- | The top boundary of the cube. See Sorokina and Zeilfelder,
 --   p. 76.
 zmax :: Cube -> Double
-zmax c = (2*k' + 1)*delta / 2
+zmax cube = (k' + 1/2)*delta
     where
-      k' = fromIntegral (k c) :: Double
-      delta = h (grid c)
+      k' = fromIntegral (k cube) :: Double
+      delta = h cube
 
 instance ThreeDimensional Cube where
     -- | The center of Cube_ijk coincides with v_ijk at
     --   (ih, jh, kh). See Sorokina and Zeilfelder, p. 76.
-    center c = (x, y, z)
+    center cube = (x, y, z)
            where
-             delta = h (grid c)
-             i' = fromIntegral (i c) :: Double
-             j' = fromIntegral (j c) :: Double
-             k' = fromIntegral (k c) :: Double
+             delta = h cube
+             i' = fromIntegral (i cube) :: Double
+             j' = fromIntegral (j cube) :: Double
+             k' = fromIntegral (k cube) :: Double
              x = delta * i'
              y = delta * j'
              z = delta * k'
 
-    contains_point c p
-        | (x_coord p) < (xmin c) = False
-        | (x_coord p) > (xmax c) = False
-        | (y_coord p) < (ymin c) = False
-        | (y_coord p) > (ymax c) = False
-        | (z_coord p) < (zmin c) = False
-        | (z_coord p) > (zmax c) = False
+    -- | It's easy to tell if a point is within a cube; just make sure
+    --   that it falls on the proper side of each of the cube's faces.
+    contains_point cube (x, y, z)
+        | x < (xmin cube) = False
+        | x > (xmax cube) = False
+        | y < (ymin cube) = False
+        | y > (ymax cube) = False
+        | z < (zmin cube) = False
+        | z > (zmax cube) = False
         | otherwise = True
 
 
-instance Num Cube where
-    (Cube g1 i1 j1 k1 d1) + (Cube _ i2 j2 k2 d2) =
-        Cube g1 (i1 + i2) (j1 + j2) (k1 + k2) (d1 + d2)
 
