src/Tests/Cube.hs

   1 module Tests.Cube
   2 where
   3
   4 import Test.QuickCheck
   5
   6 import Comparisons
   7 import Cube
   8 import FunctionValues (FunctionValues)
   9 import Tests.FunctionValues ()
  10 import Tetrahedron (b0, b1, b2, b3, c,
  11                     v0, v1, v2, v3, volume)
  12
  13 instance Arbitrary Cube where
  14     arbitrary = do
  15       (Positive h') <- arbitrary :: Gen (Positive Double)
  16       i' <- choose (coordmin, coordmax)
  17       j' <- choose (coordmin, coordmax)
  18       k' <- choose (coordmin, coordmax)
  19       fv' <- arbitrary :: Gen FunctionValues
  20       return (Cube h' i' j' k' fv')
  21         where
  22           coordmin = -268435456 -- -(2^29 / 2)
  23           coordmax = 268435456  -- +(2^29 / 2)
  24
  25
  26 -- Quickcheck tests.
  27
  28 -- | Since the grid size is necessarily positive, all tetrahedrons
  29 --   (which comprise cubes of positive volume) must have positive volume
  30 --   as well.
  31 prop_all_volumes_positive :: Cube -> Bool
  32 prop_all_volumes_positive cube =
  33     null nonpositive_volumes
  34     where
  35       ts = tetrahedrons cube
  36       volumes = map volume ts
  37       nonpositive_volumes = filter (<= 0) volumes
  38
  39 -- | In fact, since all of the tetrahedra are identical, we should
  40 --   already know their volumes. There's 24 tetrahedra to a cube, so
  41 --   we'd expect the volume of each one to be (1/24)*h^3.
  42 prop_tetrahedron0_volumes_exact :: Cube -> Bool
  43 prop_tetrahedron0_volumes_exact cube =
  44     volume (tetrahedron0 cube) ~= (1/24)*(delta^(3::Int))
  45     where
  46       delta = h cube
  47
  48 -- | In fact, since all of the tetrahedra are identical, we should
  49 --   already know their volumes. There's 24 tetrahedra to a cube, so
  50 --   we'd expect the volume of each one to be (1/24)*h^3.
  51 prop_tetrahedron1_volumes_exact :: Cube -> Bool
  52 prop_tetrahedron1_volumes_exact cube =
  53     volume (tetrahedron1 cube) ~= (1/24)*(delta^(3::Int))
  54     where
  55       delta = h cube
  56
  57 -- | In fact, since all of the tetrahedra are identical, we should
  58 --   already know their volumes. There's 24 tetrahedra to a cube, so
  59 --   we'd expect the volume of each one to be (1/24)*h^3.
  60 prop_tetrahedron2_volumes_exact :: Cube -> Bool
  61 prop_tetrahedron2_volumes_exact cube =
  62     volume (tetrahedron2 cube) ~= (1/24)*(delta^(3::Int))
  63     where
  64       delta = h cube
  65
  66 -- | In fact, since all of the tetrahedra are identical, we should
  67 --   already know their volumes. There's 24 tetrahedra to a cube, so
  68 --   we'd expect the volume of each one to be (1/24)*h^3.
  69 prop_tetrahedron3_volumes_exact :: Cube -> Bool
  70 prop_tetrahedron3_volumes_exact cube =
  71     volume (tetrahedron3 cube) ~= (1/24)*(delta^(3::Int))
  72     where
  73       delta = h cube
  74
  75 -- | In fact, since all of the tetrahedra are identical, we should
  76 --   already know their volumes. There's 24 tetrahedra to a cube, so
  77 --   we'd expect the volume of each one to be (1/24)*h^3.
  78 prop_tetrahedron4_volumes_exact :: Cube -> Bool
  79 prop_tetrahedron4_volumes_exact cube =
  80     volume (tetrahedron4 cube) ~= (1/24)*(delta^(3::Int))
  81     where
  82       delta = h cube
  83
  84 -- | In fact, since all of the tetrahedra are identical, we should
  85 --   already know their volumes. There's 24 tetrahedra to a cube, so
  86 --   we'd expect the volume of each one to be (1/24)*h^3.
