X-Git-Url: https://gitweb.michael.orlitzky.com/?p=spline3.git;a=blobdiff_plain;f=src%2FCube.hs;h=8596b11846838b91f0b4db3667d130ac4152b7ed;hp=4c70a4126c7778e711806d200f3811d64c84c261;hb=83ef0aaeae074756e4ee90d72d3e27e74e136061;hpb=a73ad12eaff8510ae18ffafdefc0ae928603d5a5

diff --git a/src/Cube.hs b/src/Cube.hs
index 4c70a41..8596b11 100644
--- a/src/Cube.hs
+++ b/src/Cube.hs
@@ -1,374 +1,1228 @@
-module Cube
+module Cube (
+  Cube(..),
+  cube_properties,
+  find_containing_tetrahedron,
+  tetrahedra,
+  tetrahedron )
 where
 
-import Cardinal
-import qualified Face (Face(Face, v0, v1, v2, v3))
-import FunctionValues
-import Point
-import Tetrahedron hiding (c)
-import ThreeDimensional
-
-data Cube = Cube { h :: Double,
-                   i :: Int,
-                   j :: Int,
-                   k :: Int,
-                   fv :: FunctionValues }
+import Data.Maybe ( fromJust )
+import qualified Data.Vector as V (
+  Vector,
+  findIndex,
+  map,
+  minimum,
+  singleton,
+  snoc,
+  unsafeIndex)
+import Prelude hiding ( LT )
+import Test.Tasty ( TestTree, testGroup )
+import Test.Tasty.QuickCheck (
+  Arbitrary( arbitrary ),
+  Gen,
+  Positive( Positive ),
+  choose,
+  testProperty )
+import Cardinal (
+  Cardinal(F, B, L, R, D, T, FL, FR, FD, FT,
+           BL, BR, BD, BT, LD, LT, RD, RT, I),
+  ccwx,
+  ccwy,
+  ccwz,
+  cwx,
+  cwy,
+  cwz )
+import Comparisons ( (~=), (~~=) )
+import qualified Face ( Face(..), center )
+import FunctionValues ( FunctionValues, eval, rotate )
+import Misc ( all_equal, disjoint )
+import Point ( Point( Point ), dot )
+import Tetrahedron (
+  Tetrahedron(Tetrahedron, function_values, v0, v1, v2, v3),
+  barycenter,
+  c,
+  volume )
+
+data Cube = Cube { i  :: !Int,
+                   j  :: !Int,
+                   k  :: !Int,
+                   fv :: !FunctionValues,
+                   tetrahedra_volume :: !Double }
             deriving (Eq)
 
 
+instance Arbitrary Cube where
+    arbitrary = do
+      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 i' j' k' fv' tet_vol)
+      where
+        -- The idea here is that, when cubed in the volume formula,
+        -- these numbers don't overflow 64 bits. This number is not
+        -- magic in any other sense than that it does not cause test
+        -- failures, while 2^23 does.
+        coordmax = 4194304 :: Int -- 2^22
+        coordmin = -coordmax
+
+
 instance Show Cube where
-    show c =
+    show cube =
         "Cube_" ++ subscript ++ "\n" ++
-        " h: " ++ (show (h c)) ++ "\n" ++
-        " Center: " ++ (show (center c)) ++ "\n" ++
-        " xmin: " ++ (show (xmin c)) ++ "\n" ++
-        " xmax: " ++ (show (xmax c)) ++ "\n" ++
-        " ymin: " ++ (show (ymin c)) ++ "\n" ++
-        " ymax: " ++ (show (ymax c)) ++ "\n" ++
-        " zmin: " ++ (show (zmin c)) ++ "\n" ++
-        " zmax: " ++ (show (zmax c)) ++ "\n" ++
-        " fv: " ++ (show (Cube.fv c)) ++ "\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"
         where
           subscript =
-              (show (i c)) ++ "," ++ (show (j c)) ++ "," ++ (show (k c))
-
-
-empty_cube :: Cube
-empty_cube = Cube 0 0 0 0 empty_values
+              (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)
     where
-      i' = fromIntegral (i c) :: Double
-      delta = h c
+      i' = fromIntegral (i cube) :: Double
 