-    (Cube g1 i1 j1 k1 d1) - (Cube _ i2 j2 k2 d2) =
-        Cube g1 (i1 - i2) (j1 - j2) (k1 - k2) (d1 - d2)
+-- Face stuff.
+
+-- | The top (in the direction of z) face of the cube.
+top_face :: Cube -> Face.Face
+top_face cube = Face.Face v0' v1' v2' v3'
+    where
+      delta = (1/2)*(h cube)
+      v0' = (center cube) + (delta, -delta, delta)
+      v1' = (center cube) + (delta, delta, delta)
+      v2' = (center cube) + (-delta, delta, delta)
+      v3' = (center cube) + (-delta, -delta, delta)
+
+
+
+-- | The back (in the direction of x) face of the cube.
+back_face :: Cube -> Face.Face
+back_face cube = Face.Face v0' v1' v2' v3'
+    where
+      delta = (1/2)*(h cube)
+      v0' = (center cube) + (delta, -delta, -delta)
+      v1' = (center cube) + (delta, delta, -delta)
+      v2' = (center cube) + (delta, delta, delta)
+      v3' = (center cube) + (delta, -delta, delta)
+
+
+-- The bottom face (in the direction of -z) of the cube.
+down_face :: Cube -> Face.Face
+down_face cube = Face.Face v0' v1' v2' v3'
+    where
+      delta = (1/2)*(h cube)
+      v0' = (center cube) + (-delta, -delta, -delta)
+      v1' = (center cube) + (-delta, delta, -delta)
+      v2' = (center cube) + (delta, delta, -delta)
+      v3' = (center cube) + (delta, -delta, -delta)
+
+
+
+-- | The front (in the direction of -x) face of the cube.
+front_face :: Cube -> Face.Face
+front_face cube = Face.Face v0' v1' v2' v3'
+    where
+      delta = (1/2)*(h cube)
+      v0' = (center cube) + (-delta, -delta, delta)
+      v1' = (center cube) + (-delta, delta, delta)
+      v2' = (center cube) + (-delta, delta, -delta)
+      v3' = (center cube) + (-delta, -delta, -delta)
+
+-- | The left (in the direction of -y) face of the cube.
+left_face :: Cube -> Face.Face
+left_face cube = Face.Face v0' v1' v2' v3'
+    where
+      delta = (1/2)*(h cube)
+      v0' = (center cube) + (delta, -delta, delta)
+      v1' = (center cube) + (-delta, -delta, delta)
+      v2' = (center cube) + (-delta, -delta, -delta)
+      v3' = (center cube) + (delta, -delta, -delta)
+
+
+-- | The right (in the direction of y) face of the cube.
+right_face :: Cube -> Face.Face
+right_face cube = Face.Face v0' v1' v2' v3'
+    where
+      delta = (1/2)*(h cube)
+      v0' = (center cube) + (-delta, delta, delta)
+      v1' = (center cube) + (delta, delta, delta)
+      v2' = (center cube) + (delta, delta, -delta)
+      v3' = (center cube) + (-delta, delta, -delta)
+
+
+tetrahedron :: Cube -> Int -> Tetrahedron
+
+tetrahedron cube 0 =
+    Tetrahedron (fv cube) v0' v1' v2' v3' vol
+    where
+      v0' = center cube
+      v1' = center (front_face cube)
+      v2' = Face.v0 (front_face cube)
+      v3' = Face.v1 (front_face cube)
+      vol = tetrahedra_volume cube
+
+tetrahedron cube 1 =
+    Tetrahedron fv' v0' v1' v2' v3' vol
+    where
+      v0' = center cube
+      v1' = center (front_face cube)
+      v2' = Face.v1 (front_face cube)
+      v3' = Face.v2 (front_face cube)
+      fv' = rotate ccwx (fv cube)
+      vol = tetrahedra_volume cube
+
+tetrahedron cube 2 =
+    Tetrahedron fv' v0' v1' v2' v3' vol
+    where
+      v0' = center cube
+      v1' = center (front_face cube)
+      v2' = Face.v2 (front_face cube)
+      v3' = Face.v3 (front_face cube)
+      fv' = rotate ccwx $ rotate ccwx $ fv cube
+      vol = tetrahedra_volume cube
+
+tetrahedron cube 3 =
+    Tetrahedron fv' v0' v1' v2' v3' vol
+    where
+      v0' = center cube
+      v1' = center (front_face cube)
+      v2' = Face.v3 (front_face cube)
+      v3' = Face.v0 (front_face cube)
+      fv' = rotate cwx (fv cube)
+      vol = tetrahedra_volume cube
+
+tetrahedron cube 4 =
+    Tetrahedron fv' v0' v1' v2' v3' vol
+    where
+      v0' = center cube
+      v1' = center (top_face cube)
+      v2' = Face.v0 (top_face cube)
+      v3' = Face.v1 (top_face cube)
+      fv' = rotate cwy (fv cube)
+      vol = tetrahedra_volume cube
+
+tetrahedron cube 5 =
+    Tetrahedron fv' v0' v1' v2' v3' vol
+    where
+      v0' = center cube
+      v1' = center (top_face cube)
+      v2' = Face.v1 (top_face cube)
+      v3' = Face.v2 (top_face cube)
+      fv' = rotate cwy $ rotate cwz $ fv cube
+      vol = tetrahedra_volume cube
+
+tetrahedron cube 6 =
+    Tetrahedron fv' v0' v1' v2' v3' vol
+    where
+      v0' = center cube
+      v1' = center (top_face cube)
+      v2' = Face.v2 (top_face cube)
+      v3' = Face.v3 (top_face cube)
+      fv' = rotate cwy $ rotate cwz
+                       $ rotate cwz
+                       $ fv cube
+      vol = tetrahedra_volume cube
+
+tetrahedron cube 7 =
+    Tetrahedron fv' v0' v1' v2' v3' vol
+    where
+      v0' = center cube
+      v1' = center (top_face cube)
+      v2' = Face.v3 (top_face cube)
+      v3' = Face.v0 (top_face cube)
+      fv' = rotate cwy $ rotate ccwz $ fv cube
+      vol = tetrahedra_volume cube
+
+tetrahedron cube 8 =
+    Tetrahedron fv' v0' v1' v2' v3' vol
+    where
+      v0' = center cube
+      v1' = center (back_face cube)
+      v2' = Face.v0 (back_face cube)
+      v3' = Face.v1 (back_face cube)
+      fv' = rotate cwy $ rotate cwy $ fv cube
+      vol = tetrahedra_volume cube
+
+tetrahedron cube 9 =
+    Tetrahedron fv' v0' v1' v2' v3' vol
+    where
+      v0' = center cube
+      v1' = center (back_face cube)
+      v2' = Face.v1 (back_face cube)
+      v3' = Face.v2 (back_face cube)
+      fv' = rotate cwy $ rotate cwy
+                       $ rotate cwx
+                       $ fv cube
+      vol = tetrahedra_volume cube
+
+tetrahedron cube 10 =
+    Tetrahedron fv' v0' v1' v2' v3' vol
+    where
+      v0' = center cube
+      v1' = center (back_face cube)
+      v2' = Face.v2 (back_face cube)
+      v3' = Face.v3 (back_face cube)
+      fv' = rotate cwy $ rotate cwy
+                       $ rotate cwx
+                       $ rotate cwx
+                       $ fv cube
+
+      vol = tetrahedra_volume cube
+
+tetrahedron cube 11 =
+    Tetrahedron fv' v0' v1' v2' v3' vol
+    where
+      v0' = center cube
+      v1' = center (back_face cube)
+      v2' = Face.v3 (back_face cube)
+      v3' = Face.v0 (back_face cube)
+      fv' = rotate cwy $ rotate cwy
+                       $ rotate ccwx
+                       $ fv cube
+      vol = tetrahedra_volume cube
+
+tetrahedron cube 12 =
+    Tetrahedron fv' v0' v1' v2' v3' vol
+    where
+      v0' = center cube
+      v1' = center (down_face cube)
+      v2' = Face.v0 (down_face cube)
+      v3' = Face.v1 (down_face cube)
+      fv' = rotate ccwy $ fv cube
+      vol = tetrahedra_volume cube
+
+tetrahedron cube 13 =
+    Tetrahedron fv' v0' v1' v2' v3' vol
+    where
+      v0' = center cube
+      v1' = center (down_face cube)
+      v2' = Face.v1 (down_face cube)
+      v3' = Face.v2 (down_face cube)
+      fv' = rotate ccwy $ rotate ccwz $ fv cube
+      vol = tetrahedra_volume cube
+
+tetrahedron cube 14 =
+    Tetrahedron fv' v0' v1' v2' v3' vol
+    where
+      v0' = center cube
+      v1' = center (down_face cube)
+      v2' = Face.v2 (down_face cube)
+      v3' = Face.v3 (down_face cube)
+      fv' = rotate ccwy $ rotate ccwz
+                        $ rotate ccwz
+                        $ fv cube
+      vol = tetrahedra_volume cube
+
+tetrahedron cube 15 =
+    Tetrahedron fv' v0' v1' v2' v3' vol
+    where
+      v0' = center cube
+      v1' = center (down_face cube)
+      v2' = Face.v3 (down_face cube)
+      v3' = Face.v0 (down_face cube)