  87 prop_tetrahedron5_volumes_exact :: Cube -> Bool
  88 prop_tetrahedron5_volumes_exact cube =
  89     volume (tetrahedron5 cube) ~= (1/24)*(delta^(3::Int))
  90     where
  91       delta = h cube
  92
  93 -- | In fact, since all of the tetrahedra are identical, we should
  94 --   already know their volumes. There's 24 tetrahedra to a cube, so
  95 --   we'd expect the volume of each one to be (1/24)*h^3.
  96 prop_tetrahedron6_volumes_exact :: Cube -> Bool
  97 prop_tetrahedron6_volumes_exact cube =
  98     volume (tetrahedron6 cube) ~= (1/24)*(delta^(3::Int))
  99     where
 100       delta = h cube
 101
 102 -- | In fact, since all of the tetrahedra are identical, we should
 103 --   already know their volumes. There's 24 tetrahedra to a cube, so
 104 --   we'd expect the volume of each one to be (1/24)*h^3.
 105 prop_tetrahedron7_volumes_exact :: Cube -> Bool
 106 prop_tetrahedron7_volumes_exact cube =
 107     volume (tetrahedron7 cube) ~= (1/24)*(delta^(3::Int))
 108     where
 109       delta = h cube
 110
 111 -- | All tetrahedron should have their v0 located at the center of the cube.
 112 prop_v0_all_equal :: Cube -> Bool
 113 prop_v0_all_equal cube = (v0 t0) == (v0 t1)
 114     where
 115       t0 = head (tetrahedrons cube) -- Doesn't matter which two we choose.
 116       t1 = head $ tail (tetrahedrons cube)
 117
 118
 119 -- | This pretty much repeats the prop_all_volumes_positive property,
 120 --   but will let me know which tetrahedrons's vertices are disoriented.
 121 prop_tetrahedron0_volumes_positive :: Cube -> Bool
 122 prop_tetrahedron0_volumes_positive cube =
 123     volume (tetrahedron0 cube) > 0
 124
 125 -- | This pretty much repeats the prop_all_volumes_positive property,
 126 --   but will let me know which tetrahedrons's vertices are disoriented.
 127 prop_tetrahedron1_volumes_positive :: Cube -> Bool
 128 prop_tetrahedron1_volumes_positive cube =
 129     volume (tetrahedron1 cube) > 0
 130
 131 -- | This pretty much repeats the prop_all_volumes_positive property,
 132 --   but will let me know which tetrahedrons's vertices are disoriented.
 133 prop_tetrahedron2_volumes_positive :: Cube -> Bool
 134 prop_tetrahedron2_volumes_positive cube =
 135     volume (tetrahedron2 cube) > 0
 136
 137 -- | This pretty much repeats the prop_all_volumes_positive property,
 138 --   but will let me know which tetrahedrons's vertices are disoriented.
 139 prop_tetrahedron3_volumes_positive :: Cube -> Bool
 140 prop_tetrahedron3_volumes_positive cube =
 141     volume (tetrahedron3 cube) > 0
 142
 143 -- | This pretty much repeats the prop_all_volumes_positive property,
 144 --   but will let me know which tetrahedrons's vertices are disoriented.
 145 prop_tetrahedron4_volumes_positive :: Cube -> Bool
 146 prop_tetrahedron4_volumes_positive cube =
 147     volume (tetrahedron4 cube) > 0
 148
 149 -- | This pretty much repeats the prop_all_volumes_positive property,
 150 --   but will let me know which tetrahedrons's vertices are disoriented.
 151 prop_tetrahedron5_volumes_positive :: Cube -> Bool
 152 prop_tetrahedron5_volumes_positive cube =
 153     volume (tetrahedron5 cube) > 0
 154
 155 -- | This pretty much repeats the prop_all_volumes_positive property,
 156 --   but will let me know which tetrahedrons's vertices are disoriented.
 157 prop_tetrahedron6_volumes_positive :: Cube -> Bool
 158 prop_tetrahedron6_volumes_positive cube =
 159     volume (tetrahedron6 cube) > 0
 160
 161 -- | This pretty much repeats the prop_all_volumes_positive property,
 162 --   but will let me know which tetrahedrons's vertices are disoriented.
 163 prop_tetrahedron7_volumes_positive :: Cube -> Bool
 164 prop_tetrahedron7_volumes_positive cube =
 165     volume (tetrahedron7 cube) > 0
 166
 167 -- | This pretty much repeats the prop_all_volumes_positive property,
 168 --   but will let me know which tetrahedrons's vertices are disoriented.
