src/Tests/Cube.hs

   1 module Tests.Cube
   2 where
   3
   4 import Prelude hiding (LT)
   5 import Test.QuickCheck
   6
   7 import Cardinal
   8 import Comparisons
   9 import Cube
  10 import FunctionValues
  11 import Misc (all_equal)
  12 import Tests.FunctionValues ()
  13 import Tetrahedron (b0, b1, b2, b3, c, fv,
  14                     v0, v1, v2, v3, volume)
  15
  16 instance Arbitrary Cube where
  17     arbitrary = do
  18       (Positive h') <- arbitrary :: Gen (Positive Double)
  19       i' <- choose (coordmin, coordmax)
  20       j' <- choose (coordmin, coordmax)
  21       k' <- choose (coordmin, coordmax)
  22       fv' <- arbitrary :: Gen FunctionValues
  23       return (Cube h' i' j' k' fv')
  24         where
  25           coordmin = -268435456 -- -(2^29 / 2)
  26           coordmax = 268435456  -- +(2^29 / 2)
  27
  28
  29 -- Quickcheck tests.
  30
  31 -- | Since the grid size is necessarily positive, all tetrahedrons
  32 --   (which comprise cubes of positive volume) must have positive volume
  33 --   as well.
  34 prop_all_volumes_positive :: Cube -> Bool
  35 prop_all_volumes_positive cube =
  36     null nonpositive_volumes
  37     where
  38       ts = tetrahedrons cube
  39       volumes = map volume ts
  40       nonpositive_volumes = filter (<= 0) volumes
  41
  42 -- | In fact, since all of the tetrahedra are identical, we should
  43 --   already know their volumes. There's 24 tetrahedra to a cube, so
  44 --   we'd expect the volume of each one to be (1/24)*h^3.
  45 prop_tetrahedron0_volumes_exact :: Cube -> Bool
  46 prop_tetrahedron0_volumes_exact cube =
  47     volume (tetrahedron0 cube) ~= (1/24)*(delta^(3::Int))
  48     where
  49       delta = h cube
  50
  51 -- | In fact, since all of the tetrahedra are identical, we should
  52 --   already know their volumes. There's 24 tetrahedra to a cube, so
  53 --   we'd expect the volume of each one to be (1/24)*h^3.
  54 prop_tetrahedron1_volumes_exact :: Cube -> Bool
  55 prop_tetrahedron1_volumes_exact cube =
  56     volume (tetrahedron1 cube) ~= (1/24)*(delta^(3::Int))
  57     where
  58       delta = h cube
  59
  60 -- | In fact, since all of the tetrahedra are identical, we should
  61 --   already know their volumes. There's 24 tetrahedra to a cube, so
  62 --   we'd expect the volume of each one to be (1/24)*h^3.
  63 prop_tetrahedron2_volumes_exact :: Cube -> Bool
  64 prop_tetrahedron2_volumes_exact cube =
  65     volume (tetrahedron2 cube) ~= (1/24)*(delta^(3::Int))
  66     where
  67       delta = h cube
  68
  69 -- | In fact, since all of the tetrahedra are identical, we should
  70 --   already know their volumes. There's 24 tetrahedra to a cube, so
  71 --   we'd expect the volume of each one to be (1/24)*h^3.
  72 prop_tetrahedron3_volumes_exact :: Cube -> Bool
  73 prop_tetrahedron3_volumes_exact cube =
  74     volume (tetrahedron3 cube) ~= (1/24)*(delta^(3::Int))
  75     where
  76       delta = h cube
  77
  78 -- | In fact, since all of the tetrahedra are identical, we should
  79 --   already know their volumes. There's 24 tetrahedra to a cube, so
  80 --   we'd expect the volume of each one to be (1/24)*h^3.
  81 prop_tetrahedron4_volumes_exact :: Cube -> Bool
  82 prop_tetrahedron4_volumes_exact cube =
  83     volume (tetrahedron4 cube) ~= (1/24)*(delta^(3::Int))
  84     where
  85       delta = h cube
  86
  87 -- | In fact, since all of the tetrahedra are identical, we should
  88 --   already know their volumes. There's 24 tetrahedra to a cube, so
  89 --   we'd expect the volume of each one to be (1/24)*h^3.