 -- | 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)
     where
-      i' = fromIntegral (i c) :: Double
-      delta = h c
+      i' = fromIntegral (i cube) :: Double
 
 -- | 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)
     where
-      j' = fromIntegral (j c) :: Double
-      delta = h c
+      j' = fromIntegral (j cube) :: Double
 
 -- | 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)
     where
-      j' = fromIntegral (j c) :: Double
-      delta = h c
+      j' = fromIntegral (j cube) :: Double
 
 -- | 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)
     where
-      k' = fromIntegral (k c) :: Double
-      delta = h c
+      k' = fromIntegral (k cube) :: Double
 
 -- | The top boundary of the cube. See Sorokina and Zeilfelder,
 --   p. 76.
 zmax :: Cube -> Double
-zmax c = (2*k' + 1)*delta / 2
-    where
-      k' = fromIntegral (k c) :: Double
-      delta = h c
-
-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)
-           where
-             delta = h c
-             i' = fromIntegral (i c) :: Double
-             j' = fromIntegral (j c) :: Double
-             k' = fromIntegral (k c) :: 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
-        | otherwise = True
+zmax cube = (k' + 1/2)
+    where
+      k' = fromIntegral (k cube) :: Double
 
 
+-- | The center of Cube_ijk coincides with v_ijk at
+--   (i, j, k). See Sorokina and Zeilfelder, p. 76.
+center :: Cube -> Point
+center cube =
+  Point x y z
+  where
+    x = fromIntegral (i cube) :: Double
+    y = fromIntegral (j cube) :: Double
+    z = fromIntegral (k cube) :: Double
+
 
 -- Face stuff.
 
 -- | The top (in the direction of z) face of the cube.
 top_face :: Cube -> Face.Face
-top_face c = Face.Face v0' v1' v2' v3'
+top_face cube = Face.Face v0' v1' v2' v3'
     where
-      delta = (1/2)*(h c)
-      v0' = (center c) + (delta, -delta, delta)
-      v1' = (center c) + (delta, delta, delta)
-      v2' = (center c) + (-delta, delta, delta)
-      v3' = (center c) + (-delta, -delta, delta)
+      delta = (1/2) :: Double
+      cc  = center cube
+      v0' = cc + ( Point delta (-delta) delta )
+      v1' = cc + ( Point delta delta delta )
+      v2' = cc + ( Point (-delta) delta delta )
+      v3' = cc + ( Point (-delta) (-delta) delta )
 
 
 
 -- | The back (in the direction of x) face of the cube.
 back_face :: Cube -> Face.Face
-back_face c = Face.Face v0' v1' v2' v3'
+back_face cube = Face.Face v0' v1' v2' v3'
     where
-      delta = (1/2)*(h c)
-      v0' = (center c) + (delta, -delta, -delta)
-      v1' = (center c) + (delta, delta, -delta)
-      v2' = (center c) + (delta, delta, delta)
-      v3' = (center c) + (delta, -delta, delta)
+      delta = (1/2) :: Double
+      cc  = center cube
+      v0' = cc + ( Point delta (-delta) (-delta) )
+      v1' = cc + ( Point delta delta (-delta) )
+      v2' = cc + ( Point delta delta delta )
+      v3' = cc + ( Point delta (-delta) delta )
 
 
 -- The bottom face (in the direction of -z) of the cube.
 down_face :: Cube -> Face.Face
-down_face c = Face.Face v0' v1' v2' v3'
+down_face cube = Face.Face v0' v1' v2' v3'
     where
-      delta = (1/2)*(h c)
-      v0' = (center c) + (-delta, -delta, -delta)
-      v1' = (center c) + (-delta, delta, -delta)
-      v2' = (center c) + (delta, delta, -delta)
-      v3' = (center c) + (delta, -delta, -delta)
+      delta = (1/2) :: Double
+      cc  = center cube
+      v0' = cc + ( Point (-delta) (-delta) (-delta) )
+      v1' = cc + ( Point (-delta) delta (-delta) )
+      v2' = cc + ( Point delta delta (-delta) )
+      v3' = cc + ( Point delta (-delta) (-delta) )
 
 
 
