X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=src%2FTests%2FCube.hs;h=e867e5d478364fb9073b9a6ec67bab82ff1940f7;hb=c864b66c83f2be395fa590321ca313e227d79fab;hp=aaafa7ca77a6cad7f9a6afc4cbab1745d120d611;hpb=edf374f084fe4e495ea4f5bd87a4646e8559c1d1;p=spline3.git

diff --git a/src/Tests/Cube.hs b/src/Tests/Cube.hs
index aaafa7c..e867e5d 100644
--- a/src/Tests/Cube.hs
+++ b/src/Tests/Cube.hs
@@ -8,6 +8,7 @@ import Cardinal
 import Comparisons
 import Cube
 import FunctionValues
+import Misc (all_equal)
 import Tests.FunctionValues ()
 import Tetrahedron (b0, b1, b2, b3, c, fv,
                     v0, v1, v2, v3, volume)
@@ -43,16 +44,17 @@ prop_all_volumes_positive cube =
 --   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))
+    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))
+    volume (tetrahedron1 cube) ~~= (1/24)*(delta^(3::Int))
     where
       delta = h cube
 
@@ -61,7 +63,7 @@ prop_tetrahedron1_volumes_exact cube =
 --   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))
+    volume (tetrahedron2 cube) ~~= (1/24)*(delta^(3::Int))
     where
       delta = h cube
 
@@ -70,7 +72,7 @@ prop_tetrahedron2_volumes_exact cube =
 --   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))
+    volume (tetrahedron3 cube) ~~= (1/24)*(delta^(3::Int))
     where
       delta = h cube
 
@@ -79,7 +81,7 @@ prop_tetrahedron3_volumes_exact cube =
 --   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))
+    volume (tetrahedron4 cube) ~~= (1/24)*(delta^(3::Int))
     where
       delta = h cube
 
@@ -88,7 +90,7 @@ prop_tetrahedron4_volumes_exact cube =
 --   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))
+    volume (tetrahedron5 cube) ~~= (1/24)*(delta^(3::Int))
     where
       delta = h cube
 
@@ -97,7 +99,7 @@ prop_tetrahedron5_volumes_exact cube =
 --   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))
+    volume (tetrahedron6 cube) ~~= (1/24)*(delta^(3::Int))
     where
       delta = h cube
 
@@ -106,7 +108,151 @@ prop_tetrahedron6_volumes_exact cube =
 --   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))
+    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))
     where
       delta = h cube
 
@@ -263,260 +409,282 @@ 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
---   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). It appears that
+--   the assumptions in sections (2.6) and (2.7) have been
+--   switched. From the description, one would expect 'tetrahedron0'
+--   and 'tetrahedron3' to share face \<v0,v1,v2\>; however, we have
+--   to use 'tetrahedron0' and 'tetahedron1' for all of the tests in
+--   section (2.6). Also 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
+   c t0 0 1 2 0 ~= (c t0 0 0 2 1 + c t1 0 0 1 2) / 2
      where
        t0 = tetrahedron0 cube
-       t3 = tetrahedron3 cube
+       t1 = tetrahedron1 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 t2 0 0 1 2) / 2
      where
-       t3 = tetrahedron3 cube
+       t1 = tetrahedron1 cube
        t2 = tetrahedron2 cube
 