 169 prop_tetrahedron8_volumes_positive :: Cube -> Bool
 170 prop_tetrahedron8_volumes_positive cube =
 171     volume (tetrahedron8 cube) > 0
 172
 173 -- | This pretty much repeats the prop_all_volumes_positive property,
 174 --   but will let me know which tetrahedrons's vertices are disoriented.
 175 prop_tetrahedron9_volumes_positive :: Cube -> Bool
 176 prop_tetrahedron9_volumes_positive cube =
 177     volume (tetrahedron9 cube) > 0
 178
 179 -- | This pretty much repeats the prop_all_volumes_positive property,
 180 --   but will let me know which tetrahedrons's vertices are disoriented.
 181 prop_tetrahedron10_volumes_positive :: Cube -> Bool
 182 prop_tetrahedron10_volumes_positive cube =
 183     volume (tetrahedron10 cube) > 0
 184
 185 -- | This pretty much repeats the prop_all_volumes_positive property,
 186 --   but will let me know which tetrahedrons's vertices are disoriented.
 187 prop_tetrahedron11_volumes_positive :: Cube -> Bool
 188 prop_tetrahedron11_volumes_positive cube =
 189     volume (tetrahedron11 cube) > 0
 190
 191 -- | This pretty much repeats the prop_all_volumes_positive property,
 192 --   but will let me know which tetrahedrons's vertices are disoriented.
 193 prop_tetrahedron12_volumes_positive :: Cube -> Bool
 194 prop_tetrahedron12_volumes_positive cube =
 195     volume (tetrahedron12 cube) > 0
 196
 197 -- | This pretty much repeats the prop_all_volumes_positive property,
 198 --   but will let me know which tetrahedrons's vertices are disoriented.
 199 prop_tetrahedron13_volumes_positive :: Cube -> Bool
 200 prop_tetrahedron13_volumes_positive cube =
 201     volume (tetrahedron13 cube) > 0
 202
 203 -- | This pretty much repeats the prop_all_volumes_positive property,
 204 --   but will let me know which tetrahedrons's vertices are disoriented.
 205 prop_tetrahedron14_volumes_positive :: Cube -> Bool
 206 prop_tetrahedron14_volumes_positive cube =
 207     volume (tetrahedron14 cube) > 0
 208
 209 -- | This pretty much repeats the prop_all_volumes_positive property,
 210 --   but will let me know which tetrahedrons's vertices are disoriented.
 211 prop_tetrahedron15_volumes_positive :: Cube -> Bool
 212 prop_tetrahedron15_volumes_positive cube =
 213     volume (tetrahedron15 cube) > 0
 214
 215 -- | This pretty much repeats the prop_all_volumes_positive property,
 216 --   but will let me know which tetrahedrons's vertices are disoriented.
 217 prop_tetrahedron16_volumes_positive :: Cube -> Bool
 218 prop_tetrahedron16_volumes_positive cube =
 219     volume (tetrahedron16 cube) > 0
 220
 221 -- | This pretty much repeats the prop_all_volumes_positive property,
 222 --   but will let me know which tetrahedrons's vertices are disoriented.
 223 prop_tetrahedron17_volumes_positive :: Cube -> Bool
 224 prop_tetrahedron17_volumes_positive cube =
 225     volume (tetrahedron17 cube) > 0
 226
 227 -- | This pretty much repeats the prop_all_volumes_positive property,
 228 --   but will let me know which tetrahedrons's vertices are disoriented.
 229 prop_tetrahedron18_volumes_positive :: Cube -> Bool
 230 prop_tetrahedron18_volumes_positive cube =
 231     volume (tetrahedron18 cube) > 0
 232
 233 -- | This pretty much repeats the prop_all_volumes_positive property,
 234 --   but will let me know which tetrahedrons's vertices are disoriented.
 235 prop_tetrahedron19_volumes_positive :: Cube -> Bool
 236 prop_tetrahedron19_volumes_positive cube =
 237     volume (tetrahedron19 cube) > 0
 238
 239 -- | This pretty much repeats the prop_all_volumes_positive property,
 240 --   but will let me know which tetrahedrons's vertices are disoriented.