 -- | The front (in the direction of -x) face of the cube.
 front_face :: Cube -> Face.Face
-front_face c = Face.Face v0' v1' v2' v3'
+front_face cube = Face.Face v0' v1' v2' v3'
     where
-      delta = (1/2)*(h c)
-      v0' = (center c) + (-delta, -delta, delta)
-      v1' = (center c) + (-delta, delta, delta)
-      v2' = (center c) + (-delta, delta, -delta)
-      v3' = (center c) + (-delta, -delta, -delta)
+      delta = (1/2) :: Double
+      cc  = center cube
+      v0' = cc + ( Point (-delta) (-delta) delta )
+      v1' = cc + ( Point (-delta) delta delta )
+      v2' = cc + ( Point (-delta) delta (-delta) )
+      v3' = cc + ( Point (-delta) (-delta) (-delta) )
 
 -- | The left (in the direction of -y) face of the cube.
 left_face :: Cube -> Face.Face
-left_face c = Face.Face v0' v1' v2' v3'
+left_face cube = Face.Face v0' v1' v2' v3'
     where
-      delta = (1/2)*(h c)
-      v0' = (center c) + (-delta, -delta, delta)
-      v1' = (center c) + (delta, -delta, delta)
-      v2' = (center c) + (delta, -delta, -delta)
-      v3' = (center c) + (-delta, -delta, -delta)
+      delta = (1/2) :: Double
+      cc  = center cube
+      v0' = cc + ( Point delta (-delta) delta )
+      v1' = cc + ( Point (-delta) (-delta) delta )
+      v2' = cc + ( Point (-delta) (-delta) (-delta) )
+      v3' = cc + ( Point delta (-delta) (-delta) )
 
