X-Git-Url: https://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=src%2FTests%2FCube.hs;h=46b2530ab82053fae93fcf68de6e839ed8cf6304;hb=b36cb982a689fa31b1d0c4e3e9994f36cc4b26b2;hp=11df961836d709565cb8f68e3f3b319a09376e4a;hpb=a032d4426427de084931d194248f4086b12c11ce;p=spline3.git

diff --git a/src/Tests/Cube.hs b/src/Tests/Cube.hs
index 11df961..46b2530 100644
--- a/src/Tests/Cube.hs
+++ b/src/Tests/Cube.hs
@@ -2,257 +2,37 @@ module Tests.Cube
 where
 
 import Prelude hiding (LT)
-import Test.QuickCheck
 
 import Cardinal
 import Comparisons
-import Cube
+import Cube hiding (i, j, k)
 import FunctionValues
 import Misc (all_equal)
 import Tests.FunctionValues ()
 import Tetrahedron (b0, b1, b2, b3, c, fv,
                     v0, v1, v2, v3, volume)
 
-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
-      return (Cube h' i' j' k' fv')
-        where
-          coordmin = -268435456 -- -(2^29 / 2)
-          coordmax = 268435456  -- +(2^29 / 2)
 
 
 -- Quickcheck tests.
 
--- | Since the grid size is necessarily positive, all tetrahedrons
+-- | 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 = tetrahedrons cube
+      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_tetrahedron0_volumes_exact :: Cube -> Bool
-prop_tetrahedron0_volumes_exact cube =
-    volume (tetrahedron0 cube) ~~= (1/24)*(delta^(3::Int))
-    where
-      delta = h cube
-
-
--- | 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_tetrahedron1_volumes_exact :: Cube -> Bool
-prop_tetrahedron1_volumes_exact cube =
-    volume (tetrahedron1 cube) ~~= (1/24)*(delta^(3::Int))
-    where
-      delta = h cube
-
--- | 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_tetrahedron2_volumes_exact :: Cube -> Bool
-prop_tetrahedron2_volumes_exact cube =
-    volume (tetrahedron2 cube) ~~= (1/24)*(delta^(3::Int))
-    where
-      delta = h cube
-
--- | 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_tetrahedron3_volumes_exact :: Cube -> Bool
-prop_tetrahedron3_volumes_exact cube =
-    volume (tetrahedron3 cube) ~~= (1/24)*(delta^(3::Int))
-    where
-      delta = h cube
-
--- | 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_tetrahedron4_volumes_exact :: Cube -> Bool
-prop_tetrahedron4_volumes_exact cube =
-    volume (tetrahedron4 cube) ~~= (1/24)*(delta^(3::Int))
-    where
-      delta = h cube
-
--- | 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_tetrahedron5_volumes_exact :: Cube -> Bool
-prop_tetrahedron5_volumes_exact cube =
-    volume (tetrahedron5 cube) ~~= (1/24)*(delta^(3::Int))
-    where
-      delta = h cube
-
--- | 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_tetrahedron6_volumes_exact :: Cube -> Bool
-prop_tetrahedron6_volumes_exact cube =
-    volume (tetrahedron6 cube) ~~= (1/24)*(delta^(3::Int))
-    where
-      delta = h cube
-
--- | 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_tetrahedron7_volumes_exact :: Cube -> Bool
-prop_tetrahedron7_volumes_exact cube =
-    volume (tetrahedron7 cube) ~~= (1/24)*(delta^(3::Int))
-    where
-      delta = h cube
-
--- | 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_tetrahedron8_volumes_exact :: Cube -> Bool
-prop_tetrahedron8_volumes_exact cube =
-    volume (tetrahedron8 cube) ~~= (1/24)*(delta^(3::Int))
-    where
-      delta = h cube
-
--- | 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_tetrahedron9_volumes_exact :: Cube -> Bool
-prop_tetrahedron9_volumes_exact cube =