 241 prop_tetrahedron20_volumes_positive :: Cube -> Bool
 242 prop_tetrahedron20_volumes_positive cube =
 243     volume (tetrahedron20 cube) > 0
 244
 245 -- | This pretty much repeats the prop_all_volumes_positive property,
 246 --   but will let me know which tetrahedrons's vertices are disoriented.
 247 prop_tetrahedron21_volumes_positive :: Cube -> Bool
 248 prop_tetrahedron21_volumes_positive cube =
 249     volume (tetrahedron21 cube) > 0
 250
 251 -- | This pretty much repeats the prop_all_volumes_positive property,
 252 --   but will let me know which tetrahedrons's vertices are disoriented.
 253 prop_tetrahedron22_volumes_positive :: Cube -> Bool
 254 prop_tetrahedron22_volumes_positive cube =
 255     volume (tetrahedron22 cube) > 0
 256
 257 -- | This pretty much repeats the prop_all_volumes_positive property,
 258 --   but will let me know which tetrahedrons's vertices are disoriented.
 259 prop_tetrahedron23_volumes_positive :: Cube -> Bool
 260 prop_tetrahedron23_volumes_positive cube =
 261     volume (tetrahedron23 cube) > 0
 262
 263
 264 -- | Given in Sorokina and Zeilfelder, p. 79. Note that the third and
 265 --   fourth indices of c-t3 have been switched. This is because we
 266 --   store the triangles oriented such that their volume is
 267 --   positive. If T and T-tilde share \<v0,v1,v2\> and v3,v3-tilde point
 268 --   in opposite directions, one of them has to have negative volume!
 269 prop_c0120_identity1 :: Cube -> Bool
 270 prop_c0120_identity1 cube =
 271    c t0 0 1 2 0 ~= (c t0 0 0 2 1 + c t3 0 0 1 2) / 2
 272      where
 273        t0 = tetrahedron0 cube
 274        t3 = tetrahedron3 cube
 275
 276
 277 -- | Given in Sorokina and Zeilfelder, p. 79. Note that the third and
 278 --   fourth indices of c-t3 have been switched. This is because we
 279 --   store the triangles oriented such that their volume is
 280 --   positive. If T and T-tilde share \<v0,v1,v2\> and v3,v3-tilde point
 281 --   in opposite directions, one of them has to have negative volume!
 282 prop_c0210_identity1 :: Cube -> Bool
 283 prop_c0210_identity1 cube =
 284     c t0 0 2 1 0 ~= (c t0 0 1 1 1 + c t3 0 1 1 1) / 2
 285       where
 286         t0 = tetrahedron0 cube
 287         t3 = tetrahedron3 cube
 288
 289
 290 -- | Given in Sorokina and Zeilfelder, p. 79. Note that the third and
 291 --   fourth indices of c-t3 have been switched. This is because we
 292 --   store the triangles oriented such that their volume is
 293 --   positive. If T and T-tilde share \<v0,v1,v2\> and v3,v3-tilde point
 294 --   in opposite directions, one of them has to have negative volume!
 295 prop_c0300_identity1 :: Cube -> Bool
 296 prop_c0300_identity1 cube =
 297     c t0 0 3 0 0 ~= (c t0 0 2 0 1 + c t3 0 2 1 0) / 2
 298       where
 299         t0 = tetrahedron0 cube
 300         t3 = tetrahedron3 cube
 301
 302
 303 -- | Given in Sorokina and Zeilfelder, p. 79. Note that the third and
 304 --   fourth indices of c-t3 have been switched. This is because we
 305 --   store the triangles oriented such that their volume is
 306 --   positive. If T and T-tilde share \<v0,v1,v2\> and v3,v3-tilde point
 307 --   in opposite directions, one of them has to have negative volume!
 308 prop_c1110_identity :: Cube -> Bool
 309 prop_c1110_identity cube =
 310     c t0 1 1 1 0 ~= (c t0 1 0 1 1 + c t3 1 0 1 1) / 2
 311       where
 312         t0 = tetrahedron0 cube
 313         t3 = tetrahedron3 cube
 314
 315
 316 -- | Given in Sorokina and Zeilfelder, p. 79. Note that the third and
 317 --   fourth indices of c-t3 have been switched. This is because we
 318 --   store the triangles oriented such that their volume is
 319 --   positive. If T and T-tilde share \<v0,v1,v2\> and v3,v3-tilde point
 320 --   in opposite directions, one of them has to have negative volume!