  90 prop_tetrahedron5_volumes_exact :: Cube -> Bool
  91 prop_tetrahedron5_volumes_exact cube =
  92     volume (tetrahedron5 cube) ~= (1/24)*(delta^(3::Int))
  93     where
  94       delta = h cube
  95
  96 -- | In fact, since all of the tetrahedra are identical, we should
  97 --   already know their volumes. There's 24 tetrahedra to a cube, so
  98 --   we'd expect the volume of each one to be (1/24)*h^3.
  99 prop_tetrahedron6_volumes_exact :: Cube -> Bool
 100 prop_tetrahedron6_volumes_exact cube =
 101     volume (tetrahedron6 cube) ~= (1/24)*(delta^(3::Int))
 102     where
 103       delta = h cube
 104
 105 -- | In fact, since all of the tetrahedra are identical, we should
 106 --   already know their volumes. There's 24 tetrahedra to a cube, so
 107 --   we'd expect the volume of each one to be (1/24)*h^3.
 108 prop_tetrahedron7_volumes_exact :: Cube -> Bool
 109 prop_tetrahedron7_volumes_exact cube =
 110     volume (tetrahedron7 cube) ~= (1/24)*(delta^(3::Int))
 111     where
 112       delta = h cube
 113
 114 -- | All tetrahedron should have their v0 located at the center of the cube.
 115 prop_v0_all_equal :: Cube -> Bool
 116 prop_v0_all_equal cube = (v0 t0) == (v0 t1)
 117     where
 118       t0 = head (tetrahedrons cube) -- Doesn't matter which two we choose.
 119       t1 = head $ tail (tetrahedrons cube)
 120
 121
 122 -- | This pretty much repeats the prop_all_volumes_positive property,
 123 --   but will let me know which tetrahedrons's vertices are disoriented.
 124 prop_tetrahedron0_volumes_positive :: Cube -> Bool
 125 prop_tetrahedron0_volumes_positive cube =
 126     volume (tetrahedron0 cube) > 0
 127
 128 -- | This pretty much repeats the prop_all_volumes_positive property,
 129 --   but will let me know which tetrahedrons's vertices are disoriented.
 130 prop_tetrahedron1_volumes_positive :: Cube -> Bool
 131 prop_tetrahedron1_volumes_positive cube =
 132     volume (tetrahedron1 cube) > 0
 133
 134 -- | This pretty much repeats the prop_all_volumes_positive property,
 135 --   but will let me know which tetrahedrons's vertices are disoriented.
 136 prop_tetrahedron2_volumes_positive :: Cube -> Bool
 137 prop_tetrahedron2_volumes_positive cube =
 138     volume (tetrahedron2 cube) > 0
 139
 140 -- | This pretty much repeats the prop_all_volumes_positive property,
 141 --   but will let me know which tetrahedrons's vertices are disoriented.
 142 prop_tetrahedron3_volumes_positive :: Cube -> Bool
 143 prop_tetrahedron3_volumes_positive cube =
 144     volume (tetrahedron3 cube) > 0
 145
 146 -- | This pretty much repeats the prop_all_volumes_positive property,
 147 --   but will let me know which tetrahedrons's vertices are disoriented.
 148 prop_tetrahedron4_volumes_positive :: Cube -> Bool
 149 prop_tetrahedron4_volumes_positive cube =
 150     volume (tetrahedron4 cube) > 0
 151
 152 -- | This pretty much repeats the prop_all_volumes_positive property,
 153 --   but will let me know which tetrahedrons's vertices are disoriented.
 154 prop_tetrahedron5_volumes_positive :: Cube -> Bool
 155 prop_tetrahedron5_volumes_positive cube =
 156     volume (tetrahedron5 cube) > 0
 157
 158 -- | This pretty much repeats the prop_all_volumes_positive property,
 159 --   but will let me know which tetrahedrons's vertices are disoriented.
 160 prop_tetrahedron6_volumes_positive :: Cube -> Bool
 161 prop_tetrahedron6_volumes_positive cube =
 162     volume (tetrahedron6 cube) > 0
 163
 164 -- | This pretty much repeats the prop_all_volumes_positive property,
 165 --   but will let me know which tetrahedrons's vertices are disoriented.
 166 prop_tetrahedron7_volumes_positive :: Cube -> Bool
 167 prop_tetrahedron7_volumes_positive cube =
 168     volume (tetrahedron7 cube) > 0
 169
 170 -- | This pretty much repeats the prop_all_volumes_positive property,
 171 --   but will let me know which tetrahedrons's vertices are disoriented.