-    volume (tetrahedron9 cube) ~~= (1/24)*(delta^(3::Int))
-    where
-      delta = h cube
-
--- | 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_tetrahedron10_volumes_exact :: Cube -> Bool
-prop_tetrahedron10_volumes_exact cube =
-    volume (tetrahedron10 cube) ~~= (1/24)*(delta^(3::Int))
-    where
-      delta = h cube
-
--- | 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_tetrahedron11_volumes_exact :: Cube -> Bool
-prop_tetrahedron11_volumes_exact cube =
-    volume (tetrahedron11 cube) ~~= (1/24)*(delta^(3::Int))
-    where
-      delta = h cube
-
--- | 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_tetrahedron12_volumes_exact :: Cube -> Bool
-prop_tetrahedron12_volumes_exact cube =
-    volume (tetrahedron12 cube) ~~= (1/24)*(delta^(3::Int))
-    where
-      delta = h cube
-
--- | 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_tetrahedron13_volumes_exact :: Cube -> Bool
-prop_tetrahedron13_volumes_exact cube =
-    volume (tetrahedron13 cube) ~~= (1/24)*(delta^(3::Int))
-    where
-      delta = h cube
-
--- | 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_tetrahedron14_volumes_exact :: Cube -> Bool
-prop_tetrahedron14_volumes_exact cube =
-    volume (tetrahedron14 cube) ~~= (1/24)*(delta^(3::Int))
-    where
-      delta = h cube
-
--- | 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_tetrahedron15_volumes_exact :: Cube -> Bool
-prop_tetrahedron15_volumes_exact cube =
-    volume (tetrahedron15 cube) ~~= (1/24)*(delta^(3::Int))
-    where
-      delta = h cube
-
--- | 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_tetrahedron16_volumes_exact :: Cube -> Bool
-prop_tetrahedron16_volumes_exact cube =
-    volume (tetrahedron16 cube) ~~= (1/24)*(delta^(3::Int))
-    where
-      delta = h cube
-
--- | 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_tetrahedron17_volumes_exact :: Cube -> Bool
-prop_tetrahedron17_volumes_exact cube =
-    volume (tetrahedron17 cube) ~~= (1/24)*(delta^(3::Int))
-    where
-      delta = h cube
-
--- | 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_tetrahedron18_volumes_exact :: Cube -> Bool
-prop_tetrahedron18_volumes_exact cube =
-    volume (tetrahedron18 cube) ~~= (1/24)*(delta^(3::Int))
-    where
-      delta = h cube
-
--- | 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_tetrahedron19_volumes_exact :: Cube -> Bool
-prop_tetrahedron19_volumes_exact cube =
-    volume (tetrahedron19 cube) ~~= (1/24)*(delta^(3::Int))
-    where
-      delta = h cube
-
--- | 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_tetrahedron20_volumes_exact :: Cube -> Bool
-prop_tetrahedron20_volumes_exact cube =
-    volume (tetrahedron20 cube) ~~= (1/24)*(delta^(3::Int))
-    where
-      delta = h cube
-
--- | 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_tetrahedron21_volumes_exact :: Cube -> Bool
-prop_tetrahedron21_volumes_exact cube =
-    volume (tetrahedron21 cube) ~~= (1/24)*(delta^(3::Int))
-    where
-      delta = h cube
-
--- | 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_tetrahedron22_volumes_exact :: Cube -> Bool
-prop_tetrahedron22_volumes_exact cube =
-    volume (tetrahedron22 cube) ~~= (1/24)*(delta^(3::Int))
-    where
-      delta = h cube
-
--- | 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_tetrahedron23_volumes_exact :: Cube -> Bool
-prop_tetrahedron23_volumes_exact cube =
-    volume (tetrahedron23 cube) ~~= (1/24)*(delta^(3::Int))
+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
 