+      fv' = rotate ccwy $ rotate cwz $ fv cube
+      vol = tetrahedra_volume cube
+
+tetrahedron cube 16 =
+    Tetrahedron fv' v0' v1' v2' v3' vol
+    where
+      v0' = center cube
+      v1' = center (right_face cube)
+      v2' = Face.v0 (right_face cube)
+      v3' = Face.v1 (right_face cube)
+      fv' = rotate ccwz $ fv cube
+      vol = tetrahedra_volume cube
+
+tetrahedron cube 17 =
+    Tetrahedron fv' v0' v1' v2' v3' vol
+    where
+      v0' = center cube
+      v1' = center (right_face cube)
+      v2' = Face.v1 (right_face cube)
+      v3' = Face.v2 (right_face cube)
+      fv' = rotate ccwz $ rotate cwy $ fv cube
+      vol = tetrahedra_volume cube
+
+tetrahedron cube 18 =
+    Tetrahedron fv' v0' v1' v2' v3' vol
+    where
+      v0' = center cube
+      v1' = center (right_face cube)
+      v2' = Face.v2 (right_face cube)
+      v3' = Face.v3 (right_face cube)
+      fv' = rotate ccwz $ rotate cwy
+                        $ rotate cwy
+                        $ fv cube
+      vol = tetrahedra_volume cube
+
+tetrahedron cube 19 =
+    Tetrahedron fv' v0' v1' v2' v3' vol
+    where
+      v0' = center cube
+      v1' = center (right_face cube)
+      v2' = Face.v3 (right_face cube)
+      v3' = Face.v0 (right_face cube)
+      fv' = rotate ccwz $ rotate ccwy
+                        $ fv cube
+      vol = tetrahedra_volume cube
+
+tetrahedron cube 20 =
+    Tetrahedron fv' v0' v1' v2' v3' vol
+    where
+      v0' = center cube
+      v1' = center (left_face cube)
+      v2' = Face.v0 (left_face cube)
+      v3' = Face.v1 (left_face cube)
+      fv' = rotate cwz $ fv cube
+      vol = tetrahedra_volume cube
+
+tetrahedron cube 21 =
+    Tetrahedron fv' v0' v1' v2' v3' vol
+    where
+      v0' = center cube
+      v1' = center (left_face cube)
+      v2' = Face.v1 (left_face cube)
+      v3' = Face.v2 (left_face cube)
+      fv' = rotate cwz $ rotate ccwy $ fv cube
+      vol = tetrahedra_volume cube
+
+tetrahedron cube 22 =
+    Tetrahedron fv' v0' v1' v2' v3' vol
+    where
+      v0' = center cube
+      v1' = center (left_face cube)
+      v2' = Face.v2 (left_face cube)
+      v3' = Face.v3 (left_face cube)
+      fv' = rotate cwz $ rotate ccwy
+                       $ rotate ccwy
+                       $ fv cube
+      vol = tetrahedra_volume cube
+
+tetrahedron cube 23 =
+    Tetrahedron fv' v0' v1' v2' v3' vol
+    where
+      v0' = center cube
+      v1' = center (left_face cube)
+      v2' = Face.v3 (left_face cube)
+      v3' = Face.v0 (left_face cube)
+      fv' = rotate cwz $ rotate cwy
+                       $ fv cube
+      vol = tetrahedra_volume cube
+
+-- Feels dirty, but whatever.
+tetrahedron _ _ = error "asked for a nonexistent tetrahedron"
+
+
+-- Only used in tests, so we don't need the added speed
+-- of Data.Vector.
+tetrahedra :: Cube -> [Tetrahedron]
+tetrahedra cube = [ tetrahedron cube n | n <- [0..23] ]
+
+front_left_top_tetrahedra :: Cube -> V.Vector Tetrahedron
+front_left_top_tetrahedra  cube =
+  V.singleton (tetrahedron cube 0) `V.snoc`
+    (tetrahedron cube 3) `V.snoc`
+    (tetrahedron cube 6) `V.snoc`
+    (tetrahedron cube 7) `V.snoc`
+    (tetrahedron cube 20) `V.snoc`
+    (tetrahedron cube 21)
+
+front_left_down_tetrahedra :: Cube -> V.Vector Tetrahedron
+front_left_down_tetrahedra  cube =
+  V.singleton (tetrahedron cube 0) `V.snoc`
+    (tetrahedron cube 2) `V.snoc`
+    (tetrahedron cube 3) `V.snoc`
+    (tetrahedron cube 12) `V.snoc`
+    (tetrahedron cube 15) `V.snoc`
+    (tetrahedron cube 21)
+
+front_right_top_tetrahedra :: Cube -> V.Vector Tetrahedron
+front_right_top_tetrahedra  cube =
+  V.singleton (tetrahedron cube 0) `V.snoc`
+    (tetrahedron cube 1) `V.snoc`
+    (tetrahedron cube 5) `V.snoc`
+    (tetrahedron cube 6) `V.snoc`
+    (tetrahedron cube 16) `V.snoc`
+    (tetrahedron cube 19)
+
+front_right_down_tetrahedra :: Cube -> V.Vector Tetrahedron
+front_right_down_tetrahedra  cube =
+  V.singleton (tetrahedron cube 1) `V.snoc`
+    (tetrahedron cube 2) `V.snoc`
+    (tetrahedron cube 12) `V.snoc`
+    (tetrahedron cube 13) `V.snoc`
+    (tetrahedron cube 18) `V.snoc`
+    (tetrahedron cube 19)
+
+back_left_top_tetrahedra :: Cube -> V.Vector Tetrahedron
+back_left_top_tetrahedra  cube =
+  V.singleton (tetrahedron cube 0) `V.snoc`
+    (tetrahedron cube 3) `V.snoc`
+    (tetrahedron cube 6) `V.snoc`
+    (tetrahedron cube 7) `V.snoc`
+    (tetrahedron cube 20) `V.snoc`
+    (tetrahedron cube 21)
+
+back_left_down_tetrahedra :: Cube -> V.Vector Tetrahedron
+back_left_down_tetrahedra  cube =
+  V.singleton (tetrahedron cube 8) `V.snoc`
+    (tetrahedron cube 11) `V.snoc`
+    (tetrahedron cube 14) `V.snoc`
+    (tetrahedron cube 15) `V.snoc`
+    (tetrahedron cube 22) `V.snoc`
+    (tetrahedron cube 23)
+
+back_right_top_tetrahedra :: Cube -> V.Vector Tetrahedron
+back_right_top_tetrahedra  cube =
+  V.singleton (tetrahedron cube 4) `V.snoc`
+    (tetrahedron cube 5) `V.snoc`
+    (tetrahedron cube 9) `V.snoc`
+    (tetrahedron cube 10) `V.snoc`
+    (tetrahedron cube 16) `V.snoc`
+    (tetrahedron cube 17)
+
+back_right_down_tetrahedra :: Cube -> V.Vector Tetrahedron
+back_right_down_tetrahedra  cube =
+  V.singleton (tetrahedron cube 8) `V.snoc`
+    (tetrahedron cube 9) `V.snoc`
+    (tetrahedron cube 13) `V.snoc`
+    (tetrahedron cube 14) `V.snoc`
+    (tetrahedron cube 17) `V.snoc`
+    (tetrahedron cube 18)
+
+in_top_half :: Cube -> Point -> Bool
+in_top_half cube (_,_,z) =
+  distance_from_top <= distance_from_bottom
+  where
+    distance_from_top = abs $ (zmax cube) - z
+    distance_from_bottom = abs $ (zmin cube) - z
+
+in_front_half :: Cube -> Point -> Bool
+in_front_half cube (x,_,_) =
+    distance_from_front <= distance_from_back
+  where
+    distance_from_front = abs $ (xmin cube) - x
+    distance_from_back = abs $ (xmax cube) - x
+
+
+in_left_half :: Cube -> Point -> Bool
+in_left_half cube (_,y,_) =
+    distance_from_left <= distance_from_right
+  where
+    distance_from_left = abs $ (ymin cube) - y
+    distance_from_right = abs $ (ymax cube) - y
+
+
+-- | Takes a 'Cube', and returns the Tetrahedra belonging to it that
+--   contain the given 'Point'. This should be faster than checking
+--   every tetrahedron individually, since we determine which half
+--   (hemisphere?) of the cube the point lies in three times: once in
+--   each dimension. This allows us to eliminate non-candidates
+--   quickly.
+--
+--   This can throw an exception, but the use of 'head' might
+--   save us some unnecessary computations.
+--
+find_containing_tetrahedron :: Cube -> Point -> Tetrahedron
+find_containing_tetrahedron cube p =
+  candidates `V.unsafeIndex` (fromJust lucky_idx)
+  where
+    front_half = in_front_half cube p
+    top_half = in_top_half cube p
+    left_half = in_left_half cube p
+
+    candidates =
+      if front_half then
+
+        if left_half then
+          if top_half then
+            front_left_top_tetrahedra cube
+          else
+            front_left_down_tetrahedra cube
+        else
+          if top_half then
+            front_right_top_tetrahedra cube
+          else
+            front_right_down_tetrahedra cube
+
+      else -- bottom half
+
+        if left_half then
+          if top_half then
+            back_left_top_tetrahedra cube
+          else
+            back_left_down_tetrahedra cube
+        else
+          if top_half then
+            back_right_top_tetrahedra cube
