X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=src%2FTests%2FCube.hs;h=2fd8cb67187c601fb63b3811bb3e7ad5cb6341fe;hb=3f0b6b7faecc561af0b7312a11c73a44a1b416f6;hp=26d4d33f5ab7e21f3b85fc032ae98113edd6b25d;hpb=2064e9a7da32813c6dce843127e2306b841df353;p=spline3.git

diff --git a/src/Tests/Cube.hs b/src/Tests/Cube.hs
index 26d4d33..2fd8cb6 100644
--- a/src/Tests/Cube.hs
+++ b/src/Tests/Cube.hs
@@ -1,291 +1,506 @@
 module Tests.Cube
 where
 
-import Debug.Trace (trace)
-import Test.QuickCheck
+import Prelude hiding (LT)
 
+import Cardinal
 import Comparisons
-import Cube
-import FunctionValues (FunctionValues)
-import Tests.FunctionValues ()
-import Tetrahedron (b0, b1, b2, b3, c,
-                    Tetrahedron(Tetrahedron),
+import Cube hiding (i, j, k)
+import FunctionValues
+import Misc (all_equal, disjoint)
+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
---   (which comprise cubes of positive volume) must have positive volume
---   as well.
-prop_all_volumes_positive :: Cube -> Bool
-prop_all_volumes_positive c =
-    null nonpositive_volumes
-    where
-      ts = tetrahedrons c
-      volumes = map volume ts
-      nonpositive_volumes = filter (<= 0) volumes
+prop_opposite_octant_tetrahedra_disjoint1 :: Cube -> Bool
+prop_opposite_octant_tetrahedra_disjoint1 c =
+  disjoint (front_left_top_tetrahedra c) (front_right_down_tetrahedra c)
 
--- | 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 c =
-    volume (tetrahedron0 c) ~= (1/24)*(delta^(3::Int))
-    where
-      delta = h c
+prop_opposite_octant_tetrahedra_disjoint2 :: Cube -> Bool
+prop_opposite_octant_tetrahedra_disjoint2 c =
+  disjoint (back_left_top_tetrahedra c) (back_right_down_tetrahedra c)
 
--- | 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 c =
-    volume (tetrahedron1 c) ~= (1/24)*(delta^(3::Int))
-    where
-      delta = h c
+prop_opposite_octant_tetrahedra_disjoint3 :: Cube -> Bool
+prop_opposite_octant_tetrahedra_disjoint3 c =
+  disjoint (front_left_top_tetrahedra c) (back_right_top_tetrahedra c)
 
--- | 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 c =
-    volume (tetrahedron2 c) ~= (1/24)*(delta^(3::Int))
-    where
-      delta = h c
+prop_opposite_octant_tetrahedra_disjoint4 :: Cube -> Bool
+prop_opposite_octant_tetrahedra_disjoint4 c =
+  disjoint (front_left_down_tetrahedra c) (back_right_down_tetrahedra c)
 
--- | 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 c =
-    volume (tetrahedron3 c) ~= (1/24)*(delta^(3::Int))
-    where
-      delta = h c
+prop_opposite_octant_tetrahedra_disjoint5 :: Cube -> Bool
+prop_opposite_octant_tetrahedra_disjoint5 c =
+  disjoint (front_left_top_tetrahedra c) (back_left_down_tetrahedra c)
 
--- | 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 c =
-    volume (tetrahedron4 c) ~= (1/24)*(delta^(3::Int))
-    where
-      delta = h c
+prop_opposite_octant_tetrahedra_disjoint6 :: Cube -> Bool
+prop_opposite_octant_tetrahedra_disjoint6 c =
+  disjoint (front_right_top_tetrahedra c) (back_right_down_tetrahedra c)
 
--- | 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 c =
-    volume (tetrahedron5 c) ~= (1/24)*(delta^(3::Int))
-    where
-      delta = h c
 