 172 prop_tetrahedron8_volumes_positive :: Cube -> Bool
 173 prop_tetrahedron8_volumes_positive cube =
 174     volume (tetrahedron8 cube) > 0
 175
 176 -- | This pretty much repeats the prop_all_volumes_positive property,
 177 --   but will let me know which tetrahedrons's vertices are disoriented.
 178 prop_tetrahedron9_volumes_positive :: Cube -> Bool
 179 prop_tetrahedron9_volumes_positive cube =
 180     volume (tetrahedron9 cube) > 0
 181
 182 -- | This pretty much repeats the prop_all_volumes_positive property,
 183 --   but will let me know which tetrahedrons's vertices are disoriented.
 184 prop_tetrahedron10_volumes_positive :: Cube -> Bool
 185 prop_tetrahedron10_volumes_positive cube =
 186     volume (tetrahedron10 cube) > 0
 187
 188 -- | This pretty much repeats the prop_all_volumes_positive property,
 189 --   but will let me know which tetrahedrons's vertices are disoriented.
 190 prop_tetrahedron11_volumes_positive :: Cube -> Bool
 191 prop_tetrahedron11_volumes_positive cube =
 192     volume (tetrahedron11 cube) > 0
 193
 194 -- | This pretty much repeats the prop_all_volumes_positive property,
 195 --   but will let me know which tetrahedrons's vertices are disoriented.
 196 prop_tetrahedron12_volumes_positive :: Cube -> Bool
 197 prop_tetrahedron12_volumes_positive cube =
 198     volume (tetrahedron12 cube) > 0
 199
 200 -- | This pretty much repeats the prop_all_volumes_positive property,
 201 --   but will let me know which tetrahedrons's vertices are disoriented.
 202 prop_tetrahedron13_volumes_positive :: Cube -> Bool
 203 prop_tetrahedron13_volumes_positive cube =
 204     volume (tetrahedron13 cube) > 0
 205
 206 -- | This pretty much repeats the prop_all_volumes_positive property,
 207 --   but will let me know which tetrahedrons's vertices are disoriented.
 208 prop_tetrahedron14_volumes_positive :: Cube -> Bool
 209 prop_tetrahedron14_volumes_positive cube =
 210     volume (tetrahedron14 cube) > 0
 211
 212 -- | This pretty much repeats the prop_all_volumes_positive property,
 213 --   but will let me know which tetrahedrons's vertices are disoriented.
 214 prop_tetrahedron15_volumes_positive :: Cube -> Bool
 215 prop_tetrahedron15_volumes_positive cube =
 216     volume (tetrahedron15 cube) > 0
 217
 218 -- | This pretty much repeats the prop_all_volumes_positive property,
 219 --   but will let me know which tetrahedrons's vertices are disoriented.
 220 prop_tetrahedron16_volumes_positive :: Cube -> Bool
 221 prop_tetrahedron16_volumes_positive cube =
 222     volume (tetrahedron16 cube) > 0
 223
 224 -- | This pretty much repeats the prop_all_volumes_positive property,
 225 --   but will let me know which tetrahedrons's vertices are disoriented.
 226 prop_tetrahedron17_volumes_positive :: Cube -> Bool
 227 prop_tetrahedron17_volumes_positive cube =
 228     volume (tetrahedron17 cube) > 0
 229
 230 -- | This pretty much repeats the prop_all_volumes_positive property,
 231 --   but will let me know which tetrahedrons's vertices are disoriented.
 232 prop_tetrahedron18_volumes_positive :: Cube -> Bool
 233 prop_tetrahedron18_volumes_positive cube =
 234     volume (tetrahedron18 cube) > 0
 235
 236 -- | This pretty much repeats the prop_all_volumes_positive property,
 237 --   but will let me know which tetrahedrons's vertices are disoriented.
 238 prop_tetrahedron19_volumes_positive :: Cube -> Bool
 239 prop_tetrahedron19_volumes_positive cube =
 240     volume (tetrahedron19 cube) > 0
 241
 242 -- | This pretty much repeats the prop_all_volumes_positive property,
 243 --   but will let me know which tetrahedrons's vertices are disoriented.
 244 prop_tetrahedron20_volumes_positive :: Cube -> Bool
 245 prop_tetrahedron20_volumes_positive cube =
 246     volume (tetrahedron20 cube) > 0
 247
 248 -- | This pretty much repeats the prop_all_volumes_positive property,
 249 --   but will let me know which tetrahedrons's vertices are disoriented.