+          else
+            back_right_down_tetrahedra cube
+
+    -- Use the dot product instead of 'distance' here to save a
+    -- sqrt(). So, "distances" below really means "distances squared."
+    distances = V.map ((dot p) . center) candidates
+    shortest_distance = V.minimum distances
+    lucky_idx = V.findIndex
+                  (\t -> (center t) `dot` p == shortest_distance)
+                  candidates
+
+
+
+
+
+
+-- Tests
+
+-- Quickcheck tests.
+
+prop_opposite_octant_tetrahedra_disjoint1 :: Cube -> Bool
+prop_opposite_octant_tetrahedra_disjoint1 cube =
+  disjoint (front_left_top_tetrahedra cube) (front_right_down_tetrahedra cube)
+
+prop_opposite_octant_tetrahedra_disjoint2 :: Cube -> Bool
+prop_opposite_octant_tetrahedra_disjoint2 cube =
+  disjoint (back_left_top_tetrahedra cube) (back_right_down_tetrahedra cube)
+
+prop_opposite_octant_tetrahedra_disjoint3 :: Cube -> Bool
+prop_opposite_octant_tetrahedra_disjoint3 cube =
+  disjoint (front_left_top_tetrahedra cube) (back_right_top_tetrahedra cube)
+
+prop_opposite_octant_tetrahedra_disjoint4 :: Cube -> Bool
+prop_opposite_octant_tetrahedra_disjoint4 cube =
+  disjoint (front_left_down_tetrahedra cube) (back_right_down_tetrahedra cube)
+
+prop_opposite_octant_tetrahedra_disjoint5 :: Cube -> Bool
+prop_opposite_octant_tetrahedra_disjoint5 cube =
+  disjoint (front_left_top_tetrahedra cube) (back_left_down_tetrahedra cube)
+
+prop_opposite_octant_tetrahedra_disjoint6 :: Cube -> Bool
+prop_opposite_octant_tetrahedra_disjoint6 cube =
+  disjoint (front_right_top_tetrahedra cube) (back_right_down_tetrahedra cube)
+
+
+-- | Since the grid size is necessarily positive, all tetrahedra
+--   (which comprise cubes of positive volume) must have positive volume
+--   as well.
+prop_all_volumes_positive :: Cube -> Bool
+prop_all_volumes_positive cube =
+    null nonpositive_volumes
+    where
+      ts = tetrahedra cube
+      volumes = map volume ts
+      nonpositive_volumes = filter (<= 0) volumes
+
+-- | In fact, since all of the tetrahedra are identical, we should
+--   already know their volumes. There's 24 tetrahedra to a cube, so
+--   we'd expect the volume of each one to be (1/24)*h^3.
+prop_all_volumes_exact :: Cube -> Bool
+prop_all_volumes_exact cube =
+    and [volume t ~~= (1/24)*(delta^(3::Int)) | t <- tetrahedra cube]
+    where
+      delta = h cube
+
+-- | All tetrahedron should have their v0 located at the center of the cube.
+prop_v0_all_equal :: Cube -> Bool
+prop_v0_all_equal cube = (v0 t0) == (v0 t1)
+    where
+      t0 = head (tetrahedra cube) -- Doesn't matter which two we choose.
+      t1 = head $ tail (tetrahedra cube)
+
+
+-- | Given in Sorokina and Zeilfelder, p. 79, (2.6). Note that the
+--   third and fourth indices of c-t3 have been switched. This is
+--   because we store the triangles oriented such that their volume is
+--   positive. If T and T-tilde share \<v0,v1,v2\> and v3,v3-tilde point
+--   in opposite directions, one of them has to have negative volume!
+prop_c0120_identity1 :: Cube -> Bool
+prop_c0120_identity1 cube =
+   c t0 0 1 2 0 ~= (c t0 0 0 2 1 + c t3 0 0 1 2) / 2
+     where
+       t0 = tetrahedron cube 0
+       t3 = tetrahedron cube 3
+
+
+-- | Given in Sorokina and Zeilfelder, p. 79, (2.6). Repeats
+--   'prop_c0120_identity1' with tetrahedrons 1 and 2.
+prop_c0120_identity2 :: Cube -> Bool
+prop_c0120_identity2 cube =
+   c t1 0 1 2 0 ~= (c t1 0 0 2 1 + c t0 0 0 1 2) / 2
+     where
+       t0 = tetrahedron cube 0
+       t1 = tetrahedron cube 1
+
+-- | Given in Sorokina and Zeilfelder, p. 79, (2.6). Repeats
+--   'prop_c0120_identity1' with tetrahedrons 1 and 2.
+prop_c0120_identity3 :: Cube -> Bool
+prop_c0120_identity3 cube =
+   c t2 0 1 2 0 ~= (c t2 0 0 2 1 + c t1 0 0 1 2) / 2
+     where
+       t1 = tetrahedron cube 1
+       t2 = tetrahedron cube 2
+
+-- | Given in Sorokina and Zeilfelder, p. 79, (2.6). Repeats
+--   'prop_c0120_identity1' with tetrahedrons 2 and 3.
+prop_c0120_identity4 :: Cube -> Bool
+prop_c0120_identity4 cube =
+   c t3 0 1 2 0 ~= (c t3 0 0 2 1 + c t2 0 0 1 2) / 2
+     where
+       t2 = tetrahedron cube 2
+       t3 = tetrahedron cube 3
+
+
+-- | Given in Sorokina and Zeilfelder, p. 79, (2.6). Repeats
+--   'prop_c0120_identity1' with tetrahedrons 4 and 5.
+prop_c0120_identity5 :: Cube -> Bool
+prop_c0120_identity5 cube =
+    c t5 0 1 2 0 ~= (c t5 0 0 2 1 + c t4 0 0 1 2) / 2
+    where
+      t4 = tetrahedron cube 4
+      t5 = tetrahedron cube 5
+
+-- | Given in Sorokina and Zeilfelder, p. 79, (2.6). Repeats
+--   'prop_c0120_identity1' with tetrahedrons 5 and 6.
+prop_c0120_identity6 :: Cube -> Bool
+prop_c0120_identity6 cube =
+   c t6 0 1 2 0 ~= (c t6 0 0 2 1 + c t5 0 0 1 2) / 2
+     where
+       t5 = tetrahedron cube 5
+       t6 = tetrahedron cube 6
+
+
+-- | Given in Sorokina and Zeilfelder, p. 79, (2.6). Repeats
+--   'prop_c0120_identity1' with tetrahedrons 6 and 7.
+prop_c0120_identity7 :: Cube -> Bool
+prop_c0120_identity7 cube =
+   c t7 0 1 2 0 ~= (c t7 0 0 2 1 + c t6 0 0 1 2) / 2
+     where
+       t6 = tetrahedron cube 6
+       t7 = tetrahedron cube 7
+
+
+-- | Given in Sorokina and Zeilfelder, p. 79, (2.6). See
+--   'prop_c0120_identity1'.
+prop_c0210_identity1 :: Cube -> Bool
+prop_c0210_identity1 cube =
+    c t0 0 2 1 0 ~= (c t0 0 1 1 1 + c t3 0 1 1 1) / 2
+      where
+        t0 = tetrahedron cube 0
+        t3 = tetrahedron cube 3
+
+
+-- | Given in Sorokina and Zeilfelder, p. 79, (2.6). See
+--   'prop_c0120_identity1'.
+prop_c0300_identity1 :: Cube -> Bool
+prop_c0300_identity1 cube =
+    c t0 0 3 0 0 ~= (c t0 0 2 0 1 + c t3 0 2 1 0) / 2
+      where
+        t0 = tetrahedron cube 0
+        t3 = tetrahedron cube 3
+
+
+-- | Given in Sorokina and Zeilfelder, p. 79, (2.6). See
+--   'prop_c0120_identity1'.
+prop_c1110_identity :: Cube -> Bool
+prop_c1110_identity cube =
+    c t0 1 1 1 0 ~= (c t0 1 0 1 1 + c t3 1 0 1 1) / 2
+      where
+        t0 = tetrahedron cube 0
+        t3 = tetrahedron cube 3
+
+
+-- | Given in Sorokina and Zeilfelder, p. 79, (2.6). See
+--   'prop_c0120_identity1'.
+prop_c1200_identity1 :: Cube -> Bool
+prop_c1200_identity1 cube =
+    c t0 1 2 0 0 ~= (c t0 1 1 0 1 + c t3 1 1 1 0) / 2
+      where
+        t0 = tetrahedron cube 0
+        t3 = tetrahedron cube 3
+
+
+-- | Given in Sorokina and Zeilfelder, p. 79, (2.6). See
+--   'prop_c0120_identity1'.
+prop_c2100_identity1 :: Cube -> Bool
+prop_c2100_identity1 cube =
+    c t0 2 1 0 0 ~= (c t0 2 0 0 1 + c t3 2 0 1 0) / 2
+      where
+        t0 = tetrahedron cube 0
+        t3 = tetrahedron cube 3
+
+
+
+-- | Given in Sorokina and Zeilfelder, p. 79, (2.7). Note that the
+--   third and fourth indices of c-t3 have been switched. This is
+--   because we store the triangles oriented such that their volume is
+--   positive. If T and T-tilde share \<v0,v1,v2\> and v3,v3-tilde
+--   point in opposite directions, one of them has to have negative
+--   volume!
+prop_c0102_identity1 :: Cube -> Bool
+prop_c0102_identity1 cube =
+    c t0 0 1 0 2 ~= (c t0 0 0 1 2 + c t1 0 0 2 1) / 2
+      where
+        t0 = tetrahedron cube 0
+        t1 = tetrahedron cube 1
+
+
+-- | Given in Sorokina and Zeilfelder, p. 79, (2.7). See
+--   'prop_c0102_identity1'.
+prop_c0201_identity1 :: Cube -> Bool
+prop_c0201_identity1 cube =
+    c t0 0 2 0 1 ~= (c t0 0 1 1 1 + c t1 0 1 1 1) / 2
+      where
+        t0 = tetrahedron cube 0
+        t1 = tetrahedron cube 1
+
+
