src/Tests/Face.hs

   1 module Tests.Face
   2 where
   3
   4 import Control.Monad (unless)
   5 import Test.HUnit
   6 import Test.QuickCheck
   7
   8 import Comparisons
   9 import Cube (Cube(grid), cube_at, top)
  10 import Face (face0,
  11              face2,
  12              face5,
  13              tetrahedron0,
  14              tetrahedron1,
  15              tetrahedron2,
  16              tetrahedron3,
  17              tetrahedrons)
  18 import Grid (Grid(h), make_grid)
  19 import Tetrahedron
  20
  21
  22 -- HUnit tests.
  23
  24 -- | An HUnit assertion that wraps the almost_equals function. Stolen
  25 --   from the definition of assertEqual in Test/HUnit/Base.hs.
  26 assertAlmostEqual :: String -> Double -> Double -> Assertion
  27 assertAlmostEqual preface expected actual =
  28   unless (actual ~= expected) (assertFailure msg)
  29  where msg = (if null preface then "" else preface ++ "\n") ++
  30              "expected: " ++ show expected ++ "\n but got: " ++ show actual
  31
  32
  33 -- | Values of the function f(x,y,z) = 1 + x + xy + xyz taken at nine
  34 --   points (hi, hj, jk) with h = 1. From example one in the paper.
  35 --   Used in the next bunch of tests.
  36 trilinear :: [[[Double]]]
  37 trilinear = [ [ [ 1, 2, 3 ],
  38                 [ 1, 3, 5 ],
  39                 [ 1, 4, 7 ] ],
  40               [ [ 1, 2, 3 ],
  41                 [ 1, 4, 7 ],
  42                 [ 1, 6, 11 ] ],
  43               [ [ 1, 2, 3 ],
  44                 [ 1, 5, 9 ],
  45                 [ 1, 8, 15 ]]]
  46
  47 -- | Check the value of c0030 for any tetrahedron belonging to the
  48 --   cube centered on (1,1,1) with a grid constructed from the
  49 --   trilinear values. See example one in the paper.
  50 test_trilinear_c0030 :: Test
  51 test_trilinear_c0030 =
  52     TestCase $ assertAlmostEqual "c0030 is correct" (c t 0 0 3 0) (17/8)
  53       where
  54         g = make_grid 1 trilinear
  55         cube = cube_at g 1 1 1
  56         t = head (tetrahedrons cube) -- Any one will do.
  57
  58
  59 -- | Check the value of c0003 for any tetrahedron belonging to the
  60 --   cube centered on (1,1,1) with a grid constructed from the
  61 --   trilinear values. See example one in the paper.
  62 test_trilinear_c0003 :: Test
  63 test_trilinear_c0003 =
  64     TestCase $ assertAlmostEqual "c0003 is correct" (c t 0 0 0 3) (27/8)
  65       where
  66         g = make_grid 1 trilinear
  67         cube = cube_at g 1 1 1
  68         t = head (tetrahedrons cube) -- Any one will do.
  69
  70
  71 -- | Check the value of c0021 for any tetrahedron belonging to the
  72 --   cube centered on (1,1,1) with a grid constructed from the
  73 --   trilinear values. See example one in the paper.
  74 test_trilinear_c0021 :: Test
  75 test_trilinear_c0021 =
  76     TestCase $ assertAlmostEqual "c0021 is correct" (c t 0 0 2 1) (61/24)
  77       where
  78         g = make_grid 1 trilinear
  79         cube = cube_at g 1 1 1
  80         t = head (tetrahedrons cube) -- Any one will do.
  81
  82
  83 -- | Check the value of c0012 for any tetrahedron belonging to the
  84 --   cube centered on (1,1,1) with a grid constructed from the
  85 --   trilinear values. See example one in the paper.
  86 test_trilinear_c0012 :: Test
  87 test_trilinear_c0012 =
  88     TestCase $ assertAlmostEqual "c0012 is correct" (c t 0 0 1 2) (71/24)
  89       where
  90         g = make_grid 1 trilinear
  91         cube = cube_at g 1 1 1
  92         t = head (tetrahedrons cube) -- Any one will do.
  93
  94
  95 -- | Check the value of c0120 for any tetrahedron belonging to the
  96 --   cube centered on (1,1,1) with a grid constructed from the
  97 --   trilinear values. See example one in the paper.