 250 prop_tetrahedron21_volumes_positive :: Cube -> Bool
 251 prop_tetrahedron21_volumes_positive cube =
 252     volume (tetrahedron21 cube) > 0
 253
 254 -- | This pretty much repeats the prop_all_volumes_positive property,
 255 --   but will let me know which tetrahedrons's vertices are disoriented.
 256 prop_tetrahedron22_volumes_positive :: Cube -> Bool
 257 prop_tetrahedron22_volumes_positive cube =
 258     volume (tetrahedron22 cube) > 0
 259
 260 -- | This pretty much repeats the prop_all_volumes_positive property,
 261 --   but will let me know which tetrahedrons's vertices are disoriented.
 262 prop_tetrahedron23_volumes_positive :: Cube -> Bool
 263 prop_tetrahedron23_volumes_positive cube =
 264     volume (tetrahedron23 cube) > 0
 265
 266
 267 -- | Given in Sorokina and Zeilfelder, p. 79. Note that the third and
 268 --   fourth indices of c-t3 have been switched. This is because we
 269 --   store the triangles oriented such that their volume is
 270 --   positive. If T and T-tilde share \<v0,v1,v2\> and v3,v3-tilde point
 271 --   in opposite directions, one of them has to have negative volume!
 272 prop_c0120_identity1 :: Cube -> Bool
 273 prop_c0120_identity1 cube =
 274    c t0 0 1 2 0 ~= (c t0 0 0 2 1 + c t3 0 0 1 2) / 2
 275      where
 276        t0 = tetrahedron0 cube
 277        t3 = tetrahedron3 cube
 278
 279
 280 -- | Given in Sorokina and Zeilfelder, p. 79. Repeats
 281 --   prop_c0120_identity2 with tetrahedrons 3 and 2.
 282 prop_c0120_identity2 :: Cube -> Bool
 283 prop_c0120_identity2 cube =
 284    c t3 0 1 2 0 ~= (c t3 0 0 2 1 + c t2 0 0 1 2) / 2
 285      where
 286        t3 = tetrahedron3 cube
 287        t2 = tetrahedron2 cube
 288
 289 -- | Given in Sorokina and Zeilfelder, p. 79. Repeats
 290 --   prop_c0120_identity1 with tetrahedrons 2 and 1.
 291 prop_c0120_identity3 :: Cube -> Bool
 292 prop_c0120_identity3 cube =
 293    c t2 0 1 2 0 ~= (c t2 0 0 2 1 + c t1 0 0 1 2) / 2
 294      where
 295        t2 = tetrahedron2 cube
 296        t1 = tetrahedron1 cube
 297
 298
 299 -- | Given in Sorokina and Zeilfelder, p. 79. Repeats
 300 --   prop_c0120_identity1 with tetrahedrons 4 and 7.
 301 prop_c0120_identity4 :: Cube -> Bool
 302 prop_c0120_identity4 cube =
 303    c t4 0 1 2 0 ~= (c t4 0 0 2 1 + c t7 0 0 1 2) / 2
 304      where
 305        t4 = tetrahedron4 cube
 306        t7 = tetrahedron7 cube
 307
 308
 309 -- | Given in Sorokina and Zeilfelder, p. 79. Repeats
 310 --   prop_c0120_identity1 with tetrahedrons 7 and 6.
 311 prop_c0120_identity5 :: Cube -> Bool
 312 prop_c0120_identity5 cube =
 313    c t7 0 1 2 0 ~= (c t7 0 0 2 1 + c t6 0 0 1 2) / 2
 314      where
 315        t7 = tetrahedron7 cube
 316        t6 = tetrahedron6 cube
 317
 318
 319 -- | Given in Sorokina and Zeilfelder, p. 79. Repeats
 320 --   prop_c0120_identity1 with tetrahedrons 6 and 5.
 321 prop_c0120_identity6 :: Cube -> Bool
 322 prop_c0120_identity6 cube =
 323    c t6 0 1 2 0 ~= (c t6 0 0 2 1 + c t5 0 0 1 2) / 2
 324      where
 325        t6 = tetrahedron6 cube
 326        t5 = tetrahedron5 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_c0210_identity1 :: Cube -> Bool
 335 prop_c0210_identity1 cube =
 336     c t0 0 2 1 0 ~= (c t0 0 1 1 1 + c t3 0 1 1 1) / 2
 337       where
 338         t0 = tetrahedron0 cube
 339         t3 = tetrahedron3 cube
 340
 341
 342 -- | Given in Sorokina and Zeilfelder, p. 79. Note that the third and
 343 --   fourth indices of c-t3 have been switched. This is because we
 344 --   store the triangles oriented such that their volume is
 345 --   positive. If T and T-tilde share \<v0,v1,v2\> and v3,v3-tilde point
 346 --   in opposite directions, one of them has to have negative volume!