@@ -260,8 +40,8 @@ prop_tetrahedron23_volumes_exact cube =
 prop_v0_all_equal :: Cube -> Bool
 prop_v0_all_equal cube = (v0 t0) == (v0 t1)
     where
-      t0 = head (tetrahedrons cube) -- Doesn't matter which two we choose.
-      t1 = head $ tail (tetrahedrons cube)
+      t0 = head (tetrahedra cube) -- Doesn't matter which two we choose.
+      t1 = head $ tail (tetrahedra cube)
 
 
 -- | This pretty much repeats the prop_all_volumes_positive property,
@@ -409,9 +189,9 @@ prop_tetrahedron23_volumes_positive cube =
     volume (tetrahedron23 cube) > 0
 
 
--- | Given in Sorokina and Zeilfelder, p. 79. 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
+-- | Given in Sorokina and Zeilfelder, p. 79, (2.6). Note that the
+--   third and fourth indices of c-t1 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
@@ -422,60 +202,65 @@ prop_c0120_identity1 cube =
        t3 = tetrahedron3 cube
 
 
--- | Given in Sorokina and Zeilfelder, p. 79. Repeats
---   prop_c0120_identity2 with tetrahedrons 3 and 2.
+-- | 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 t3 0 1 2 0 ~= (c t3 0 0 2 1 + c t2 0 0 1 2) / 2
+   c t1 0 1 2 0 ~= (c t1 0 0 2 1 + c t0 0 0 1 2) / 2
      where
-       t3 = tetrahedron3 cube
-       t2 = tetrahedron2 cube
-
--- | Given in Sorokina and Zeilfelder, p. 79. Repeats
---   prop_c0120_identity1 with tetrahedrons 2 and 1.
+       t0 = tetrahedron0 cube
+       t1 = tetrahedron1 cube
+            
+-- | 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
-       t2 = tetrahedron2 cube
        t1 = tetrahedron1 cube
+       t2 = tetrahedron2 cube
 
-
--- | Given in Sorokina and Zeilfelder, p. 79. Repeats
---   prop_c0120_identity1 with tetrahedrons 4 and 7.
+-- | 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 t4 0 1 2 0 ~= (c t4 0 0 2 1 + c t7 0 0 1 2) / 2
+   c t3 0 1 2 0 ~= (c t3 0 0 2 1 + c t2 0 0 1 2) / 2
      where
-       t4 = tetrahedron4 cube
-       t7 = tetrahedron7 cube
+       t2 = tetrahedron2 cube
+       t3 = tetrahedron3 cube
 
 
--- | Given in Sorokina and Zeilfelder, p. 79. Repeats
---   prop_c0120_identity1 with tetrahedrons 7 and 6.
+-- | 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 t7 0 1 2 0 ~= (c t7 0 0 2 1 + c t6 0 0 1 2) / 2
-     where
-       t7 = tetrahedron7 cube
-       t6 = tetrahedron6 cube
-
+    c t5 0 1 2 0 ~= (c t5 0 0 2 1 + c t4 0 0 1 2) / 2
+    where
+      t4 = tetrahedron4 cube
+      t5 = tetrahedron5 cube
 
--- | Given in Sorokina and Zeilfelder, p. 79. Repeats
---   prop_c0120_identity1 with tetrahedrons 6 and 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
-       t6 = tetrahedron6 cube
        t5 = tetrahedron5 cube
+       t6 = tetrahedron6 cube
 
 
--- | Given in Sorokina and Zeilfelder, p. 79. 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!
+-- -- | 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 = tetrahedron6 cube
+       t7 = tetrahedron7 cube
+
+
+-- | 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
@@ -484,11 +269,8 @@ prop_c0210_identity1 cube =
         t3 = tetrahedron3 cube
 
 
--- | Given in Sorokina and Zeilfelder, p. 79. 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!
+-- | 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
@@ -497,11 +279,8 @@ prop_c0300_identity1 cube =
         t3 = tetrahedron3 cube
 
 
--- | Given in Sorokina and Zeilfelder, p. 79. 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!
+-- | 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
@@ -510,11 +289,8 @@ prop_c1110_identity cube =
         t3 = tetrahedron3 cube
 
 
--- | Given in Sorokina and Zeilfelder, p. 79. 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!
+-- | 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
@@ -523,11 +299,8 @@ prop_c1200_identity1 cube =
         t3 = tetrahedron3 cube
 
 
--- | Given in Sorokina and Zeilfelder, p. 79. 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!
+-- | 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
@@ -537,11 +310,12 @@ prop_c2100_identity1 cube =
 
 
 
--- | Given in Sorokina and Zeilfelder, p. 79. Note that the third and
---   fourth indices of c-t1 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,v3\> and v2,v2-tilde point
---   in opposite directions, one of them has to have negative volume!
+-- | 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
@@ -550,11 +324,8 @@ prop_c0102_identity1 cube =
         t1 = tetrahedron1 cube
 
 
--- | Given in Sorokina and Zeilfelder, p. 79. Note that the third and
---   fourth indices of c-t1 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,v3\> and v2,v2-tilde point
---   in opposite directions, one of them has to have negative volume!
+-- | 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
@@ -563,11 +334,8 @@ prop_c0201_identity1 cube =
         t1 = tetrahedron1 cube
 
 
--- | Given in Sorokina and Zeilfelder, p. 79. Note that the third and
---   fourth indices of c-t1 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,v3\> and v2,v2-tilde point
---   in opposite directions, one of them has to have negative volume!
+-- | 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
@@ -576,11 +344,8 @@ prop_c0300_identity2 cube =
         t1 = tetrahedron1 cube
 
 
--- | Given in Sorokina and Zeilfelder, p. 79. Note that the third and
---   fourth indices of c-t1 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,v3\> and v2,v2-tilde point
---   in opposite directions, one of them has to have negative volume!
+-- | 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
@@ -589,11 +354,8 @@ prop_c1101_identity cube =
         t1 = tetrahedron1 cube
 