 
 -- | The right (in the direction of y) face of the cube.
 right_face :: Cube -> Face.Face
-right_face c = Face.Face v0' v1' v2' v3'
-    where
-      delta = (1/2)*(h c)
-      v0' = (center c) + (-delta, delta, -delta)
-      v1' = (center c) + (delta, delta, -delta)
-      v2' = (center c) + (delta, delta, delta)
-      v3' = (center c) + (-delta, delta, delta)
-
-
-tetrahedron0 :: Cube -> Tetrahedron
-tetrahedron0 c =
-    Tetrahedron (Cube.fv c) v0' v1' v2' v3'
-    where
-      v0' = center c
-      v1' = center (front_face c)
-      v2' = Face.v0 (front_face c)
-      v3' = Face.v1 (front_face c)
-
-tetrahedron1 :: Cube -> Tetrahedron
-tetrahedron1 c =
-    Tetrahedron fv' v0' v1' v2' v3'
-    where
-      v0' = center c
-      v1' = center (front_face c)
-      v2' = Face.v1 (front_face c)
-      v3' = Face.v2 (front_face c)
-      fv' = rotate (Cube.fv c) ccwx
-
-tetrahedron2 :: Cube -> Tetrahedron
-tetrahedron2 c =
-    Tetrahedron fv' v0' v1' v2' v3'
-    where
-      v0' = center c
-      v1' = center (front_face c)
-      v2' = Face.v2 (front_face c)
-      v3' = Face.v3 (front_face c)
-      fv' = rotate (Cube.fv c) (ccwx . ccwx)
-
-tetrahedron3 :: Cube -> Tetrahedron
-tetrahedron3 c =
-    Tetrahedron fv' v0' v1' v2' v3'
-    where
-      v0' = center c
-      v1' = center (front_face c)
-      v2' = Face.v3 (front_face c)
-      v3' = Face.v0 (front_face c)
-      fv' = rotate (Cube.fv c) cwx
-
-tetrahedron4 :: Cube -> Tetrahedron
-tetrahedron4 c =
-    Tetrahedron fv' v0' v1' v2' v3'
-    where
-      v0' = center c
-      v1' = center (top_face c)
-      v2' = Face.v0 (top_face c)
-      v3' = Face.v1 (top_face c)
-      fv' = rotate (Cube.fv c) cwy
-
-tetrahedron5 :: Cube -> Tetrahedron
-tetrahedron5 c =
-    Tetrahedron fv' v0' v1' v2' v3'
-    where
-      v0' = center c
-      v1' = center (top_face c)
-      v2' = Face.v1 (top_face c)
-      v3' = Face.v2 (top_face c)
-      fv' = rotate (Tetrahedron.fv (tetrahedron4 c)) ccwz
-
-tetrahedron6 :: Cube -> Tetrahedron
-tetrahedron6 c =
-    Tetrahedron fv' v0' v1' v2' v3'
-    where
-      v0' = center c
-      v1' = center (top_face c)
-      v2' = Face.v2 (top_face c)
-      v3' = Face.v3 (top_face c)
-      fv' = rotate (Tetrahedron.fv (tetrahedron4 c)) (ccwz . ccwz)
-
-tetrahedron7 :: Cube -> Tetrahedron
-tetrahedron7 c =
-    Tetrahedron fv' v0' v1' v2' v3'
-    where
-      v0' = center c
-      v1' = center (top_face c)
-      v2' = Face.v3 (top_face c)
-      v3' = Face.v0 (top_face c)
-      fv' = rotate (Tetrahedron.fv (tetrahedron4 c)) cwz
-
-tetrahedron8 :: Cube -> Tetrahedron
-tetrahedron8 c =
-    Tetrahedron fv' v0' v1' v2' v3'
-    where
-      v0' = center c
-      v1' = center (back_face c)
-      v2' = Face.v0 (back_face c)
-      v3' = Face.v1 (back_face c)
-      fv' = rotate (Tetrahedron.fv (tetrahedron4 c)) cwy
-
-tetrahedron9 :: Cube -> Tetrahedron
-tetrahedron9 c =
-    Tetrahedron fv' v0' v1' v2' v3'
-    where
-      v0' = center c
-      v1' = center (back_face c)
-      v2' = Face.v1 (back_face c)
-      v3' = Face.v2 (back_face c)
-      fv' = rotate (Tetrahedron.fv (tetrahedron8 c)) ccwx
-
-tetrahedron10 :: Cube -> Tetrahedron
-tetrahedron10 c =
-    Tetrahedron fv' v0' v1' v2' v3'
-    where
-      v0' = center c
-      v1' = center (back_face c)
-      v2' = Face.v2 (back_face c)
-      v3' = Face.v3 (back_face c)
-      fv' = rotate (Tetrahedron.fv (tetrahedron8 c)) (ccwx . ccwx)
-
-
-tetrahedron11 :: Cube -> Tetrahedron
-tetrahedron11 c =
-    Tetrahedron fv' v0' v1' v2' v3'
-    where
-      v0' = center c
-      v1' = center (back_face c)
-      v2' = Face.v3 (back_face c)
-      v3' = Face.v0 (back_face c)
-      fv' = rotate (Tetrahedron.fv (tetrahedron8 c)) cwx
-
-
-tetrahedron12 :: Cube -> Tetrahedron
-tetrahedron12 c =
-    Tetrahedron fv' v0' v1' v2' v3'
-    where
-      v0' = center c
-      v1' = center (down_face c)
-      v2' = Face.v0 (down_face c)
-      v3' = Face.v1 (down_face c)
-      fv' = rotate (Tetrahedron.fv (tetrahedron8 c)) cwy
-
-
-tetrahedron13 :: Cube -> Tetrahedron
-tetrahedron13 c =
-    Tetrahedron fv' v0' v1' v2' v3'
-    where
-      v0' = center c
-      v1' = center (down_face c)
-      v2' = Face.v1 (down_face c)
-      v3' = Face.v2 (down_face c)
-      fv' = rotate (Tetrahedron.fv (tetrahedron12 c)) ccwz
-
-
-tetrahedron14 :: Cube -> Tetrahedron
-tetrahedron14 c =
-    Tetrahedron fv' v0' v1' v2' v3'
-    where
-      v0' = center c
-      v1' = center (down_face c)