 321 prop_c1200_identity1 :: Cube -> Bool
 322 prop_c1200_identity1 cube =
 323     c t0 1 2 0 0 ~= (c t0 1 1 0 1 + c t3 1 1 1 0) / 2
 324       where
 325         t0 = tetrahedron0 cube
 326         t3 = tetrahedron3 cube
 327
 328
 329 -- | Given in Sorokina and Zeilfelder, p. 79. Note that the third and
 330 --   fourth indices of c-t3 have been switched. This is because we
 331 --   store the triangles oriented such that their volume is
 332 --   positive. If T and T-tilde share \<v0,v1,v2\> and v3,v3-tilde point
 333 --   in opposite directions, one of them has to have negative volume!
 334 prop_c2100_identity1 :: Cube -> Bool
 335 prop_c2100_identity1 cube =
 336     c t0 2 1 0 0 ~= (c t0 2 0 0 1 + c t3 2 0 1 0) / 2
 337       where
 338         t0 = tetrahedron0 cube
 339         t3 = tetrahedron3 cube
 340
 341
 342
 343 -- | Given in Sorokina and Zeilfelder, p. 79. Note that the third and
 344 --   fourth indices of c-t1 have been switched. This is because we
 345 --   store the triangles oriented such that their volume is
 346 --   positive. If T and T-tilde share \<v0,v1,v3\> and v2,v2-tilde point
 347 --   in opposite directions, one of them has to have negative volume!
 348 prop_c0102_identity1 :: Cube -> Bool
 349 prop_c0102_identity1 cube =
 350     c t0 0 1 0 2 ~= (c t0 0 0 1 2 + c t1 0 0 2 1) / 2
 351       where
 352         t0 = tetrahedron0 cube
 353         t1 = tetrahedron1 cube
 354
 355
 356 -- | Given in Sorokina and Zeilfelder, p. 79. Note that the third and
 357 --   fourth indices of c-t1 have been switched. This is because we
 358 --   store the triangles oriented such that their volume is
 359 --   positive. If T and T-tilde share \<v0,v1,v3\> and v2,v2-tilde point
 360 --   in opposite directions, one of them has to have negative volume!
 361 prop_c0201_identity1 :: Cube -> Bool
 362 prop_c0201_identity1 cube =
 363     c t0 0 2 0 1 ~= (c t0 0 1 1 1 + c t1 0 1 1 1) / 2
 364       where
 365         t0 = tetrahedron0 cube
 366         t1 = tetrahedron1 cube
 367
 368
 369 -- | Given in Sorokina and Zeilfelder, p. 79. Note that the third and
 370 --   fourth indices of c-t1 have been switched. This is because we
 371 --   store the triangles oriented such that their volume is
 372 --   positive. If T and T-tilde share \<v0,v1,v3\> and v2,v2-tilde point
 373 --   in opposite directions, one of them has to have negative volume!
 374 prop_c0300_identity2 :: Cube -> Bool
 375 prop_c0300_identity2 cube =
 376     c t0 0 3 0 0 ~= (c t0 0 2 1 0 + c t1 0 2 0 1) / 2
 377       where
 378         t0 = tetrahedron0 cube
 379         t1 = tetrahedron1 cube
 380
 381
 382 -- | Given in Sorokina and Zeilfelder, p. 79. Note that the third and
 383 --   fourth indices of c-t1 have been switched. This is because we
 384 --   store the triangles oriented such that their volume is
 385 --   positive. If T and T-tilde share \<v0,v1,v3\> and v2,v2-tilde point
 386 --   in opposite directions, one of them has to have negative volume!
 387 prop_c1101_identity :: Cube -> Bool
 388 prop_c1101_identity cube =
 389     c t0 1 1 0 1 ~= (c t0 1 0 1 1 + c t1 1 0 1 1) / 2
 390       where
 391         t0 = tetrahedron0 cube
 392         t1 = tetrahedron1 cube
 393
 394
 395 -- | Given in Sorokina and Zeilfelder, p. 79. Note that the third and
 396 --   fourth indices of c-t1 have been switched. This is because we
 397 --   store the triangles oriented such that their volume is
 398 --   positive. If T and T-tilde share \<v0,v1,v3\> and v2,v2-tilde point
 399 --   in opposite directions, one of them has to have negative volume!