 347 prop_c0300_identity1 :: Cube -> Bool
 348 prop_c0300_identity1 cube =
 349     c t0 0 3 0 0 ~= (c t0 0 2 0 1 + c t3 0 2 1 0) / 2
 350       where
 351         t0 = tetrahedron0 cube
 352         t3 = tetrahedron3 cube
 353
 354
 355 -- | Given in Sorokina and Zeilfelder, p. 79. Note that the third and
 356 --   fourth indices of c-t3 have been switched. This is because we
 357 --   store the triangles oriented such that their volume is
 358 --   positive. If T and T-tilde share \<v0,v1,v2\> and v3,v3-tilde point
 359 --   in opposite directions, one of them has to have negative volume!
 360 prop_c1110_identity :: Cube -> Bool
 361 prop_c1110_identity cube =
 362     c t0 1 1 1 0 ~= (c t0 1 0 1 1 + c t3 1 0 1 1) / 2
 363       where
 364         t0 = tetrahedron0 cube
 365         t3 = tetrahedron3 cube
 366
 367
 368 -- | Given in Sorokina and Zeilfelder, p. 79. Note that the third and
 369 --   fourth indices of c-t3 have been switched. This is because we
 370 --   store the triangles oriented such that their volume is
 371 --   positive. If T and T-tilde share \<v0,v1,v2\> and v3,v3-tilde point
 372 --   in opposite directions, one of them has to have negative volume!
 373 prop_c1200_identity1 :: Cube -> Bool
 374 prop_c1200_identity1 cube =
 375     c t0 1 2 0 0 ~= (c t0 1 1 0 1 + c t3 1 1 1 0) / 2
 376       where
 377         t0 = tetrahedron0 cube
 378         t3 = tetrahedron3 cube
 379
 380
 381 -- | Given in Sorokina and Zeilfelder, p. 79. Note that the third and
 382 --   fourth indices of c-t3 have been switched. This is because we
 383 --   store the triangles oriented such that their volume is
 384 --   positive. If T and T-tilde share \<v0,v1,v2\> and v3,v3-tilde point
 385 --   in opposite directions, one of them has to have negative volume!
 386 prop_c2100_identity1 :: Cube -> Bool
 387 prop_c2100_identity1 cube =
 388     c t0 2 1 0 0 ~= (c t0 2 0 0 1 + c t3 2 0 1 0) / 2
 389       where
 390         t0 = tetrahedron0 cube
 391         t3 = tetrahedron3 cube
 392
 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_c0102_identity1 :: Cube -> Bool
 401 prop_c0102_identity1 cube =
 402     c t0 0 1 0 2 ~= (c t0 0 0 1 2 + c t1 0 0 2 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_c0201_identity1 :: Cube -> Bool
 414 prop_c0201_identity1 cube =
 415     c t0 0 2 0 1 ~= (c t0 0 1 1 1 + c t1 0 1 1 1) / 2
 416       where
 417         t0 = tetrahedron0 cube
 418         t1 = tetrahedron1 cube
 419
 420
 421 -- | Given in Sorokina and Zeilfelder, p. 79. Note that the third and
 422 --   fourth indices of c-t1 have been switched. This is because we
 423 --   store the triangles oriented such that their volume is
 424 --   positive. If T and T-tilde share \<v0,v1,v3\> and v2,v2-tilde point
 425 --   in opposite directions, one of them has to have negative volume!
 426 prop_c0300_identity2 :: Cube -> Bool
 427 prop_c0300_identity2 cube =
 428     c t0 0 3 0 0 ~= (c t0 0 2 1 0 + c t1 0 2 0 1) / 2
 429       where
 430         t0 = tetrahedron0 cube
 431         t1 = tetrahedron1 cube
 432
 433
 434 -- | Given in Sorokina and Zeilfelder, p. 79. Note that the third and
 435 --   fourth indices of c-t1 have been switched. This is because we
 436 --   store the triangles oriented such that their volume is
 437 --   positive. If T and T-tilde share \<v0,v1,v3\> and v2,v2-tilde point
 438 --   in opposite directions, one of them has to have negative volume!
