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
   9 import Tests.FunctionValues ()
  10 import Tetrahedron (b0, b1, b2, b3, c, fv,
  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. Repeats
 278 --   prop_c0120_identity2 with tetrahedrons 3 and 2.
 279 prop_c0120_identity2 :: Cube -> Bool
 280 prop_c0120_identity2 cube =
 281    c t3 0 1 2 0 ~= (c t3 0 0 2 1 + c t2 0 0 1 2) / 2
 282      where
 283        t3 = tetrahedron3 cube
 284        t2 = tetrahedron2 cube
 285
 286 -- | Given in Sorokina and Zeilfelder, p. 79. Repeats
 287 --   prop_c0120_identity1 with tetrahedrons 2 and 1.
 288 prop_c0120_identity3 :: Cube -> Bool
 289 prop_c0120_identity3 cube =
 290    c t2 0 1 2 0 ~= (c t2 0 0 2 1 + c t1 0 0 1 2) / 2
 291      where
 292        t2 = tetrahedron2 cube
 293        t1 = tetrahedron1 cube
 294
 295
 296 -- | Given in Sorokina and Zeilfelder, p. 79. Repeats
 297 --   prop_c0120_identity1 with tetrahedrons 4 and 7.
 298 prop_c0120_identity4 :: Cube -> Bool
 299 prop_c0120_identity4 cube =
 300    c t4 0 1 2 0 ~= (c t4 0 0 2 1 + c t7 0 0 1 2) / 2
 301      where
 302        t4 = tetrahedron4 cube
 303        t7 = tetrahedron7 cube
 304
 305
 306 -- | Given in Sorokina and Zeilfelder, p. 79. Repeats
 307 --   prop_c0120_identity1 with tetrahedrons 7 and 6.
 308 prop_c0120_identity5 :: Cube -> Bool
 309 prop_c0120_identity5 cube =
 310    c t7 0 1 2 0 ~= (c t7 0 0 2 1 + c t6 0 0 1 2) / 2
 311      where
 312        t7 = tetrahedron7 cube
 313        t6 = tetrahedron6 cube
 314
 315
 316 -- | Given in Sorokina and Zeilfelder, p. 79. Repeats
 317 --   prop_c0120_identity1 with tetrahedrons 6 and 5.
 318 prop_c0120_identity6 :: Cube -> Bool
 319 prop_c0120_identity6 cube =
 320    c t6 0 1 2 0 ~= (c t6 0 0 2 1 + c t5 0 0 1 2) / 2
 321      where
 322        t6 = tetrahedron6 cube
 323        t5 = tetrahedron5 cube
 324
 325
 326 -- | Given in Sorokina and Zeilfelder, p. 79. Note that the third and
 327 --   fourth indices of c-t3 have been switched. This is because we
 328 --   store the triangles oriented such that their volume is
 329 --   positive. If T and T-tilde share \<v0,v1,v2\> and v3,v3-tilde point
 330 --   in opposite directions, one of them has to have negative volume!
 331 prop_c0210_identity1 :: Cube -> Bool
 332 prop_c0210_identity1 cube =
 333     c t0 0 2 1 0 ~= (c t0 0 1 1 1 + c t3 0 1 1 1) / 2
 334       where
 335         t0 = tetrahedron0 cube
 336         t3 = tetrahedron3 cube
 337
 338
 339 -- | Given in Sorokina and Zeilfelder, p. 79. Note that the third and
 340 --   fourth indices of c-t3 have been switched. This is because we
 341 --   store the triangles oriented such that their volume is
 342 --   positive. If T and T-tilde share \<v0,v1,v2\> and v3,v3-tilde point
 343 --   in opposite directions, one of them has to have negative volume!