+-- | Given in Sorokina and Zeilfelder, p. 79, (2.7). See
+--   'prop_c0102_identity1'.
+prop_c0300_identity2 :: Cube -> Bool
+prop_c0300_identity2 cube =
+    c t0 0 3 0 0 ~= (c t0 0 2 1 0 + c t1 0 2 0 1) / 2
+      where
+        t0 = tetrahedron cube 0
+        t1 = tetrahedron cube 1
+
+
+-- | Given in Sorokina and Zeilfelder, p. 79, (2.7). See
+--   'prop_c0102_identity1'.
+prop_c1101_identity :: Cube -> Bool
+prop_c1101_identity cube =
+    c t0 1 1 0 1 ~= (c t0 1 0 1 1 + c t1 1 0 1 1) / 2
+      where
+        t0 = tetrahedron cube 0
+        t1 = tetrahedron cube 1
+
+
+-- | Given in Sorokina and Zeilfelder, p. 79, (2.7). See
+--   'prop_c0102_identity1'.
+prop_c1200_identity2 :: Cube -> Bool
+prop_c1200_identity2 cube =
+    c t0 1 2 0 0 ~= (c t0 1 1 1 0 + c t1 1 1 0 1) / 2
+      where
+        t0 = tetrahedron cube 0
+        t1 = tetrahedron cube 1
+
+
+-- | Given in Sorokina and Zeilfelder, p. 79, (2.7). See
+--   'prop_c0102_identity1'.
+prop_c2100_identity2 :: Cube -> Bool
+prop_c2100_identity2 cube =
+    c t0 2 1 0 0 ~= (c t0 2 0 1 0 + c t1 2 0 0 1) / 2
+      where
+        t0 = tetrahedron cube 0
+        t1 = tetrahedron cube 1
+
+
+-- | Given in Sorokina and Zeilfelder, p. 79, (2.8). The third and
+--   fourth indices of c-t6 have been switched. This is because we
+--   store the triangles oriented such that their volume is
+--   positive. If T and T-tilde share \<v0,v1,v2\> and v3,v3-tilde
+--   point in opposite directions, one of them has to have negative
+--   volume!
+prop_c3000_identity :: Cube -> Bool
+prop_c3000_identity cube =
+    c t0 3 0 0 0 ~= c t0 2 1 0 0 + c t6 2 1 0 0
+                    - ((c t0 2 0 1 0 + c t0 2 0 0 1)/ 2)
+      where
+        t0 = tetrahedron cube 0
+        t6 = tetrahedron cube 6
+
+
+-- | Given in Sorokina and Zeilfelder, p. 79, (2.8). See
+--   'prop_c3000_identity'.
+prop_c2010_identity :: Cube -> Bool
+prop_c2010_identity cube =
+    c t0 2 0 1 0 ~= c t0 1 1 1 0 + c t6 1 1 0 1
+                    - ((c t0 1 0 2 0 + c t0 1 0 1 1)/ 2)
+      where
+        t0 = tetrahedron cube 0
+        t6 = tetrahedron cube 6
+
+
+-- | Given in Sorokina and Zeilfelder, p. 79, (2.8). See
+--   'prop_c3000_identity'.
+prop_c2001_identity :: Cube -> Bool
+prop_c2001_identity cube =
+    c t0 2 0 0 1 ~= c t0 1 1 0 1 + c t6 1 1 1 0
+                    - ((c t0 1 0 0 2 + c t0 1 0 1 1)/ 2)
+      where
+        t0 = tetrahedron cube 0
+        t6 = tetrahedron cube 6
+
+
+-- | Given in Sorokina and Zeilfelder, p. 79, (2.8). See
+--   'prop_c3000_identity'.
+prop_c1020_identity :: Cube -> Bool
+prop_c1020_identity cube =
+    c t0 1 0 2 0 ~= c t0 0 1 2 0 + c t6 0 1 0 2
+                    - ((c t0 0 0 3 0 + c t0 0 0 2 1)/ 2)
+      where
+        t0 = tetrahedron cube 0
+        t6 = tetrahedron cube 6
+
+
+-- | Given in Sorokina and Zeilfelder, p. 79, (2.8). See
+--   'prop_c3000_identity'.
+prop_c1002_identity :: Cube -> Bool
+prop_c1002_identity cube =
+    c t0 1 0 0 2 ~= c t0 0 1 0 2 + c t6 0 1 2 0
+                    - ((c t0 0 0 0 3 + c t0 0 0 1 2)/ 2)
+      where
+        t0 = tetrahedron cube 0
+        t6 = tetrahedron cube 6
+
+
+-- | Given in Sorokina and Zeilfelder, p. 79, (2.8). See
+--   'prop_c3000_identity'.
+prop_c1011_identity :: Cube -> Bool
+prop_c1011_identity cube =
+    c t0 1 0 1 1 ~= c t0 0 1 1 1 + c t6 0 1 1 1 -
+                    ((c t0 0 0 1 2 + c t0 0 0 2 1)/ 2)
+      where
+        t0 = tetrahedron cube 0
+        t6 = tetrahedron cube 6
+
+
+-- | The function values at the interior should be the same for all
+--   tetrahedra.
+prop_interior_values_all_identical :: Cube -> Bool
+prop_interior_values_all_identical cube =
+    all_equal [ eval (function_values tet) I | tet <- tetrahedra cube ]
+
+
+-- | We know what (c t6 2 1 0 0) should be from Sorokina and Zeilfelder, p. 87.
+--   This test checks the rotation works as expected.
+prop_c_tilde_2100_rotation_correct :: Cube -> Bool
+prop_c_tilde_2100_rotation_correct cube =
+    expr1 == expr2
+    where
+      t0 = tetrahedron cube 0
+      t6 = tetrahedron cube 6
+
+      -- What gets computed for c2100 of t6.
+      expr1 = eval (function_values t6) $
+          (3/8)*I +
+          (1/12)*(T + R + L + D) +
+          (1/64)*(FT + FR + FL + FD) +
+          (7/48)*F +
+          (1/48)*B +
+          (1/96)*(RT + LD + LT + RD) +
+          (1/192)*(BT + BR + BL + BD)
+
+      -- What should be computed for c2100 of t6.
+      expr2 = eval (function_values t0) $
+              (3/8)*I +
+              (1/12)*(F + R + L + B) +
+              (1/64)*(FT + RT + LT + BT) +
+              (7/48)*T +
+              (1/48)*D +
+              (1/96)*(FR + FL + BR + BL) +
+              (1/192)*(FD + RD + LD + BD)
+
+
+-- | We know what (c t6 2 1 0 0) should be from Sorokina and
+--   Zeilfelder, p. 87.  This test checks the actual value based on
+--   the FunctionValues of the cube.
+--
+--   If 'prop_c_tilde_2100_rotation_correct' passes, then this test is
+--   even meaningful!
+prop_c_tilde_2100_correct :: Cube -> Bool
+prop_c_tilde_2100_correct cube =
+    c t6 2 1 0 0 == expected
+    where
+      t0 = tetrahedron cube 0
+      t6 = tetrahedron cube 6
+      fvs = function_values t0
+      expected = eval fvs $
+                  (3/8)*I +
+                  (1/12)*(F + R + L + B) +
+                  (1/64)*(FT + RT + LT + BT) +
+                  (7/48)*T +
+                  (1/48)*D +
+                  (1/96)*(FR + FL + BR + BL) +
+                  (1/192)*(FD + RD + LD + BD)
+
+
+-- Tests to check that the correct edges are incidental.
+prop_t0_shares_edge_with_t1 :: Cube -> Bool
+prop_t0_shares_edge_with_t1 cube =
+    (v1 t0) == (v1 t1) && (v3 t0) == (v2 t1)
+      where
+        t0 = tetrahedron cube 0
+        t1 = tetrahedron cube 1
+
+prop_t0_shares_edge_with_t3 :: Cube -> Bool
+prop_t0_shares_edge_with_t3 cube =
+    (v1 t0) == (v1 t3) && (v2 t0) == (v3 t3)
+      where
+        t0 = tetrahedron cube 0
+        t3 = tetrahedron cube 3
+
+prop_t0_shares_edge_with_t6 :: Cube -> Bool
+prop_t0_shares_edge_with_t6 cube =
+    (v2 t0) == (v3 t6) && (v3 t0) == (v2 t6)
+      where
+        t0 = tetrahedron cube 0
+        t6 = tetrahedron cube 6
+
+prop_t1_shares_edge_with_t2 :: Cube -> Bool
+prop_t1_shares_edge_with_t2 cube =
+    (v1 t1) == (v1 t2) && (v3 t1) == (v2 t2)
+      where
+        t1 = tetrahedron cube 1
+        t2 = tetrahedron cube 2
+
+prop_t1_shares_edge_with_t19 :: Cube -> Bool
+prop_t1_shares_edge_with_t19 cube =
+    (v2 t1) == (v3 t19) && (v3 t1) == (v2 t19)
+      where
+        t1 = tetrahedron cube 1
+        t19 = tetrahedron cube 19
+
+prop_t2_shares_edge_with_t3 :: Cube -> Bool
+prop_t2_shares_edge_with_t3 cube =
+    (v1 t1) == (v1 t2) && (v3 t1) == (v2 t2)
+      where
+        t1 = tetrahedron cube 1
+        t2 = tetrahedron cube 2
+
+prop_t2_shares_edge_with_t12 :: Cube -> Bool
+prop_t2_shares_edge_with_t12 cube =
+    (v2 t2) == (v3 t12) && (v3 t2) == (v2 t12)
+      where
+        t2 = tetrahedron cube 2
+        t12 = tetrahedron cube 12
+
+prop_t3_shares_edge_with_t21 :: Cube -> Bool
+prop_t3_shares_edge_with_t21 cube =
+    (v2 t3) == (v3 t21) && (v3 t3) == (v2 t21)
+      where
+        t3 = tetrahedron cube 3
+        t21 = tetrahedron cube 21
+
+prop_t4_shares_edge_with_t5 :: Cube -> Bool
+prop_t4_shares_edge_with_t5 cube =
+    (v1 t4) == (v1 t5) && (v3 t4) == (v2 t5)
+      where
+        t4 = tetrahedron cube 4
+        t5 = tetrahedron cube 5
 