 439 prop_c1101_identity :: Cube -> Bool
 440 prop_c1101_identity cube =
 441     c t0 1 1 0 1 ~= (c t0 1 0 1 1 + c t1 1 0 1 1) / 2
 442       where
 443         t0 = tetrahedron0 cube
 444         t1 = tetrahedron1 cube
 445
 446
 447 -- | Given in Sorokina and Zeilfelder, p. 79. Note that the third and
 448 --   fourth indices of c-t1 have been switched. This is because we
 449 --   store the triangles oriented such that their volume is
 450 --   positive. If T and T-tilde share \<v0,v1,v3\> and v2,v2-tilde point
 451 --   in opposite directions, one of them has to have negative volume!
 452 prop_c1200_identity2 :: Cube -> Bool
 453 prop_c1200_identity2 cube =
 454     c t0 1 2 0 0 ~= (c t0 1 1 1 0 + c t1 1 1 0 1) / 2
 455       where
 456         t0 = tetrahedron0 cube
 457         t1 = tetrahedron1 cube
 458
 459
 460 -- | Given in Sorokina and Zeilfelder, p. 79. Note that the third and
 461 --   fourth indices of c-t1 have been switched. This is because we
 462 --   store the triangles oriented such that their volume is
 463 --   positive. If T and T-tilde share \<v0,v1,v3\> and v2,v2-tilde point
 464 --   in opposite directions, one of them has to have negative volume!
 465 prop_c2100_identity2 :: Cube -> Bool
 466 prop_c2100_identity2 cube =
 467     c t0 2 1 0 0 ~= (c t0 2 0 1 0 + c t1 2 0 0 1) / 2
 468       where
 469         t0 = tetrahedron0 cube
 470         t1 = tetrahedron1 cube
 471
 472
 473 -- | Given in Sorokina and Zeilfelder, p. 79.
 474 prop_c3000_identity :: Cube -> Bool
 475 prop_c3000_identity cube =
 476     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)
 477       where
 478         t0 = tetrahedron0 cube
 479         t6 = (tetrahedron6 cube) { v2 = (v3 t6), v3 = (v2 t6) }
 480
 481
 482 -- | Given in Sorokina and Zeilfelder, p. 79.
 483 prop_c2010_identity :: Cube -> Bool
 484 prop_c2010_identity cube =
 485     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)
 486       where
 487         t0 = tetrahedron0 cube
 488         t6 = tetrahedron6 cube
 489
 490
 491 -- | Given in Sorokina and Zeilfelder, p. 79.
 492 prop_c2001_identity :: Cube -> Bool
 493 prop_c2001_identity cube =
 494     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)
 495       where
 496         t0 = tetrahedron0 cube
 497         t6 = tetrahedron6 cube
 498
 499 -- | Given in Sorokina and Zeilfelder, p. 79.
 500 prop_c1020_identity :: Cube -> Bool
 501 prop_c1020_identity cube =
 502     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)
 503       where
 504         t0 = tetrahedron0 cube
 505         t6 = tetrahedron6 cube
 506
 507
 508 -- | Given in Sorokina and Zeilfelder, p. 79.
 509 prop_c1002_identity :: Cube -> Bool
 510 prop_c1002_identity cube =
 511     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)
 512       where
 513         t0 = tetrahedron0 cube
 514         t6 = tetrahedron6 cube
 515
 516
 517 -- | Given in Sorokina and Zeilfelder, p. 79.
 518 prop_c1011_identity :: Cube -> Bool
 519 prop_c1011_identity cube =
 520     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)
 521       where
 522         t0 = tetrahedron0 cube
 523         t6 = tetrahedron6 cube
 524
 525
 526
 527 -- | Given in Sorokina and Zeilfelder, p. 78.
 528 -- prop_cijk1_identity :: Cube -> Bool
 529 -- prop_cijk1_identity cube =
 530 --      and [ c t0 i j k 1 ~=
 531 --                  (c t1 (i+1) j k 0) * ((b0 t0) (v3 t1)) +
 532 --                  (c t1 i (j+1) k 0) * ((b1 t0) (v3 t1)) +
 533 --                  (c t1 i j (k+1) 0) * ((b2 t0) (v3 t1)) +
 534 --                  (c t1 i j k 1) * ((b3 t0) (v3 t1)) | i <- [0..2],
 535 --                                                       j <- [0..2],
 536 --                                                       k <- [0..2],
 537 --                                                       i + j + k == 2]
 538 --       where
 539 --         t0 = tetrahedron0 cube
 540 --         t1 = tetrahedron1 cube
 541
 542
 543
 544 -- | The function values at the interior should be the same for all tetrahedra.