 344 prop_c0300_identity1 :: Cube -> Bool
 345 prop_c0300_identity1 cube =
 346     c t0 0 3 0 0 ~= (c t0 0 2 0 1 + c t3 0 2 1 0) / 2
 347       where
 348         t0 = tetrahedron0 cube
 349         t3 = tetrahedron3 cube
 350
 351
 352 -- | Given in Sorokina and Zeilfelder, p. 79. Note that the third and
 353 --   fourth indices of c-t3 have been switched. This is because we
 354 --   store the triangles oriented such that their volume is
 355 --   positive. If T and T-tilde share \<v0,v1,v2\> and v3,v3-tilde point
 356 --   in opposite directions, one of them has to have negative volume!
 357 prop_c1110_identity :: Cube -> Bool
 358 prop_c1110_identity cube =
 359     c t0 1 1 1 0 ~= (c t0 1 0 1 1 + c t3 1 0 1 1) / 2
 360       where
 361         t0 = tetrahedron0 cube
 362         t3 = tetrahedron3 cube
 363
 364
 365 -- | Given in Sorokina and Zeilfelder, p. 79. Note that the third and
 366 --   fourth indices of c-t3 have been switched. This is because we
 367 --   store the triangles oriented such that their volume is
 368 --   positive. If T and T-tilde share \<v0,v1,v2\> and v3,v3-tilde point
 369 --   in opposite directions, one of them has to have negative volume!
 370 prop_c1200_identity1 :: Cube -> Bool
 371 prop_c1200_identity1 cube =
 372     c t0 1 2 0 0 ~= (c t0 1 1 0 1 + c t3 1 1 1 0) / 2
 373       where
 374         t0 = tetrahedron0 cube
 375         t3 = tetrahedron3 cube
 376
 377
 378 -- | Given in Sorokina and Zeilfelder, p. 79. Note that the third and
 379 --   fourth indices of c-t3 have been switched. This is because we
 380 --   store the triangles oriented such that their volume is
 381 --   positive. If T and T-tilde share \<v0,v1,v2\> and v3,v3-tilde point
 382 --   in opposite directions, one of them has to have negative volume!
 383 prop_c2100_identity1 :: Cube -> Bool
 384 prop_c2100_identity1 cube =
 385     c t0 2 1 0 0 ~= (c t0 2 0 0 1 + c t3 2 0 1 0) / 2
 386       where
 387         t0 = tetrahedron0 cube
 388         t3 = tetrahedron3 cube
 389
 390
 391
 392 -- | Given in Sorokina and Zeilfelder, p. 79. Note that the third and
 393 --   fourth indices of c-t1 have been switched. This is because we
 394 --   store the triangles oriented such that their volume is
 395 --   positive. If T and T-tilde share \<v0,v1,v3\> and v2,v2-tilde point
 396 --   in opposite directions, one of them has to have negative volume!
 397 prop_c0102_identity1 :: Cube -> Bool
 398 prop_c0102_identity1 cube =
 399     c t0 0 1 0 2 ~= (c t0 0 0 1 2 + c t1 0 0 2 1) / 2
 400       where
 401         t0 = tetrahedron0 cube
 402         t1 = tetrahedron1 cube
 403
 404
 405 -- | Given in Sorokina and Zeilfelder, p. 79. Note that the third and
 406 --   fourth indices of c-t1 have been switched. This is because we
 407 --   store the triangles oriented such that their volume is
 408 --   positive. If T and T-tilde share \<v0,v1,v3\> and v2,v2-tilde point
 409 --   in opposite directions, one of them has to have negative volume!
 410 prop_c0201_identity1 :: Cube -> Bool
 411 prop_c0201_identity1 cube =
 412     c t0 0 2 0 1 ~= (c t0 0 1 1 1 + c t1 0 1 1 1) / 2
 413       where
 414         t0 = tetrahedron0 cube
 415         t1 = tetrahedron1 cube
 416
 417
 418 -- | Given in Sorokina and Zeilfelder, p. 79. Note that the third and
 419 --   fourth indices of c-t1 have been switched. This is because we
 420 --   store the triangles oriented such that their volume is
 421 --   positive. If T and T-tilde share \<v0,v1,v3\> and v2,v2-tilde point
 422 --   in opposite directions, one of them has to have negative volume!