 400 prop_c1200_identity2 :: Cube -> Bool
 401 prop_c1200_identity2 cube =
 402     c t0 1 2 0 0 ~= (c t0 1 1 1 0 + c t1 1 1 0 1) / 2
 403       where
 404         t0 = tetrahedron0 cube
 405         t1 = tetrahedron1 cube
 406
 407
 408 -- | Given in Sorokina and Zeilfelder, p. 79. Note that the third and
 409 --   fourth indices of c-t1 have been switched. This is because we
 410 --   store the triangles oriented such that their volume is
 411 --   positive. If T and T-tilde share \<v0,v1,v3\> and v2,v2-tilde point
 412 --   in opposite directions, one of them has to have negative volume!
 413 prop_c2100_identity2 :: Cube -> Bool
 414 prop_c2100_identity2 cube =
 415     c t0 2 1 0 0 ~= (c t0 2 0 1 0 + c t1 2 0 0 1) / 2
 416       where
 417         t0 = tetrahedron0 cube
 418         t1 = tetrahedron1 cube
 419
 420
 421 prop_t0_shares_edge_with_t6 :: Cube -> Bool
 422 prop_t0_shares_edge_with_t6 cube =
 423     (v2 t0) == (v3 t6) && (v3 t0) == (v2 t6)
 424       where
 425         t0 = tetrahedron0 cube
 426         t6 = tetrahedron6 cube
 427
 428
 429 -- | Given in Sorokina and Zeilfelder, p. 79.
 430 prop_c3000_identity :: Cube -> Bool
 431 prop_c3000_identity cube =
 432     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)
 433       where
 434         t0 = tetrahedron0 cube
 435         t6 = tetrahedron6 cube
 436
 437
 438 -- | Given in Sorokina and Zeilfelder, p. 79.
 439 prop_c2010_identity :: Cube -> Bool
 440 prop_c2010_identity cube =
 441     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)
 442       where
 443         t0 = tetrahedron0 cube
 444         t6 = tetrahedron6 cube
 445
 446
 447 -- | Given in Sorokina and Zeilfelder, p. 79.
 448 prop_c2001_identity :: Cube -> Bool
 449 prop_c2001_identity cube =
 450     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)
 451       where
 452         t0 = tetrahedron0 cube
 453         t6 = tetrahedron6 cube
 454
 455 -- | Given in Sorokina and Zeilfelder, p. 79.
 456 prop_c1020_identity :: Cube -> Bool
 457 prop_c1020_identity cube =
 458     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)
 459       where
 460         t0 = tetrahedron0 cube
 461         t6 = tetrahedron6 cube
 462
 463
 464 -- | Given in Sorokina and Zeilfelder, p. 79.
 465 prop_c1002_identity :: Cube -> Bool
 466 prop_c1002_identity cube =
 467     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)
 468       where
 469         t0 = tetrahedron0 cube
 470         t6 = tetrahedron6 cube
 471
 472
 473 -- | Given in Sorokina and Zeilfelder, p. 79.
 474 prop_c1011_identity :: Cube -> Bool
 475 prop_c1011_identity cube =
 476     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)
 477       where
 478         t0 = tetrahedron0 cube
 479         t6 = tetrahedron6 cube
 480
 481
 482
 483 -- | Given in Sorokina and Zeilfelder, p. 78.
 484 -- prop_cijk1_identity :: Cube -> Bool
 485 -- prop_cijk1_identity cube =
 486 --      and [ c t0 i j k 1 ~=
 487 --                  (c t1 (i+1) j k 0) * ((b0 t0) (v3 t1)) +
 488 --                  (c t1 i (j+1) k 0) * ((b1 t0) (v3 t1)) +
 489 --                  (c t1 i j (k+1) 0) * ((b2 t0) (v3 t1)) +
 490 --                  (c t1 i j k 1) * ((b3 t0) (v3 t1)) | i <- [0..2],
 491 --                                                       j <- [0..2],
 492 --                                                       k <- [0..2],
 493 --                                                       i + j + k == 2]
 494 --       where
 495 --         t0 = tetrahedron0 cube
 496 --         t1 = tetrahedron1 cube