 545 prop_interior_values_all_identical :: Cube -> Bool
 546 prop_interior_values_all_identical cube =
 547     all_equal [i0, i1, i2, i3, i4, i5, i6, i7, i8,
 548                i9, i10, i11, i12, i13, i14, i15, i16,
 549                i17, i18, i19, i20, i21, i22, i23]
 550     where
 551       i0 = eval (Tetrahedron.fv (tetrahedron0 cube)) $ I
 552       i1 = eval (Tetrahedron.fv (tetrahedron1 cube)) $ I
 553       i2 = eval (Tetrahedron.fv (tetrahedron2 cube)) $ I
 554       i3 = eval (Tetrahedron.fv (tetrahedron3 cube)) $ I
 555       i4 = eval (Tetrahedron.fv (tetrahedron4 cube)) $ I
 556       i5 = eval (Tetrahedron.fv (tetrahedron5 cube)) $ I
 557       i6 = eval (Tetrahedron.fv (tetrahedron6 cube)) $ I
 558       i7 = eval (Tetrahedron.fv (tetrahedron7 cube)) $ I
 559       i8 = eval (Tetrahedron.fv (tetrahedron8 cube)) $ I
 560       i9 = eval (Tetrahedron.fv (tetrahedron9 cube)) $ I
 561       i10 = eval (Tetrahedron.fv (tetrahedron10 cube)) $ I
 562       i11 = eval (Tetrahedron.fv (tetrahedron11 cube)) $ I
 563       i12 = eval (Tetrahedron.fv (tetrahedron12 cube)) $ I
 564       i13 = eval (Tetrahedron.fv (tetrahedron13 cube)) $ I
 565       i14 = eval (Tetrahedron.fv (tetrahedron14 cube)) $ I
 566       i15 = eval (Tetrahedron.fv (tetrahedron15 cube)) $ I
 567       i16 = eval (Tetrahedron.fv (tetrahedron16 cube)) $ I
 568       i17 = eval (Tetrahedron.fv (tetrahedron17 cube)) $ I
 569       i18 = eval (Tetrahedron.fv (tetrahedron18 cube)) $ I
 570       i19 = eval (Tetrahedron.fv (tetrahedron19 cube)) $ I
 571       i20 = eval (Tetrahedron.fv (tetrahedron20 cube)) $ I
 572       i21 = eval (Tetrahedron.fv (tetrahedron21 cube)) $ I
 573       i22 = eval (Tetrahedron.fv (tetrahedron22 cube)) $ I
 574       i23 = eval (Tetrahedron.fv (tetrahedron23 cube)) $ I
 575
 576
 577 -- | We know what (c t6 2 1 0 0) should be from Sorokina and Zeilfelder, p. 87.
 578 --   This test checks the rotation works as expected.
 579 prop_c_tilde_2100_rotation_correct :: Cube -> Bool
 580 prop_c_tilde_2100_rotation_correct cube =
 581     expr1 == expr2
 582     where
 583       t0 = tetrahedron0 cube
 584       t6 = tetrahedron6 cube
 585
 586       -- What gets computed for c2100 of t6.
 587       expr1 = eval (Tetrahedron.fv t6) $
 588           (3/8)*I +
 589           (1/12)*(T + R + L + D) +
 590           (1/64)*(FT + FR + FL + FD) +
 591           (7/48)*F +
 592           (1/48)*B +
 593           (1/96)*(RT + LD + LT + RD) +
 594           (1/192)*(BT + BR + BL + BD)
 595
 596       -- What should be computed for c2100 of t6.
 597       expr2 = eval (Tetrahedron.fv t0) $
 598               (3/8)*I +
 599               (1/12)*(F + R + L + B) +
 600               (1/64)*(FT + RT + LT + BT) +
 601               (7/48)*T +
 602               (1/48)*D +
 603               (1/96)*(FR + FL + BR + BL) +
 604               (1/192)*(FD + RD + LD + BD)
 605
 606
 607 -- | We know what (c t6 2 1 0 0) should be from Sorokina and Zeilfelder, p. 87.
 608 --   This test checks the actual value based on the FunctionValues of the cube.