  98 test_trilinear_c0120 :: Test
  99 test_trilinear_c0120 =
 100     TestCase $ assertAlmostEqual "c0120 is correct" (c t 0 1 2 0) (55/24)
 101       where
 102         g = make_grid 1 trilinear
 103         cube = cube_at g 1 1 1
 104         t = head (tetrahedrons cube) -- Any one will do.
 105
 106
 107 -- | Check the value of c0102 for any tetrahedron belonging to the
 108 --   cube centered on (1,1,1) with a grid constructed from the
 109 --   trilinear values. See example one in the paper.
 110 test_trilinear_c0102 :: Test
 111 test_trilinear_c0102 =
 112     TestCase $ assertAlmostEqual "c0102 is correct" (c t 0 1 0 2) (73/24)
 113       where
 114         g = make_grid 1 trilinear
 115         cube = cube_at g 1 1 1
 116         t = head (tetrahedrons cube) -- Any one will do.
 117
 118
 119 -- | Check the value of c0111 for any tetrahedron belonging to the
 120 --   cube centered on (1,1,1) with a grid constructed from the
 121 --   trilinear values. See example one in the paper.
 122 test_trilinear_c0111 :: Test
 123 test_trilinear_c0111 =
 124     TestCase $ assertAlmostEqual "c0111 is correct" (c t 0 1 1 1) (8/3)
 125       where
 126         g = make_grid 1 trilinear
 127         cube = cube_at g 1 1 1
 128         t = head (tetrahedrons cube) -- Any one will do.
 129
 130
 131 -- | Check the value of c0210 for any tetrahedron belonging to the
 132 --   cube centered on (1,1,1) with a grid constructed from the
 133 --   trilinear values. See example one in the paper.
 134 test_trilinear_c0210 :: Test
 135 test_trilinear_c0210 =
 136     TestCase $ assertAlmostEqual "c0210 is correct" (c t 0 2 1 0) (29/12)
 137       where
 138         g = make_grid 1 trilinear
 139         cube = cube_at g 1 1 1
 140         t = head (tetrahedrons cube) -- Any one will do.
 141
 142
 143 -- | Check the value of c0201 for any tetrahedron belonging to the
 144 --   cube centered on (1,1,1) with a grid constructed from the
 145 --   trilinear values. See example one in the paper.
 146 test_trilinear_c0201 :: Test
 147 test_trilinear_c0201 =
 148     TestCase $ assertAlmostEqual "c0201 is correct" (c t 0 2 0 1) (11/4)
 149       where
 150         g = make_grid 1 trilinear
 151         cube = cube_at g 1 1 1
 152         t = head (tetrahedrons cube) -- Any one will do.
 153
 154
 155 -- | Check the value of c0300 for any tetrahedron belonging to the
 156 --   cube centered on (1,1,1) with a grid constructed from the
 157 --   trilinear values. See example one in the paper.
 158 test_trilinear_c0300 :: Test
 159 test_trilinear_c0300 =
 160     TestCase $ assertAlmostEqual "c0300 is correct" (c t 0 3 0 0) (5/2)
 161       where
 162         g = make_grid 1 trilinear
 163         cube = cube_at g 1 1 1
 164         t = head (tetrahedrons cube) -- Any one will do.
 165
 166
 167 -- | Check the value of c1020 for any tetrahedron belonging to the
 168 --   cube centered on (1,1,1) with a grid constructed from the
 169 --   trilinear values. See example one in the paper.
 170 test_trilinear_c1020 :: Test
 171 test_trilinear_c1020 =
 172     TestCase $ assertAlmostEqual "c1020 is correct" (c t 1 0 2 0) (8/3)
 173       where
 174         g = make_grid 1 trilinear
 175         cube = cube_at g 1 1 1
 176         t = head (tetrahedrons cube) -- Any one will do.
 177
 178
 179 face_tests :: [Test]
 180 face_tests = [test_trilinear_c0030,
 181               test_trilinear_c0003,
 182               test_trilinear_c0021,
 183               test_trilinear_c0012,
 184               test_trilinear_c0120,
 185               test_trilinear_c0102,
 186               test_trilinear_c0111,
 187               test_trilinear_c0210,
 188               test_trilinear_c0201,
 189               test_trilinear_c0300,
 190               test_trilinear_c1020]
 191
 192
 193 -- QuickCheck Tests.
 194
 195 -- | Since the grid size is necessarily positive, all tetrahedrons
 196 --   (which comprise cubes of positive volume) must have positive volume
 197 --   as well.