-    (Cube g1 i1 j1 k1 d1) * (Cube _ i2 j2 k2 d2) =
-        Cube g1 (i1 * i2) (j1 * j2) (k1 * k2) (d1 * d2)
+prop_t4_shares_edge_with_t7 :: Cube -> Bool
+prop_t4_shares_edge_with_t7 cube =
+    (v1 t4) == (v1 t7) && (v2 t4) == (v3 t7)
+      where
+        t4 = tetrahedron cube 4
+        t7 = tetrahedron cube 7
 
-    abs (Cube g1 i1 j1 k1 d1) =
-        Cube g1 (abs i1) (abs j1) (abs k1) (abs d1)
+prop_t4_shares_edge_with_t10 :: Cube -> Bool
+prop_t4_shares_edge_with_t10 cube =
+    (v2 t4) == (v3 t10) && (v3 t4) == (v2 t10)
+      where
+        t4 = tetrahedron cube 4
+        t10 = tetrahedron cube 10
 
-    signum (Cube g1 i1 j1 k1 d1) =
-        Cube g1 (signum i1) (signum j1) (signum k1) (signum d1)
+prop_t5_shares_edge_with_t6 :: Cube -> Bool
+prop_t5_shares_edge_with_t6 cube =
+    (v1 t5) == (v1 t6) && (v3 t5) == (v2 t6)
+      where
+        t5 = tetrahedron cube 5
+        t6 = tetrahedron cube 6
 