 423 prop_c0300_identity2 :: Cube -> Bool
 424 prop_c0300_identity2 cube =
 425     c t0 0 3 0 0 ~= (c t0 0 2 1 0 + c t1 0 2 0 1) / 2
 426       where
 427         t0 = tetrahedron0 cube
 428         t1 = tetrahedron1 cube
 429
 430
 431 -- | Given in Sorokina and Zeilfelder, p. 79. Note that the third and
 432 --   fourth indices of c-t1 have been switched. This is because we
 433 --   store the triangles oriented such that their volume is
 434 --   positive. If T and T-tilde share \<v0,v1,v3\> and v2,v2-tilde point
 435 --   in opposite directions, one of them has to have negative volume!
 436 prop_c1101_identity :: Cube -> Bool
 437 prop_c1101_identity cube =
 438     c t0 1 1 0 1 ~= (c t0 1 0 1 1 + c t1 1 0 1 1) / 2
 439       where
 440         t0 = tetrahedron0 cube
 441         t1 = tetrahedron1 cube
 442
 443
 444 -- | Given in Sorokina and Zeilfelder, p. 79. Note that the third and
 445 --   fourth indices of c-t1 have been switched. This is because we
 446 --   store the triangles oriented such that their volume is
 447 --   positive. If T and T-tilde share \<v0,v1,v3\> and v2,v2-tilde point
 448 --   in opposite directions, one of them has to have negative volume!
 449 prop_c1200_identity2 :: Cube -> Bool
 450 prop_c1200_identity2 cube =
 451     c t0 1 2 0 0 ~= (c t0 1 1 1 0 + c t1 1 1 0 1) / 2
 452       where
 453         t0 = tetrahedron0 cube
 454         t1 = tetrahedron1 cube
 455
 456
 457 -- | Given in Sorokina and Zeilfelder, p. 79. Note that the third and
 458 --   fourth indices of c-t1 have been switched. This is because we
 459 --   store the triangles oriented such that their volume is
 460 --   positive. If T and T-tilde share \<v0,v1,v3\> and v2,v2-tilde point
 461 --   in opposite directions, one of them has to have negative volume!
 462 prop_c2100_identity2 :: Cube -> Bool
 463 prop_c2100_identity2 cube =
 464     c t0 2 1 0 0 ~= (c t0 2 0 1 0 + c t1 2 0 0 1) / 2
 465       where
 466         t0 = tetrahedron0 cube
 467         t1 = tetrahedron1 cube
 468
 469
 470 -- | Given in Sorokina and Zeilfelder, p. 79.
 471 prop_c3000_identity :: Cube -> Bool
 472 prop_c3000_identity cube =
 473     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)
 474       where
 475         t0 = tetrahedron0 cube
 476         t6 = (tetrahedron6 cube) { v2 = (v3 t6), v3 = (v2 t6) }
 477
 478
 479 -- | Given in Sorokina and Zeilfelder, p. 79.
 480 prop_c2010_identity :: Cube -> Bool
 481 prop_c2010_identity cube =
 482     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)
 483       where
 484         t0 = tetrahedron0 cube
 485         t6 = tetrahedron6 cube
 486
 487
 488 -- | Given in Sorokina and Zeilfelder, p. 79.
 489 prop_c2001_identity :: Cube -> Bool
 490 prop_c2001_identity cube =
 491     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)
 492       where
 493         t0 = tetrahedron0 cube
 494         t6 = tetrahedron6 cube
 495
 496 -- | Given in Sorokina and Zeilfelder, p. 79.
 497 prop_c1020_identity :: Cube -> Bool
 498 prop_c1020_identity cube =
 499     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)
 500       where
 501         t0 = tetrahedron0 cube
 502         t6 = tetrahedron6 cube
 503
 504
 505 -- | Given in Sorokina and Zeilfelder, p. 79.