--- | Given in Sorokina and Zeilfelder, p. 79. Repeats
---   prop_c0120_identity1 with tetrahedrons 2 and 1.
+-- | Given in Sorokina and Zeilfelder, p. 79, (2.6). Repeats
+--   'prop_c0120_identity1' with tetrahedrons 2 and 3.
 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
+   c t2 0 1 2 0 ~= (c t2 0 0 2 1 + c t3 0 0 1 2) / 2
      where
        t2 = tetrahedron2 cube
-       t1 = tetrahedron1 cube
-
-
--- | Given in Sorokina and Zeilfelder, p. 79. Repeats
---   prop_c0120_identity1 with tetrahedrons 4 and 7.
-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
-     where
-       t4 = tetrahedron4 cube
-       t7 = tetrahedron7 cube
-
-
--- | Given in Sorokina and Zeilfelder, p. 79. Repeats
---   prop_c0120_identity1 with tetrahedrons 7 and 6.
-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
-
-
--- | Given in Sorokina and Zeilfelder, p. 79. Repeats
---   prop_c0120_identity1 with tetrahedrons 6 and 5.
-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
+       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). Repeats
+--   'prop_c0120_identity1' with tetrahedrons 4 and 5.
+-- prop_c0120_identity4 :: Cube -> Bool
+-- prop_c0120_identity4 cube =
+--     sum [trace ("c_t4_0120: " ++ (show tmp1)) tmp1,
+--          trace ("c_t5_0012: " ++ (show tmp2)) tmp2,
+--          trace ("c_t5_0102: " ++ (show tmp3)) tmp3,
+--          trace ("c_t5_1002: " ++ (show tmp4)) tmp4,
+--          trace ("c_t5_0120: " ++ (show tmp5)) tmp5,
+--          trace ("c_t5_1020: " ++ (show tmp6)) tmp6,
+--          trace ("c_t5_1200: " ++ (show tmp7)) tmp7,
+--          trace ("c_t5_0021: " ++ (show tmp8)) tmp8,
+--          trace ("c_t5_0201: " ++ (show tmp9)) tmp9,
+--          trace ("c_t5_2001: " ++ (show tmp10)) tmp10,
+--          trace ("c_t5_0210: " ++ (show tmp11)) tmp11,
+--          trace ("c_t5_2010: " ++ (show tmp12)) tmp12,
+--          trace ("c_t5_2100: " ++ (show tmp13)) tmp13] == 10
+-- --   c t4 0 1 2 0 ~= (c t4 0 0 2 1 + c t5 0 0 1 2) / 2
+--      where
+--        t4 = tetrahedron4 cube
+--        t5 = tetrahedron5 cube
+--        tmp1 = c t4 0 1 2 0
+--        tmp2 = (c t4 0 0 2 1 + c t5 0 0 1 2) / 2
+--        tmp3 = (c t4 0 0 2 1 + c t5 0 1 0 2) / 2
+--        tmp4 = (c t4 0 0 2 1 + c t5 1 0 0 2) / 2
+--        tmp5 = (c t4 0 0 2 1 + c t5 0 1 2 0) / 2
+--        tmp6 = (c t4 0 0 2 1 + c t5 1 0 2 0) / 2
+--        tmp7 = (c t4 0 0 2 1 + c t5 1 2 0 0) / 2
+--        tmp8 = (c t4 0 0 2 1 + c t5 0 0 2 1) / 2
+--        tmp9 = (c t4 0 0 2 1 + c t5 0 2 0 1) / 2
+--        tmp10 = (c t4 0 0 2 1 + c t5 2 0 0 1) / 2
+--        tmp11 = (c t4 0 0 2 1 + c t5 0 2 1 0) / 2
+--        tmp12 = (c t4 0 0 2 1 + c t5 2 0 1 0) / 2
+--        tmp13 = (c t4 0 0 2 1 + c t5 2 1 0 0) / 2
+
+-- -- | Given in Sorokina and Zeilfelder, p. 79, (2.6). Repeats
+-- --   'prop_c0120_identity1' with tetrahedrons 5 and 6.
+-- prop_c0120_identity5 :: Cube -> Bool
+-- prop_c0120_identity5 cube =
+--    c t5 0 1 2 0 ~= (c t5 0 0 2 1 + c t6 0 0 1 2) / 2
+--      where
+--        t5 = tetrahedron5 cube
+--        t6 = tetrahedron6 cube
+
+
+-- -- | Given in Sorokina and Zeilfelder, p. 79, (2.6). Repeats
+-- --   'prop_c0120_identity1' with tetrahedrons 6 and 7.
+-- prop_c0120_identity6 :: Cube -> Bool
+-- prop_c0120_identity6 cube =
+--    c t6 0 1 2 0 ~= (c t6 0 0 2 1 + c t7 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
+    c t0 0 2 1 0 ~= (c t0 0 1 1 1 + c t1 0 1 1 1) / 2
       where
         t0 = tetrahedron0 cube
-        t3 = tetrahedron3 cube
+        t1 = tetrahedron1 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
+    c t0 0 3 0 0 ~= (c t0 0 2 0 1 + c t1 0 2 1 0) / 2
       where
         t0 = tetrahedron0 cube
-        t3 = tetrahedron3 cube
+        t1 = tetrahedron1 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
+    c t0 1 1 1 0 ~= (c t0 1 0 1 1 + c t1 1 0 1 1) / 2
       where
         t0 = tetrahedron0 cube
-        t3 = tetrahedron3 cube
+        t1 = tetrahedron1 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
+    c t0 1 2 0 0 ~= (c t0 1 1 0 1 + c t1 1 1 1 0) / 2
       where
         t0 = tetrahedron0 cube
-        t3 = tetrahedron3 cube
+        t1 = tetrahedron1 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
+    c t0 2 1 0 0 ~= (c t0 2 0 0 1 + c t1 2 0 1 0) / 2
       where
         t0 = tetrahedron0 cube
-        t3 = tetrahedron3 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). It appears that
+--   the assumptions in sections (2.6) and (2.7) have been
+--   switched. From the description, one would expect 'tetrahedron0'
+--   and 'tetrahedron1' to share face \<v0,v1,v3\>; however, we have
+--   to use 'tetrahedron0' and 'tetahedron3' for all of the tests in
+--   section (2.7). Also 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
+    c t0 0 1 0 2 ~= (c t0 0 0 1 2 + c t3 0 0 2 1) / 2
       where
         t0 = tetrahedron0 cube
-        t1 = tetrahedron1 cube
+        t3 = tetrahedron3 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
+    c t0 0 2 0 1 ~= (c t0 0 1 1 1 + c t3 0 1 1 1) / 2
       where
         t0 = tetrahedron0 cube
-        t1 = tetrahedron1 cube
+        t3 = tetrahedron3 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
+    c t0 0 3 0 0 ~= (c t0 0 2 1 0 + c t3 0 2 0 1) / 2
       where
         t0 = tetrahedron0 cube
-        t1 = tetrahedron1 cube
+        t3 = tetrahedron3 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
+    c t0 1 1 0 1 ~= (c t0 1 0 1 1 + c t3 1 0 1 1) / 2
       where
         t0 = tetrahedron0 cube
-        t1 = tetrahedron1 cube
+        t3 = tetrahedron3 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
+    c t0 1 2 0 0 ~= (c t0 1 1 1 0 + c t3 1 1 0 1) / 2
       where
         t0 = tetrahedron0 cube
-        t1 = tetrahedron1 cube
+        t3 = tetrahedron3 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
+    c t0 2 1 0 0 ~= (c t0 2 0 1 0 + c t3 2 0 0 1) / 2
       where
         t0 = tetrahedron0 cube
-        t1 = tetrahedron1 cube
+        t3 = tetrahedron3 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
@@ -524,21 +692,53 @@ 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.
+prop_interior_values_all_identical :: Cube -> Bool
+prop_interior_values_all_identical cube =
+    all_equal [i0, i1, i2, i3, i4, i5, i6, i7, i8,
+               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
+
 
 -- | 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.
@@ -574,9 +774,11 @@ 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