-    fromInteger x = empty_cube { datum = (fromIntegral x) }
+prop_t5_shares_edge_with_t16 :: Cube -> Bool
+prop_t5_shares_edge_with_t16 cube =
+    (v2 t5) == (v3 t16) && (v3 t5) == (v2 t16)
+      where
+        t5 = tetrahedron cube 5
+        t16 = tetrahedron cube 16
 
-instance Fractional Cube where
-    (Cube g1 i1 j1 k1 d1) / (Cube _ _ _ _ d2) =
-        Cube g1 i1 j1 k1 (d1 / d2)
+prop_t6_shares_edge_with_t7 :: Cube -> Bool
+prop_t6_shares_edge_with_t7 cube =
+    (v1 t6) == (v1 t7) && (v3 t6) == (v2 t7)
+      where
+        t6 = tetrahedron cube 6
+        t7 = tetrahedron cube 7
 
-    recip (Cube g1 i1 j1 k1 d1) =
-        Cube g1 i1 j1 k1 (recip d1)
+prop_t7_shares_edge_with_t20 :: Cube -> Bool
+prop_t7_shares_edge_with_t20 cube =
+    (v2 t7) == (v3 t20) && (v2 t7) == (v3 t20)
+      where
+        t7 = tetrahedron cube 7
+        t20 = tetrahedron cube 20
 
-    fromRational q = empty_cube { datum = fromRational q }
 
--- | Constructs a cube, switching the i and k axes.
-reverse_cube :: Grid -> Int -> Int -> Int -> Double -> Cube
-reverse_cube g k' j' i' = Cube g i' j' k'
+p79_26_properties :: Test.Framework.Test
+p79_26_properties =
+    testGroup "p. 79, Section (2.6) Properties" [
+      testProperty "c0120 identity1" prop_c0120_identity1,
+      testProperty "c0120 identity2" prop_c0120_identity2,
+      testProperty "c0120 identity3" prop_c0120_identity3,
+      testProperty "c0120 identity4" prop_c0120_identity4,
+      testProperty "c0120 identity5" prop_c0120_identity5,
+      testProperty "c0120 identity6" prop_c0120_identity6,
+      testProperty "c0120 identity7" prop_c0120_identity7,
+      testProperty "c0210 identity1" prop_c0210_identity1,
+      testProperty "c0300 identity1" prop_c0300_identity1,
+      testProperty "c1110 identity" prop_c1110_identity,
+      testProperty "c1200 identity1" prop_c1200_identity1,
+      testProperty "c2100 identity1" prop_c2100_identity1]
 