 198 prop_all_volumes_positive :: Cube -> Property
 199 prop_all_volumes_positive c =
 200     (delta > 0) ==> (null nonpositive_volumes)
 201     where
 202       delta = h (grid c)
 203       ts = tetrahedrons c
 204       volumes = map volume ts
 205       nonpositive_volumes = filter (<= 0) volumes
 206
 207
 208 -- | Given in Sorokina and Zeilfelder, p. 78.
 209 prop_cijk1_identity :: Cube -> Bool
 210 prop_cijk1_identity cube =
 211      and [ c t0' i j k 1 ~= (c t1' (i+1) j k 0) * ((b0 t0') (v3 t1')) +
 212                             (c t1' i (j+1) k 0) * ((b1 t0') (v3 t1')) +
 213                             (c t1' i j (k+1) 0) * ((b2 t0') (v3 t1')) +
 214                             (c t1' i j k 1) * ((b3 t0') (v3 t1')) | i <- [0..2],
 215                                                                     j <- [0..2],
 216                                                                     k <- [0..2],
 217                                                                     i + j + k == 2]
 218       where
 219         t0 = tetrahedron0 (face0 cube)
 220         t1 = tetrahedron1 (face0 cube)
 221         t0' = Tetrahedron cube (v3 t0) (v2 t0) (v1 t0) (v0 t0)
 222         t1' = Tetrahedron cube (v3 t1) (v2 t1) (v0 t1) (v1 t1)
 223
 224 -- | Given in Sorokina and Zeilfelder, p. 79.
 225 prop_c0120_identity1 :: Cube -> Bool
 226 prop_c0120_identity1 cube =
 227     c t0' 0 1 2 0 ~= (c t0' 0 0 2 1 + c t1' 0 0 2 1) / 2
 228       where
 229         t0 = tetrahedron0 (face0 cube)
 230         t1 = tetrahedron1 (face0 cube)
 231         t0' = Tetrahedron cube (v3 t0) (v2 t0) (v1 t0) (v0 t0)
 232         t1' = Tetrahedron cube (v3 t1) (v2 t1) (v0 t1) (v1 t1)
 233
 234
 235 -- | Given in Sorokina and Zeilfelder, p. 79.
 236 prop_c0210_identity1 :: Cube -> Bool
 237 prop_c0210_identity1 cube =
 238     c t0' 0 2 1 0 ~= (c t0' 0 1 1 1 + c t1' 0 1 1 1) / 2
 239       where
 240         t0 = tetrahedron0 (face0 cube)
 241         t1 = tetrahedron1 (face0 cube)
 242         t0' = Tetrahedron cube (v3 t0) (v2 t0) (v1 t0) (v0 t0)
 243         t1' = Tetrahedron cube (v3 t1) (v2 t1) (v0 t1) (v1 t1)
 244
 245
 246 -- | Given in Sorokina and Zeilfelder, p. 79.
 247 prop_c0300_identity1 :: Cube -> Bool
 248 prop_c0300_identity1 cube =
 249     c t0' 0 3 0 0 ~= (c t0' 0 2 0 1 + c t1' 0 2 0 1) / 2
 250       where
 251         t0 = tetrahedron0 (face0 cube)
 252         t1 = tetrahedron1 (face0 cube)
 253         t0' = Tetrahedron cube (v3 t0) (v2 t0) (v1 t0) (v0 t0)
 254         t1' = Tetrahedron cube (v3 t1) (v2 t1) (v0 t1) (v1 t1)
 255
 256 -- | Given in Sorokina and Zeilfelder, p. 79.
 257 prop_c1110_identity :: Cube -> Bool
 258 prop_c1110_identity cube =
 259     c t0' 1 1 1 0 ~= (c t0' 1 0 1 1 + c t1' 1 0 1 1) / 2
 260       where
 261         t0 = tetrahedron0 (face0 cube)
 262         t1 = tetrahedron1 (face0 cube)
 263         t0' = Tetrahedron cube (v3 t0) (v2 t0) (v1 t0) (v0 t0)
 264         t1' = Tetrahedron cube (v3 t1) (v2 t1) (v0 t1) (v1 t1)
 265
 266
 267 -- | Given in Sorokina and Zeilfelder, p. 79.
 268 prop_c1200_identity1 :: Cube -> Bool
 269 prop_c1200_identity1 cube =
 270     c t0' 1 2 0 0 ~= (c t0' 1 1 0 1 + c t1' 1 1 0 1) / 2
 271       where
 272         t0 = tetrahedron0 (face0 cube)
 273         t1 = tetrahedron1 (face0 cube)
 274         t0' = Tetrahedron cube (v3 t0) (v2 t0) (v1 t0) (v0 t0)
 275         t1' = Tetrahedron cube (v3 t1) (v2 t1) (v0 t1) (v1 t1)
 276
 277
 278 -- | Given in Sorokina and Zeilfelder, p. 79.