--- | 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 c =
-    volume (tetrahedron6 c) ~= (1/24)*(delta^(3::Int))
+-- | 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
-      delta = h c
+      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_tetrahedron7_volumes_exact :: Cube -> Bool
-prop_tetrahedron7_volumes_exact c =
-    volume (tetrahedron7 c) ~= (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 c
+      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 c = (v0 t0) == (v0 t1)
+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-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
+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
-      t0 = head (tetrahedrons c) -- Doesn't matter which two we choose.
-      t1 = head $ tail (tetrahedrons c)
-
-
--- | This pretty much repeats the prop_all_volumes_positive property,
---   but will let me know which tetrahedrons's vertices are disoriented.
-prop_tetrahedron0_volumes_positive :: Cube -> Bool
-prop_tetrahedron0_volumes_positive c =
-    volume (tetrahedron0 c) > 0
-
--- | This pretty much repeats the prop_all_volumes_positive property,
---   but will let me know which tetrahedrons's vertices are disoriented.
-prop_tetrahedron1_volumes_positive :: Cube -> Bool
-prop_tetrahedron1_volumes_positive c =
-    volume (tetrahedron1 c) > 0
-
--- | This pretty much repeats the prop_all_volumes_positive property,
---   but will let me know which tetrahedrons's vertices are disoriented.
-prop_tetrahedron2_volumes_positive :: Cube -> Bool
-prop_tetrahedron2_volumes_positive c =
-    volume (tetrahedron2 c) > 0
-
--- | This pretty much repeats the prop_all_volumes_positive property,
---   but will let me know which tetrahedrons's vertices are disoriented.
-prop_tetrahedron3_volumes_positive :: Cube -> Bool
-prop_tetrahedron3_volumes_positive c =
-    volume (tetrahedron3 c) > 0
-
--- | This pretty much repeats the prop_all_volumes_positive property,
---   but will let me know which tetrahedrons's vertices are disoriented.
-prop_tetrahedron4_volumes_positive :: Cube -> Bool
-prop_tetrahedron4_volumes_positive c =
-    volume (tetrahedron4 c) > 0
-
--- | This pretty much repeats the prop_all_volumes_positive property,
---   but will let me know which tetrahedrons's vertices are disoriented.
-prop_tetrahedron5_volumes_positive :: Cube -> Bool
-prop_tetrahedron5_volumes_positive c =
-    volume (tetrahedron5 c) > 0
-
--- | This pretty much repeats the prop_all_volumes_positive property,
---   but will let me know which tetrahedrons's vertices are disoriented.
-prop_tetrahedron6_volumes_positive :: Cube -> Bool
-prop_tetrahedron6_volumes_positive c =
-    volume (tetrahedron6 c) > 0
-
--- | This pretty much repeats the prop_all_volumes_positive property,
---   but will let me know which tetrahedrons's vertices are disoriented.
-prop_tetrahedron7_volumes_positive :: Cube -> Bool
-prop_tetrahedron7_volumes_positive c =
-    volume (tetrahedron7 c) > 0
-
--- | This pretty much repeats the prop_all_volumes_positive property,
---   but will let me know which tetrahedrons's vertices are disoriented.
-prop_tetrahedron8_volumes_positive :: Cube -> Bool
-prop_tetrahedron8_volumes_positive c =
-    volume (tetrahedron8 c) > 0
-
--- | This pretty much repeats the prop_all_volumes_positive property,
---   but will let me know which tetrahedrons's vertices are disoriented.
-prop_tetrahedron9_volumes_positive :: Cube -> Bool
-prop_tetrahedron9_volumes_positive c =
-    volume (tetrahedron9 c) > 0
-
--- | This pretty much repeats the prop_all_volumes_positive property,
---   but will let me know which tetrahedrons's vertices are disoriented.
-prop_tetrahedron10_volumes_positive :: Cube -> Bool
-prop_tetrahedron10_volumes_positive c =
-    volume (tetrahedron10 c) > 0
-
--- | This pretty much repeats the prop_all_volumes_positive property,
---   but will let me know which tetrahedrons's vertices are disoriented.
-prop_tetrahedron11_volumes_positive :: Cube -> Bool
-prop_tetrahedron11_volumes_positive c =
-    volume (tetrahedron11 c) > 0
-
--- | This pretty much repeats the prop_all_volumes_positive property,