 
--- | Given in Sorokina and Zeilfelder, p. 79. Note that the third and
---   fourth indices of c-t1 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,v3\> and v2,v2-tilde point
---   in opposite directions, one of them has to have negative volume!
+-- | 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
@@ -602,11 +364,8 @@ prop_c1200_identity2 cube =
         t1 = tetrahedron1 cube
 
 
--- | Given in Sorokina and Zeilfelder, p. 79. Note that the third and
---   fourth indices of c-t1 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,v3\> and v2,v2-tilde point
---   in opposite directions, one of them has to have negative volume!
+-- | 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
@@ -615,54 +374,71 @@ prop_c2100_identity2 cube =
         t1 = tetrahedron1 cube
 
 
--- | Given in Sorokina and Zeilfelder, p. 79.
+-- | 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)
+    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 = tetrahedron0 cube
-        t6 = (tetrahedron6 cube) { v2 = (v3 t6), v3 = (v2 t6) }
+        t6 = tetrahedron6 cube
 
 
--- | Given in Sorokina and Zeilfelder, p. 79.
+-- | 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 1 0 - ((c t0 1 0 2 0 + c t0 1 0 1 1)/ 2)
+    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 = tetrahedron0 cube
         t6 = tetrahedron6 cube
 
 
--- | Given in Sorokina and Zeilfelder, p. 79.
+-- | 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 0 1 - ((c t0 1 0 0 2 + c t0 1 0 1 1)/ 2)
+    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 = tetrahedron0 cube
         t6 = tetrahedron6 cube
 
--- | Given in Sorokina and Zeilfelder, p. 79.
+
+-- | 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 2 0 - ((c t0 0 0 3 0 + c t0 0 0 2 1)/ 2)
+    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 = tetrahedron0 cube
         t6 = tetrahedron6 cube
 
 
--- | Given in Sorokina and Zeilfelder, p. 79.
+-- | 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 0 2 - ((c t0 0 0 0 3 + c t0 0 0 1 2)/ 2)
+    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 = tetrahedron0 cube
         t6 = tetrahedron6 cube
 
 
--- | Given in Sorokina and Zeilfelder, p. 79.
+-- | 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)
+    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 = tetrahedron0 cube
         t6 = tetrahedron6 cube
@@ -670,20 +446,19 @@ prop_c1011_identity cube =
 
 
 -- | Given in Sorokina and Zeilfelder, p. 78.
--- prop_cijk1_identity :: Cube -> Bool
--- prop_cijk1_identity cube =
---      and [ c t0 i j k 1 ~=
---                  (c t1 (i+1) j k 0) * ((b0 t0) (v3 t1)) +
---                  (c t1 i (j+1) k 0) * ((b1 t0) (v3 t1)) +
---                  (c t1 i j (k+1) 0) * ((b2 t0) (v3 t1)) +
---                  (c t1 i j k 1) * ((b3 t0) (v3 t1)) | i <- [0..2],
---                                                       j <- [0..2],
---                                                       k <- [0..2],
---                                                       i + j + k == 2]
---       where
---         t0 = tetrahedron0 cube
---         t1 = tetrahedron1 cube
-
+prop_cijk1_identity :: Cube -> Bool
+prop_cijk1_identity cube =
+     and [ c t0 i j k 1 ~=
+                 (c t1 (i+1) j k 0) * ((b0 t0) (v3 t1)) +
+                 (c t1 i (j+1) k 0) * ((b1 t0) (v3 t1)) +
+                 (c t1 i j (k+1) 0) * ((b2 t0) (v3 t1)) +
+                 (c t1 i j k 1) * ((b3 t0) (v3 t1)) | i <- [0..2],
+                                                      j <- [0..2],
+                                                      k <- [0..2],
+                                                      i + j + k == 2]
+      where
+        t0 = tetrahedron0 cube
+        t1 = tetrahedron1 cube
 