 279 prop_c2100_identity1 :: Cube -> Bool
 280 prop_c2100_identity1 cube =
 281     c t0' 2 1 0 0 ~= (c t0' 2 0 0 1 + c t1' 2 0 0 1) / 2
 282       where
 283         t0 = tetrahedron0 (face0 cube)
 284         t1 = tetrahedron1 (face0 cube)
 285         t0' = Tetrahedron cube (v3 t0) (v2 t0) (v1 t0) (v0 t0)
 286         t1' = Tetrahedron cube (v3 t1) (v2 t1) (v0 t1) (v1 t1)
 287
 288
 289 -- | Given in Sorokina and Zeilfelder, p. 79.
 290 prop_c0102_identity1 :: Cube -> Bool
 291 prop_c0102_identity1 cube =
 292     c t0' 0 1 0 2 ~= (c t0' 0 0 1 2 + c t3' 0 0 1 2) / 2
 293       where
 294         t0 = tetrahedron0 (face0 cube)
 295         t3 = tetrahedron3 (face0 cube)
 296         t0' = Tetrahedron cube (v3 t0) (v2 t0) (v1 t0) (v0 t0)
 297         t3' = Tetrahedron cube (v3 t3) (v2 t3) (v1 t3) (v0 t3)
 298
 299
 300 -- | Given in Sorokina and Zeilfelder, p. 79.
 301 prop_c0201_identity1 :: Cube -> Bool
 302 prop_c0201_identity1 cube =
 303     c t0' 0 2 0 1 ~= (c t0' 0 1 1 1 + c t3' 0 1 1 1) / 2
 304       where
 305         t0 = tetrahedron0 (face0 cube)
 306         t3 = tetrahedron3 (face0 cube)
 307         t0' = Tetrahedron cube (v3 t0) (v2 t0) (v1 t0) (v0 t0)
 308         t3' = Tetrahedron cube (v3 t3) (v2 t3) (v1 t3) (v0 t3)
 309
 310
 311 -- | Given in Sorokina and Zeilfelder, p. 79.
 312 prop_c0300_identity2 :: Cube -> Bool
 313 prop_c0300_identity2 cube =
 314     c t0' 3 0 0 0 ~= (c t0' 0 2 1 0 + c t3' 0 2 1 0) / 2
 315       where
 316         t0 = tetrahedron0 (face0 cube)
 317         t3 = tetrahedron3 (face0 cube)
 318         t0' = Tetrahedron cube (v3 t0) (v2 t0) (v1 t0) (v0 t0)
 319         t3' = Tetrahedron cube (v3 t3) (v2 t3) (v1 t3) (v0 t3)
 320
 321 -- | Given in Sorokina and Zeilfelder, p. 79.
 322 prop_c1101_identity :: Cube -> Bool
 323 prop_c1101_identity cube =
 324     c t0' 1 1 0 1 ~= (c t0' 1 1 0 1 + c t3' 1 1 0 1) / 2
 325       where
 326         t0 = tetrahedron0 (face0 cube)
 327         t3 = tetrahedron3 (face0 cube)
 328         t0' = Tetrahedron cube (v3 t0) (v2 t0) (v1 t0) (v0 t0)
 329         t3' = Tetrahedron cube (v3 t3) (v2 t3) (v1 t3) (v0 t3)
 330
 331
 332 -- | Given in Sorokina and Zeilfelder, p. 79.
 333 prop_c1200_identity2 :: Cube -> Bool
 334 prop_c1200_identity2 cube =
 335     c t0' 1 1 1 0 ~= (c t0' 1 1 1 0 + c t3' 1 1 1 0) / 2
 336       where
 337         t0 = tetrahedron0 (face0 cube)
 338         t3 = tetrahedron3 (face0 cube)
 339         t0' = Tetrahedron cube (v3 t0) (v2 t0) (v1 t0) (v0 t0)
 340         t3' = Tetrahedron cube (v3 t3) (v2 t3) (v1 t3) (v0 t3)
 341
 342
 343 -- | Given in Sorokina and Zeilfelder, p. 79.