 506 prop_c1002_identity :: Cube -> Bool
 507 prop_c1002_identity cube =
 508     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)
 509       where
 510         t0 = tetrahedron0 cube
 511         t6 = tetrahedron6 cube
 512
 513
 514 -- | Given in Sorokina and Zeilfelder, p. 79.
 515 prop_c1011_identity :: Cube -> Bool
 516 prop_c1011_identity cube =
 517     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)
 518       where
 519         t0 = tetrahedron0 cube
 520         t6 = tetrahedron6 cube
 521
 522
 523
 524 -- | Given in Sorokina and Zeilfelder, p. 78.
 525 -- prop_cijk1_identity :: Cube -> Bool
 526 -- prop_cijk1_identity cube =
 527 --      and [ c t0 i j k 1 ~=
 528 --                  (c t1 (i+1) j k 0) * ((b0 t0) (v3 t1)) +
 529 --                  (c t1 i (j+1) k 0) * ((b1 t0) (v3 t1)) +
 530 --                  (c t1 i j (k+1) 0) * ((b2 t0) (v3 t1)) +
 531 --                  (c t1 i j k 1) * ((b3 t0) (v3 t1)) | i <- [0..2],
 532 --                                                       j <- [0..2],
 533 --                                                       k <- [0..2],
 534 --                                                       i + j + k == 2]
 535 --       where
 536 --         t0 = tetrahedron0 cube
 537 --         t1 = tetrahedron1 cube
 538
 539
 540 -- | We know what (c t6 2 1 0 0) should be from Sorokina and Zeilfelder, p. 87.
 541 prop_c_tilde_2100_correct :: Cube -> Bool
 542 prop_c_tilde_2100_correct cube =
 543     c t6 2 1 0 0 == (3/8)*int + (1/12)*(f + r + l + b) + (1/64)*(ft + rt + lt + bt)
 544                               + (7/48)*t + (1/48)*d + (1/96)*(fr + fl + br + bl)
 545                               + (1/192)*(fd + rd + ld + bd)
 546     where
 547       t6 = tetrahedron6 cube
 548       fvs = Tetrahedron.fv t6
 549       int = interior fvs
 550       f = front fvs
 551       r = right fvs
 552       l = left fvs
 553       b = back fvs
 554       ft = front_top fvs
 555       rt = right_top fvs
 556       lt = left_top fvs
 557       bt = back_top fvs
 558       t = top fvs
 559       d = down fvs
 560       fr = front_right fvs
 561       fl = front_left fvs
 562       br = back_right fvs
 563       bl = back_left fvs
 564       fd = front_down fvs
 565       rd = right_down fvs
 566       ld = left_down fvs
 567       bd = back_down fvs
 568
 569 -- Tests to check that the correct edges are incidental.