-      v2' = Face.v2 (down_face c)
-      v3' = Face.v3 (down_face c)
-      fv' = rotate (Tetrahedron.fv (tetrahedron13 c)) (ccwz . ccwz)
-
-
-tetrahedron15 :: Cube -> Tetrahedron
-tetrahedron15 c =
-    Tetrahedron fv' v0' v1' v2' v3'
-    where
-      v0' = center c
-      v1' = center (down_face c)
-      v2' = Face.v3 (down_face c)
-      v3' = Face.v0 (down_face c)
-      fv' = rotate (Tetrahedron.fv (tetrahedron12 c)) cwz
-
-
-tetrahedrons :: Cube -> [Tetrahedron]
-tetrahedrons c =
-    [tetrahedron0 c,
-     tetrahedron1 c,
-     tetrahedron2 c,
-     tetrahedron3 c,
-     tetrahedron4 c,
-     tetrahedron5 c,
-     tetrahedron6 c,
-     tetrahedron7 c,
-     tetrahedron8 c,
-     tetrahedron9 c,
-     tetrahedron10 c,
-     tetrahedron11 c,
-     tetrahedron12 c,
-     tetrahedron13 c,
-     tetrahedron14 c,
-     tetrahedron15 c
-                --  tetrahedron16 c,
-                --  tetrahedron17 c,
-                --  tetrahedron18 c,
-                --  tetrahedron19 c,
-                --  tetrahedron20 c,
-                --  tetrahedron21 c,
-                --  tetrahedron21 c,
-                --  tetrahedron22 c,
-                --  tetrahedron23 c,
-                --  tetrahedron24 c
-    ]
+right_face cube = Face.Face v0' v1' v2' v3'
+    where
+      delta = (1/2) :: Double
+      cc  = center cube
+      v0' = cc + ( Point (-delta) delta delta)
+      v1' = cc + ( Point delta  delta delta )
+      v2' = cc + ( Point delta delta (-delta) )
+      v3' = cc + ( Point (-delta) delta (-delta) )
+
+
+tetrahedron :: Cube -> Int -> Tetrahedron
+
+tetrahedron cube 0 =
+    Tetrahedron (fv cube) v0' v1' v2' v3' vol
+    where
+      v0' = center cube
+      ff  = front_face cube
+      v1' = Face.center ff
+      v2' = Face.v0 ff
+      v3' = Face.v1 ff
+      vol = tetrahedra_volume cube
+
+tetrahedron cube 1 =
+    Tetrahedron fv' v0' v1' v2' v3' vol
+    where
+      v0' = center cube
+      ff  = front_face cube
+      v1' = Face.center ff
+      v2' = Face.v1 ff
+      v3' = Face.v2 ff
+      fv' = rotate ccwx (fv cube)
+      vol = tetrahedra_volume cube
+
+tetrahedron cube 2 =
+    Tetrahedron fv' v0' v1' v2' v3' vol
+    where
+      v0' = center cube
+      ff  = front_face cube
+      v1' = Face.center ff
+      v2' = Face.v2 ff
+      v3' = Face.v3 ff
+      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
+      ff  = front_face cube
+      v1' = Face.center ff
+      v2' = Face.v3 ff
+      v3' = Face.v0 ff
+      fv' = rotate cwx (fv cube)
+      vol = tetrahedra_volume cube
+
+tetrahedron cube 4 =
+    Tetrahedron fv' v0' v1' v2' v3' vol
+    where
+      v0' = center cube
+      tf  = top_face cube
+      v1' = Face.center tf
+      v2' = Face.v0 tf
+      v3' = Face.v1 tf
+      fv' = rotate cwy (fv cube)
+      vol = tetrahedra_volume cube
+
+tetrahedron cube 5 =
+    Tetrahedron fv' v0' v1' v2' v3' vol
+    where
+      v0' = center cube
+      tf  = top_face cube
+      v1' = Face.center tf
+      v2' = Face.v1 tf
+      v3' = Face.v2 tf
+      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
+      tf  = top_face cube
+      v1' = Face.center tf
+      v2' = Face.v2 tf
+      v3' = Face.v3 tf
+      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
+      tf  = top_face cube
+      v1' = Face.center tf
+      v2' = Face.v3 tf
+      v3' = Face.v0 tf
+      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
+      bf  = back_face cube
+      v1' = Face.center bf
+      v2' = Face.v0 bf
+      v3' = Face.v1 bf
+      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
+      bf  = back_face cube
+      v1' = Face.center bf
+      v2' = Face.v1 bf
+      v3' = Face.v2 bf
+      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
+      bf  = back_face cube
+      v1' = Face.center bf
+      v2' = Face.v2 bf
+      v3' = Face.v3 bf
+      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
+      bf  = back_face cube
+      v1' = Face.center bf
+      v2' = Face.v3 bf
+      v3' = Face.v0 bf
+      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
+      df  = down_face cube
+      v1' = Face.center df
+      v2' = Face.v0 df
+      v3' = Face.v1 df
+      fv' = rotate ccwy $ fv cube
+      vol = tetrahedra_volume cube
+
+tetrahedron cube 13 =
+    Tetrahedron fv' v0' v1' v2' v3' vol
+    where
+      v0' = center cube
+      df  = down_face cube
+      v1' = Face.center df
+      v2' = Face.v1 df
+      v3' = Face.v2 df