 344 prop_c2100_identity2 :: Cube -> Bool
 345 prop_c2100_identity2 cube =
 346     c t0' 2 1 0 0 ~= (c t0' 2 0 1 0 + c t3' 2 0 1 0) / 2
 347       where
 348         t0 = tetrahedron0 (face0 cube)
 349         t3 = tetrahedron3 (face0 cube)
 350         t0' = Tetrahedron cube (v3 t0) (v2 t0) (v1 t0) (v0 t0)
 351         t3' = Tetrahedron cube (v3 t3) (v2 t3) (v1 t3) (v0 t3)
 352
 353
 354 -- | Given in Sorokina and Zeilfelder, p. 79.
 355 prop_c3000_identity :: Cube -> Bool
 356 prop_c3000_identity cube =
 357     c t0' 3 0 0 0 ~= c t0' 2 1 0 0 + c t2' 2 1 0 0 - ((c t0' 2 0 1 0 + c t0' 2 0 0 1)/ 2)
 358       where
 359         t0 = tetrahedron0 (face0 cube)
 360         t2 = tetrahedron2 (face5 cube)
 361         t0' = Tetrahedron cube (v3 t0) (v2 t0) (v1 t0) (v0 t0)
 362         t2' = Tetrahedron cube (v3 t2) (v2 t2) (v1 t2) (v0 t2)
 363
 364
 365 -- | Given in Sorokina and Zeilfelder, p. 79.
 366 prop_c2010_identity :: Cube -> Bool
 367 prop_c2010_identity cube =
 368     c t0' 2 0 1 0 ~= c t0' 1 1 1 0 + c t2' 1 1 1 0 - ((c t0' 1 0 2 0 + c t0' 1 0 1 1)/ 2)
 369       where
 370         t0 = tetrahedron0 (face0 cube)
 371         t2 = tetrahedron2 (face5 cube)
 372         t0' = Tetrahedron cube (v3 t0) (v2 t0) (v1 t0) (v0 t0)
 373         t2' = Tetrahedron cube (v3 t2) (v2 t2) (v1 t2) (v0 t2)
 374
 375
 376 -- | Given in Sorokina and Zeilfelder, p. 79.
 377 prop_c2001_identity :: Cube -> Bool
 378 prop_c2001_identity cube =
 379     c t0' 2 0 0 1 ~= c t0' 1 1 0 1 + c t2' 1 1 0 1 - ((c t0' 1 0 0 2 + c t0' 1 0 1 1)/ 2)
 380       where
 381         t0 = tetrahedron0 (face0 cube)
 382         t2 = tetrahedron2 (face5 cube)
 383         t0' = Tetrahedron cube (v3 t0) (v2 t0) (v1 t0) (v0 t0)
 384         t2' = Tetrahedron cube (v3 t2) (v2 t2) (v1 t2) (v0 t2)
 385
 386 -- | Given in Sorokina and Zeilfelder, p. 79.
 387 prop_c1020_identity :: Cube -> Bool
 388 prop_c1020_identity cube =
 389     c t0' 1 0 2 0 ~= c t0' 0 1 2 0 + c t2' 0 1 2 0 - ((c t0' 0 0 3 0 + c t0' 0 0 2 1)/ 2)
 390       where
 391         t0 = tetrahedron0 (face0 cube)
 392         t2 = tetrahedron2 (face5 cube)
 393         t0' = Tetrahedron cube (v3 t0) (v2 t0) (v1 t0) (v0 t0)
 394         t2' = Tetrahedron cube (v3 t2) (v2 t2) (v1 t2) (v0 t2)
 395
 396
 397 -- | Given in Sorokina and Zeilfelder, p. 79.
 398 prop_c1002_identity :: Cube -> Bool
 399 prop_c1002_identity cube =
 400     c t0' 1 0 0 2 ~= c t0' 0 1 0 2 + c t2' 0 1 0 2 - ((c t0' 0 0 0 3 + c t0' 0 0 1 2)/ 2)
 401       where
 402         t0 = tetrahedron0 (face0 cube)
 403         t2 = tetrahedron2 (face5 cube)
 404         t0' = Tetrahedron cube (v3 t0) (v2 t0) (v1 t0) (v0 t0)
 405         t2' = Tetrahedron cube (v3 t2) (v2 t2) (v1 t2) (v0 t2)
 406
 407
 408 -- | Given in Sorokina and Zeilfelder, p. 79.