 570 prop_t0_shares_edge_with_t1 :: Cube -> Bool
 571 prop_t0_shares_edge_with_t1 cube =
 572     (v1 t0) == (v1 t1) && (v3 t0) == (v2 t1)
 573       where
 574         t0 = tetrahedron0 cube
 575         t1 = tetrahedron1 cube
 576
 577 prop_t0_shares_edge_with_t3 :: Cube -> Bool
 578 prop_t0_shares_edge_with_t3 cube =
 579     (v1 t0) == (v1 t3) && (v2 t0) == (v3 t3)
 580       where
 581         t0 = tetrahedron0 cube
 582         t3 = tetrahedron3 cube
 583
 584 prop_t0_shares_edge_with_t6 :: Cube -> Bool
 585 prop_t0_shares_edge_with_t6 cube =
 586     (v2 t0) == (v3 t6) && (v3 t0) == (v2 t6)
 587       where
 588         t0 = tetrahedron0 cube
 589         t6 = tetrahedron6 cube
 590
 591 prop_t1_shares_edge_with_t2 :: Cube -> Bool
 592 prop_t1_shares_edge_with_t2 cube =
 593     (v1 t1) == (v1 t2) && (v3 t1) == (v2 t2)
 594       where
 595         t1 = tetrahedron1 cube
 596         t2 = tetrahedron2 cube
 597
 598 prop_t1_shares_edge_with_t19 :: Cube -> Bool
 599 prop_t1_shares_edge_with_t19 cube =
 600     (v2 t1) == (v3 t19) && (v3 t1) == (v2 t19)
 601       where
 602         t1 = tetrahedron1 cube
 603         t19 = tetrahedron19 cube
 604
 605 prop_t2_shares_edge_with_t3 :: Cube -> Bool
 606 prop_t2_shares_edge_with_t3 cube =
 607     (v1 t1) == (v1 t2) && (v3 t1) == (v2 t2)
 608       where
 609         t1 = tetrahedron1 cube
 610         t2 = tetrahedron2 cube
 611
 612 prop_t2_shares_edge_with_t12 :: Cube -> Bool
 613 prop_t2_shares_edge_with_t12 cube =
 614     (v2 t2) == (v3 t12) && (v3 t2) == (v2 t12)
 615       where
 616         t2 = tetrahedron2 cube
 617         t12 = tetrahedron12 cube
 618
 619 prop_t3_shares_edge_with_t21 :: Cube -> Bool
 620 prop_t3_shares_edge_with_t21 cube =
 621     (v2 t3) == (v3 t21) && (v3 t3) == (v2 t21)
 622       where
 623         t3 = tetrahedron3 cube
 624         t21 = tetrahedron21 cube
 625
 626 prop_t4_shares_edge_with_t5 :: Cube -> Bool
 627 prop_t4_shares_edge_with_t5 cube =
 628     (v1 t4) == (v1 t5) && (v3 t4) == (v2 t5)
 629       where
 630         t4 = tetrahedron4 cube
 631         t5 = tetrahedron5 cube
 632
 633 prop_t4_shares_edge_with_t7 :: Cube -> Bool
 634 prop_t4_shares_edge_with_t7 cube =
 635     (v1 t4) == (v1 t7) && (v2 t4) == (v3 t7)
 636       where
 637         t4 = tetrahedron4 cube
 638         t7 = tetrahedron7 cube
 639
 640 prop_t4_shares_edge_with_t10 :: Cube -> Bool
 641 prop_t4_shares_edge_with_t10 cube =
 642     (v2 t4) == (v3 t10) && (v3 t4) == (v2 t10)
 643       where
 644         t4 = tetrahedron4 cube
 645         t10 = tetrahedron10 cube
 646
 647 prop_t5_shares_edge_with_t6 :: Cube -> Bool
 648 prop_t5_shares_edge_with_t6 cube =
 649     (v1 t5) == (v1 t6) && (v3 t5) == (v2 t6)
 650       where
 651         t5 = tetrahedron5 cube
 652         t6 = tetrahedron6 cube
 653
 654 prop_t5_shares_edge_with_t16 :: Cube -> Bool
 655 prop_t5_shares_edge_with_t16 cube =
 656     (v2 t5) == (v3 t16) && (v3 t5) == (v2 t16)
 657       where
 658         t5 = tetrahedron5 cube
 659         t16 = tetrahedron16 cube
 660
 661 prop_t6_shares_edge_with_t7 :: Cube -> Bool
 662 prop_t6_shares_edge_with_t7 cube =
 663     (v1 t6) == (v1 t7) && (v3 t6) == (v2 t7)
 664       where
 665         t6 = tetrahedron6 cube
 666         t7 = tetrahedron7 cube
 667
 668 prop_t7_shares_edge_with_t20 :: Cube -> Bool
 669 prop_t7_shares_edge_with_t20 cube =
 670     (v2 t7) == (v3 t20) && (v2 t7) == (v3 t20)
 671       where
 672         t7 = tetrahedron7 cube
 673         t20 = tetrahedron20 cube