+p79_27_properties :: Test.Framework.Test
+p79_27_properties =
+    testGroup "p. 79, Section (2.7) Properties" [
+      testProperty "c0102 identity1" prop_c0102_identity1,
+      testProperty "c0201 identity1" prop_c0201_identity1,
+      testProperty "c0300 identity2" prop_c0300_identity2,
+      testProperty "c1101 identity" prop_c1101_identity,
+      testProperty "c1200 identity2" prop_c1200_identity2,
+      testProperty "c2100 identity2" prop_c2100_identity2 ]
 
-cube_at :: Grid -> Int -> Int -> Int -> Cube
-cube_at g i' j' k'
-        | i' >= length (function_values g) = Cube g i' j' k' 0
-        | i' < 0                          = Cube g i' j' k' 0
-        | j' >= length ((function_values g) !! i') = Cube g i' j' k' 0
-        | j' < 0                                  = Cube g i' j' k' 0
-        | k' >= length (((function_values g) !! i') !! j') = Cube g i' j' k' 0
-        | k' < 0                                          = Cube g i' j' k' 0
-        | otherwise =
-            Cube g i' j' k' ((((function_values g) !! i') !! j') !! k')
 
--- These next three functions basically form a 'for' loop, looping
--- through the xs, ys, and zs in that order.
+p79_28_properties :: Test.Framework.Test
+p79_28_properties =
+  testGroup "p. 79, Section (2.8) Properties" [
+    testProperty "c3000 identity" prop_c3000_identity,
+    testProperty "c2010 identity" prop_c2010_identity,
+    testProperty "c2001 identity" prop_c2001_identity,
+    testProperty "c1020 identity" prop_c1020_identity,
+    testProperty "c1002 identity" prop_c1002_identity,
+    testProperty "c1011 identity" prop_c1011_identity ]
 
--- | The cubes_from_values function will return a function that takes
---   a list of values and returns a list of cubes. It could just as
---   well be written to take the values as a parameter; the omission
---   of the last parameter is known as an eta reduce.
-cubes_from_values :: Grid -> Int -> Int -> ([Double] -> [Cube])
-cubes_from_values g i' j' =
-    zipWith (reverse_cube g i' j') [0..]
 
--- | The cubes_from_planes function will return a function that takes
---   a list of planes and returns a list of cubes. It could just as
---   well be written to take the planes as a parameter; the omission
---   of the last parameter is known as an eta reduce.
-cubes_from_planes :: Grid -> Int -> ([[Double]] -> [[Cube]])
-cubes_from_planes g i' =
-    zipWith (cubes_from_values g i') [0..]
+edge_incidence_tests :: Test.Framework.Test
+edge_incidence_tests =
+    testGroup "Edge Incidence Tests" [
+      testProperty "t0 shares edge with t6" prop_t0_shares_edge_with_t6,
+      testProperty "t0 shares edge with t1" prop_t0_shares_edge_with_t1,
+      testProperty "t0 shares edge with t3" prop_t0_shares_edge_with_t3,
+      testProperty "t1 shares edge with t2" prop_t1_shares_edge_with_t2,
+      testProperty "t1 shares edge with t19" prop_t1_shares_edge_with_t19,
+      testProperty "t2 shares edge with t3" prop_t2_shares_edge_with_t3,
+      testProperty "t2 shares edge with t12" prop_t2_shares_edge_with_t12,
+      testProperty "t3 shares edge with t21" prop_t3_shares_edge_with_t21,
+      testProperty "t4 shares edge with t5" prop_t4_shares_edge_with_t5,
+      testProperty "t4 shares edge with t7" prop_t4_shares_edge_with_t7,
+      testProperty "t4 shares edge with t10" prop_t4_shares_edge_with_t10,
+      testProperty "t5 shares edge with t6" prop_t5_shares_edge_with_t6,
+      testProperty "t5 shares edge with t16" prop_t5_shares_edge_with_t16,
+      testProperty "t6 shares edge with t7" prop_t6_shares_edge_with_t7,
+      testProperty "t7 shares edge with t20" prop_t7_shares_edge_with_t20 ]
 
--- | Takes a grid as an argument, and returns a three-dimensional list
---   of cubes centered on its grid points.
-cubes :: Grid -> [[[Cube]]]
-cubes g = zipWith (cubes_from_planes g) [0..] (function_values g)
+cube_properties :: Test.Framework.Test
+cube_properties =
+  testGroup "Cube Properties" [
+    p79_26_properties,
+    p79_27_properties,
+    p79_28_properties,
+    edge_incidence_tests,
+    testProperty "opposite octant tetrahedra are disjoint (1)"
+      prop_opposite_octant_tetrahedra_disjoint1,
+    testProperty "opposite octant tetrahedra are disjoint (2)"
+      prop_opposite_octant_tetrahedra_disjoint2,
+    testProperty "opposite octant tetrahedra are disjoint (3)"
+     prop_opposite_octant_tetrahedra_disjoint3,
+    testProperty "opposite octant tetrahedra are disjoint (4)"
+      prop_opposite_octant_tetrahedra_disjoint4,
+    testProperty "opposite octant tetrahedra are disjoint (5)"
+      prop_opposite_octant_tetrahedra_disjoint5,
+    testProperty "opposite octant tetrahedra are disjoint (6)"
+      prop_opposite_octant_tetrahedra_disjoint6,
+    testProperty "all volumes positive" prop_all_volumes_positive,
+    testProperty "all volumes exact" prop_all_volumes_exact,
+    testProperty "v0 all equal" prop_v0_all_equal,
+    testProperty "interior values all identical"
+      prop_interior_values_all_identical,
+    testProperty "c-tilde_2100 rotation correct"
+      prop_c_tilde_2100_rotation_correct,
+    testProperty "c-tilde_2100 correct"
+      prop_c_tilde_2100_correct ]