 409 prop_c1011_identity :: Cube -> Bool
 410 prop_c1011_identity cube =
 411     c t0' 1 0 1 1 ~= c t0' 0 1 1 1 + c t2' 0 1 1 1 - ((c t0' 0 0 1 2 + c t0' 0 0 2 1)/ 2)
 412       where
 413         t0 = tetrahedron0 (face0 cube)
 414         t2 = tetrahedron2 (face5 cube)
 415         t0' = Tetrahedron cube (v3 t0) (v2 t0) (v1 t0) (v0 t0)
 416         t2' = Tetrahedron cube (v3 t2) (v2 t2) (v1 t2) (v0 t2)
 417
 418
 419 -- | Given in Sorokina and Zeilfelder, p. 80.
 420 prop_c0120_identity2 :: Cube -> Bool
 421 prop_c0120_identity2 cube =
 422         c t0' 0 1 2 0 ~= (c t0' 1 0 2 0 + c t1' 1 0 2 0) / 2
 423       where
 424         t0 = tetrahedron0 (face0 cube)
 425         t1 = tetrahedron0 (face2 (top cube))
 426         t0' = Tetrahedron cube (v3 t0) (v2 t0) (v1 t0) (v0 t0)
 427         t1' = Tetrahedron cube (v3 t1) (v2 t1) (v0 t1) (v1 t1)
 428
 429
 430 -- | Given in Sorokina and Zeilfelder, p. 80.
 431 prop_c0102_identity2 :: Cube -> Bool
 432 prop_c0102_identity2 cube =
 433     c t0' 0 1 0 2 ~= (c t0' 1 0 0 2 + c t1' 1 0 0 2) / 2
 434       where
 435         t0 = tetrahedron0 (face0 cube)
 436         t1 = tetrahedron0 (face2 (top cube))
 437         t0' = Tetrahedron cube (v3 t0) (v2 t0) (v1 t0) (v0 t0)
 438         t1' = Tetrahedron cube (v3 t1) (v2 t1) (v0 t1) (v1 t1)
 439
 440
 441 -- | Given in Sorokina and Zeilfelder, p. 80.
 442 prop_c0111_identity :: Cube -> Bool
 443 prop_c0111_identity cube =
 444     c t0' 0 1 1 1 ~= (c t0' 1 0 1 1 + c t1' 1 0 1 1) / 2
 445       where
 446         t0 = tetrahedron0 (face0 cube)
 447         t1 = tetrahedron0 (face2 (top cube))
 448         t0' = Tetrahedron cube (v3 t0) (v2 t0) (v1 t0) (v0 t0)
 449         t1' = Tetrahedron cube (v3 t1) (v2 t1) (v0 t1) (v1 t1)
 450
 451
 452 -- | Given in Sorokina and Zeilfelder, p. 80.
 453 prop_c0210_identity2 :: Cube -> Bool
 454 prop_c0210_identity2 cube =
 455     c t0 0 2 1 0 ~= (c t0 1 1 1 0 + c t1 1 1 1 0) / 2
 456       where
 457         t0 = tetrahedron0 (face0 cube)
 458         t1 = tetrahedron0 (face2 (top cube))
 459         t0' = Tetrahedron cube (v3 t0) (v2 t0) (v1 t0) (v0 t0)
 460         t1' = Tetrahedron cube (v3 t1) (v2 t1) (v0 t1) (v1 t1)
 461
 462
 463 -- | Given in Sorokina and Zeilfelder, p. 80.
 464 prop_c0201_identity2 :: Cube -> Bool
 465 prop_c0201_identity2 cube =
 466     c t0 0 2 0 1 ~= (c t0 1 1 0 1 + c t1 1 1 0 1) / 2
 467       where
 468         t0 = tetrahedron0 (face0 cube)
 469         t1 = tetrahedron0 (face2 (top cube))
 470         t0' = Tetrahedron cube (v3 t0) (v2 t0) (v1 t0) (v0 t0)
 471         t1' = Tetrahedron cube (v3 t1) (v2 t1) (v0 t1) (v1 t1)
 472
 473
 474 -- | Given in Sorokina and Zeilfelder, p. 80.
 475 prop_c0300_identity3 :: Cube -> Bool
 476 prop_c0300_identity3 cube =
 477     c t0 0 3 0 0 ~= (c t0 1 2 0 0 + c t1 1 2 0 0) / 2
 478       where
 479         t0 = tetrahedron0 (face0 cube)
 480         t1 = tetrahedron0 (face2 (top cube))
 481         t0' = Tetrahedron cube (v3 t0) (v2 t0) (v1 t0) (v0 t0)
 482         t1' = Tetrahedron cube (v3 t1) (v2 t1) (v0 t1) (v1 t1)