module Tests.Cube
where
+import Prelude hiding (LT)
import Test.QuickCheck
+import Cardinal
import Comparisons
import Cube
-import FunctionValues (FunctionValues)
+import FunctionValues
+import Misc (all_equal)
import Tests.FunctionValues ()
-import Tetrahedron (b0, b1, b2, b3, c,
+import Tetrahedron (b0, b1, b2, b3, c, fv,
v0, v1, v2, v3, volume)
instance Arbitrary Cube where
-- 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
-- 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
-- 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
-- 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
-- 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
-- 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
-- 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
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! We also switch the third and fourth vertices of t6, but
+-- as of now, why this works is a mystery.
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
+ t6 = (tetrahedron6 cube) { v2 = (v3 t6), v3 = (v2 t6) }
--- | 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
+ t6 = (tetrahedron6 cube) { v2 = (v3 t6), v3 = (v2 t6) }
--- | 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
+ t6 = (tetrahedron6 cube) { v2 = (v3 t6), v3 = (v2 t6) }
--- | 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
+ t6 = (tetrahedron6 cube) { v2 = (v3 t6), v3 = (v2 t6) }
--- | 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
+ t6 = (tetrahedron6 cube) { v2 = (v3 t6), v3 = (v2 t6) }
--- | 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
+ t6 = (tetrahedron6 cube) { v2 = (v3 t6), v3 = (v2 t6) }
-- 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.
+prop_c_tilde_2100_rotation_correct :: Cube -> Bool
+prop_c_tilde_2100_rotation_correct cube =
+ expr1 == expr2
+ where
+ t0 = tetrahedron0 cube
+ t6 = tetrahedron6 cube
+
+ -- 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.
+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)
+ where
+ t0 = tetrahedron0 cube
+ t6 = tetrahedron6 cube
+ fvs = Tetrahedron.fv t0
+ int = interior fvs
+ f = front fvs
+ r = right fvs
+ l = left fvs
+ b = back fvs
+ ft = front_top fvs
+ rt = right_top fvs
+ lt = left_top fvs
+ bt = back_top fvs
+ t = top fvs
+ d = down fvs
+ fr = front_right fvs
+ fl = front_left fvs
+ br = back_right fvs
+ bl = back_left fvs
+ fd = front_down fvs
+ rd = right_down fvs
+ ld = left_down fvs
+ bd = back_down fvs
+
-- Tests to check that the correct edges are incidental.
prop_t0_shares_edge_with_t1 :: Cube -> Bool
prop_t0_shares_edge_with_t1 cube =
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