---   but will let me know which tetrahedrons's vertices are disoriented.
-prop_tetrahedron12_volumes_positive :: Cube -> Bool
-prop_tetrahedron12_volumes_positive c =
-    volume (tetrahedron12 c) > 0
-
--- | This pretty much repeats the prop_all_volumes_positive property,
---   but will let me know which tetrahedrons's vertices are disoriented.
-prop_tetrahedron13_volumes_positive :: Cube -> Bool
-prop_tetrahedron13_volumes_positive c =
-    volume (tetrahedron13 c) > 0
-
--- | This pretty much repeats the prop_all_volumes_positive property,
---   but will let me know which tetrahedrons's vertices are disoriented.
-prop_tetrahedron14_volumes_positive :: Cube -> Bool
-prop_tetrahedron14_volumes_positive c =
-    volume (tetrahedron14 c) > 0
-
--- | This pretty much repeats the prop_all_volumes_positive property,
---   but will let me know which tetrahedrons's vertices are disoriented.
-prop_tetrahedron15_volumes_positive :: Cube -> Bool
-prop_tetrahedron15_volumes_positive c =
-    volume (tetrahedron15 c) > 0
-
--- | This pretty much repeats the prop_all_volumes_positive property,
---   but will let me know which tetrahedrons's vertices are disoriented.
-prop_tetrahedron16_volumes_positive :: Cube -> Bool
-prop_tetrahedron16_volumes_positive c =
-    volume (tetrahedron16 c) > 0
-
--- | This pretty much repeats the prop_all_volumes_positive property,
---   but will let me know which tetrahedrons's vertices are disoriented.
-prop_tetrahedron17_volumes_positive :: Cube -> Bool
-prop_tetrahedron17_volumes_positive c =
-    volume (tetrahedron17 c) > 0
-
--- | This pretty much repeats the prop_all_volumes_positive property,
---   but will let me know which tetrahedrons's vertices are disoriented.
-prop_tetrahedron18_volumes_positive :: Cube -> Bool
-prop_tetrahedron18_volumes_positive c =
-    volume (tetrahedron18 c) > 0
-
--- | This pretty much repeats the prop_all_volumes_positive property,
---   but will let me know which tetrahedrons's vertices are disoriented.
-prop_tetrahedron19_volumes_positive :: Cube -> Bool
-prop_tetrahedron19_volumes_positive c =
-    volume (tetrahedron19 c) > 0
-
--- | This pretty much repeats the prop_all_volumes_positive property,
---   but will let me know which tetrahedrons's vertices are disoriented.
-prop_tetrahedron20_volumes_positive :: Cube -> Bool
-prop_tetrahedron20_volumes_positive c =
-    volume (tetrahedron20 c) > 0
-
--- | This pretty much repeats the prop_all_volumes_positive property,
---   but will let me know which tetrahedrons's vertices are disoriented.
-prop_tetrahedron21_volumes_positive :: Cube -> Bool
-prop_tetrahedron21_volumes_positive c =
-    volume (tetrahedron21 c) > 0
-
--- | This pretty much repeats the prop_all_volumes_positive property,
---   but will let me know which tetrahedrons's vertices are disoriented.
-prop_tetrahedron22_volumes_positive :: Cube -> Bool
-prop_tetrahedron22_volumes_positive c =
-    volume (tetrahedron22 c) > 0
-
--- | This pretty much repeats the prop_all_volumes_positive property,
---   but will let me know which tetrahedrons's vertices are disoriented.
-prop_tetrahedron23_volumes_positive :: Cube -> Bool
-prop_tetrahedron23_volumes_positive c =
-    volume (tetrahedron23 c) > 0
-
-
--- | Given in Sorokina and Zeilfelder, p. 79.
---prop_c0120_identity1 :: Cube -> Bool
---prop_c0120_identity1 cube =
---   c0 ~= (c1 + c2) / 2
---     where
---       c0 = trace ("c0 :" ++ (show (c t0 0 1 2 0))) (c t0 0 1 2 0)
---       c1 = trace ("c1 :" ++ (show (c t0 0 0 2 1))) (c t0 0 0 2 1)
---       c2 = trace ("c2 :" ++ (show (c t1 0 0 2 1))) (c t1 0 0 2 1)
---       t0 = tetrahedron0 cube
---       t1 = tetrahedron1 cube
+      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
+
 
 
 -- | 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 = tetrahedron cube 0
+        t1 = tetrahedron cube 1
+
+
+-- | 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 (Tetrahedron.fv 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 (Tetrahedron.fv 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 (Tetrahedron.fv 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 = Tetrahedron.fv 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