 
 -- | The function values at the interior should be the same for all tetrahedra.
@@ -693,30 +468,30 @@ prop_interior_values_all_identical cube =
                i9, i10, i11, i12, i13, i14, i15, i16,
                i17, i18, i19, i20, i21, i22, i23]
     where
-      i0 = eval (Tetrahedron.fv (tetrahedron0 cube)) $ I
-      i1 = eval (Tetrahedron.fv (tetrahedron1 cube)) $ I
-      i2 = eval (Tetrahedron.fv (tetrahedron2 cube)) $ I
-      i3 = eval (Tetrahedron.fv (tetrahedron3 cube)) $ I
-      i4 = eval (Tetrahedron.fv (tetrahedron4 cube)) $ I
-      i5 = eval (Tetrahedron.fv (tetrahedron5 cube)) $ I
-      i6 = eval (Tetrahedron.fv (tetrahedron6 cube)) $ I
-      i7 = eval (Tetrahedron.fv (tetrahedron7 cube)) $ I
-      i8 = eval (Tetrahedron.fv (tetrahedron8 cube)) $ I
-      i9 = eval (Tetrahedron.fv (tetrahedron9 cube)) $ I
-      i10 = eval (Tetrahedron.fv (tetrahedron10 cube)) $ I
-      i11 = eval (Tetrahedron.fv (tetrahedron11 cube)) $ I
-      i12 = eval (Tetrahedron.fv (tetrahedron12 cube)) $ I
-      i13 = eval (Tetrahedron.fv (tetrahedron13 cube)) $ I
-      i14 = eval (Tetrahedron.fv (tetrahedron14 cube)) $ I
-      i15 = eval (Tetrahedron.fv (tetrahedron15 cube)) $ I
-      i16 = eval (Tetrahedron.fv (tetrahedron16 cube)) $ I
-      i17 = eval (Tetrahedron.fv (tetrahedron17 cube)) $ I
-      i18 = eval (Tetrahedron.fv (tetrahedron18 cube)) $ I
-      i19 = eval (Tetrahedron.fv (tetrahedron19 cube)) $ I
-      i20 = eval (Tetrahedron.fv (tetrahedron20 cube)) $ I
-      i21 = eval (Tetrahedron.fv (tetrahedron21 cube)) $ I
-      i22 = eval (Tetrahedron.fv (tetrahedron22 cube)) $ I
-      i23 = eval (Tetrahedron.fv (tetrahedron23 cube)) $ I
+      i0 = eval (Tetrahedron.fv (tetrahedron0 cube)) I
+      i1 = eval (Tetrahedron.fv (tetrahedron1 cube)) I
+      i2 = eval (Tetrahedron.fv (tetrahedron2 cube)) I
+      i3 = eval (Tetrahedron.fv (tetrahedron3 cube)) I
+      i4 = eval (Tetrahedron.fv (tetrahedron4 cube)) I
+      i5 = eval (Tetrahedron.fv (tetrahedron5 cube)) I
+      i6 = eval (Tetrahedron.fv (tetrahedron6 cube)) I
+      i7 = eval (Tetrahedron.fv (tetrahedron7 cube)) I
+      i8 = eval (Tetrahedron.fv (tetrahedron8 cube)) I
+      i9 = eval (Tetrahedron.fv (tetrahedron9 cube)) I
+      i10 = eval (Tetrahedron.fv (tetrahedron10 cube)) I
+      i11 = eval (Tetrahedron.fv (tetrahedron11 cube)) I
+      i12 = eval (Tetrahedron.fv (tetrahedron12 cube)) I
+      i13 = eval (Tetrahedron.fv (tetrahedron13 cube)) I
+      i14 = eval (Tetrahedron.fv (tetrahedron14 cube)) I
+      i15 = eval (Tetrahedron.fv (tetrahedron15 cube)) I
+      i16 = eval (Tetrahedron.fv (tetrahedron16 cube)) I
+      i17 = eval (Tetrahedron.fv (tetrahedron17 cube)) I
+      i18 = eval (Tetrahedron.fv (tetrahedron18 cube)) I
+      i19 = eval (Tetrahedron.fv (tetrahedron19 cube)) I
+      i20 = eval (Tetrahedron.fv (tetrahedron20 cube)) I
+      i21 = eval (Tetrahedron.fv (tetrahedron21 cube)) I
+      i22 = eval (Tetrahedron.fv (tetrahedron22 cube)) I
+      i23 = eval (Tetrahedron.fv (tetrahedron23 cube)) I
 
 
 -- | We know what (c t6 2 1 0 0) should be from Sorokina and Zeilfelder, p. 87.
@@ -753,9 +528,11 @@ prop_c_tilde_2100_rotation_correct cube =
 --   This test checks the actual value based on the FunctionValues of the cube.
 prop_c_tilde_2100_correct :: Cube -> Bool
 prop_c_tilde_2100_correct cube =
-    c t6 2 1 0 0 == (3/8)*int + (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)
+    c t6 2 1 0 0 == (3/8)*int
+                    + (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)
     where
       t0 = tetrahedron0 cube
       t6 = tetrahedron6 cube