+      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
+      df  = down_face cube
+      v1' = Face.center df
+      v2' = Face.v2 df
+      v3' = Face.v3 df
+      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
+      df  = down_face cube
+      v1' = Face.center df
+      v2' = Face.v3 df
+      v3' = Face.v0 df
+      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
+      rf  = right_face cube
+      v1' = Face.center rf
+      v2' = Face.v0 rf
+      v3' = Face.v1 rf
+      fv' = rotate ccwz $ fv cube
+      vol = tetrahedra_volume cube
+
+tetrahedron cube 17 =
+    Tetrahedron fv' v0' v1' v2' v3' vol
+    where
+      v0' = center cube
+      rf  = right_face cube
+      v1' = Face.center rf
+      v2' = Face.v1 rf
+      v3' = Face.v2 rf
+      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
+      rf  = right_face cube
+      v1' = Face.center rf
+      v2' = Face.v2 rf
+      v3' = Face.v3 rf
+      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
+      rf  = right_face cube
+      v1' = Face.center rf
+      v2' = Face.v3 rf
+      v3' = Face.v0 rf
+      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
+      lf  = left_face cube
+      v1' = Face.center lf
+      v2' = Face.v0 lf
+      v3' = Face.v1 lf
+      fv' = rotate cwz $ fv cube
+      vol = tetrahedra_volume cube
+
+tetrahedron cube 21 =
+    Tetrahedron fv' v0' v1' v2' v3' vol
+    where
+      v0' = center cube
+      lf  = left_face cube
+      v1' = Face.center lf
+      v2' = Face.v1 lf
+      v3' = Face.v2 lf
+      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
+      lf  = left_face cube
+      v1' = Face.center lf
+      v2' = Face.v2 lf
+      v3' = Face.v3 lf
+      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
+      lf  = left_face cube
+      v1' = Face.center lf
+      v2' = Face.v3 lf
+      v3' = Face.v0 lf
+      fv' = rotate cwz $ rotate cwy
+                       $ fv cube
+      vol = tetrahedra_volume cube
+
+
+-- 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 (Point _ _ 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 (Point 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 (Point _ 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.
+--
+{-# INLINE find_containing_tetrahedron #-}
+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 :: V.Vector Tetrahedron
+    candidates
+      | front_half =
+        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
+
+      | otherwise = -- back 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 Euclidean distance here to save
+    -- a sqrt(). So, "distances" below really means "distances
+    -- squared."
+    distances :: V.Vector Double
+    distances = V.map ((dot p) . barycenter) candidates
+
+    shortest_distance :: Double
+    shortest_distance = V.minimum distances
+
+    -- Compute the index of the tetrahedron with the center closest to
+    -- p. This is a bad algorithm, but don't change it! If you make it
+    -- smarter by finding the index of shortest_distance in distances
+    -- (this should give the same answer and avoids recomputing the
+    -- dot product), the program gets slower. Seriously!
+    lucky_idx :: Maybe Int
+    lucky_idx = V.findIndex
+                  (\t -> (barycenter t) `dot` p == shortest_distance)
+                  candidates
+
+
+
+
+
+
+-- * 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 =
+    all (>= 0) volumes
+    where
+      ts = tetrahedra cube
+      volumes = map volume ts
+
+
+-- | 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.
+prop_all_volumes_exact :: Cube -> Bool
+prop_all_volumes_exact cube =
+    and [volume t ~~= 1/24 | t <- tetrahedra 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
+
+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
+
+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
+
+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
+
+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
+
+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
+
+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
+
+
+p79_26_properties :: TestTree
+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 :: TestTree
+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 ]
+
+
+p79_28_properties :: TestTree
+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 ]
+
+
+edge_incidence_tests :: TestTree
+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 ]
+
+cube_properties :: TestTree
+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 ]