 609 prop_c_tilde_2100_correct :: Cube -> Bool
 610 prop_c_tilde_2100_correct cube =
 611     c t6 2 1 0 0 == (3/8)*int + (1/12)*(f + r + l + b) + (1/64)*(ft + rt + lt + bt)
 612                               + (7/48)*t + (1/48)*d + (1/96)*(fr + fl + br + bl)
 613                               + (1/192)*(fd + rd + ld + bd)
 614     where
 615       t0 = tetrahedron0 cube
 616       t6 = tetrahedron6 cube
 617       fvs = Tetrahedron.fv t0
 618       int = interior fvs
 619       f = front fvs
 620       r = right fvs
 621       l = left fvs
 622       b = back fvs
 623       ft = front_top fvs
 624       rt = right_top fvs
 625       lt = left_top fvs
 626       bt = back_top fvs
 627       t = top fvs
 628       d = down fvs
 629       fr = front_right fvs
 630       fl = front_left fvs
 631       br = back_right fvs
 632       bl = back_left fvs
 633       fd = front_down fvs
 634       rd = right_down fvs
 635       ld = left_down fvs
 636       bd = back_down fvs
 637
 638 -- Tests to check that the correct edges are incidental.
 639 prop_t0_shares_edge_with_t1 :: Cube -> Bool
 640 prop_t0_shares_edge_with_t1 cube =
 641     (v1 t0) == (v1 t1) && (v3 t0) == (v2 t1)
 642       where
 643         t0 = tetrahedron0 cube
 644         t1 = tetrahedron1 cube
 645
 646 prop_t0_shares_edge_with_t3 :: Cube -> Bool
 647 prop_t0_shares_edge_with_t3 cube =
 648     (v1 t0) == (v1 t3) && (v2 t0) == (v3 t3)
 649       where
 650         t0 = tetrahedron0 cube
 651         t3 = tetrahedron3 cube
 652
 653 prop_t0_shares_edge_with_t6 :: Cube -> Bool
 654 prop_t0_shares_edge_with_t6 cube =
 655     (v2 t0) == (v3 t6) && (v3 t0) == (v2 t6)
 656       where
 657         t0 = tetrahedron0 cube
 658         t6 = tetrahedron6 cube
 659
 660 prop_t1_shares_edge_with_t2 :: Cube -> Bool
 661 prop_t1_shares_edge_with_t2 cube =
 662     (v1 t1) == (v1 t2) && (v3 t1) == (v2 t2)
 663       where
 664         t1 = tetrahedron1 cube
 665         t2 = tetrahedron2 cube
 666
 667 prop_t1_shares_edge_with_t19 :: Cube -> Bool
 668 prop_t1_shares_edge_with_t19 cube =
 669     (v2 t1) == (v3 t19) && (v3 t1) == (v2 t19)
 670       where
 671         t1 = tetrahedron1 cube
 672         t19 = tetrahedron19 cube
 673
 674 prop_t2_shares_edge_with_t3 :: Cube -> Bool
 675 prop_t2_shares_edge_with_t3 cube =
 676     (v1 t1) == (v1 t2) && (v3 t1) == (v2 t2)
 677       where
 678         t1 = tetrahedron1 cube
 679         t2 = tetrahedron2 cube
 680
 681 prop_t2_shares_edge_with_t12 :: Cube -> Bool
 682 prop_t2_shares_edge_with_t12 cube =
 683     (v2 t2) == (v3 t12) && (v3 t2) == (v2 t12)
 684       where
 685         t2 = tetrahedron2 cube
 686         t12 = tetrahedron12 cube
 687
 688 prop_t3_shares_edge_with_t21 :: Cube -> Bool
 689 prop_t3_shares_edge_with_t21 cube =
 690     (v2 t3) == (v3 t21) && (v3 t3) == (v2 t21)
 691       where
 692         t3 = tetrahedron3 cube
 693         t21 = tetrahedron21 cube
 694
 695 prop_t4_shares_edge_with_t5 :: Cube -> Bool
 696 prop_t4_shares_edge_with_t5 cube =
 697     (v1 t4) == (v1 t5) && (v3 t4) == (v2 t5)
 698       where
 699         t4 = tetrahedron4 cube
 700         t5 = tetrahedron5 cube
 701
 702 prop_t4_shares_edge_with_t7 :: Cube -> Bool
 703 prop_t4_shares_edge_with_t7 cube =
 704     (v1 t4) == (v1 t7) && (v2 t4) == (v3 t7)
 705       where
 706         t4 = tetrahedron4 cube
 707         t7 = tetrahedron7 cube
 708
 709 prop_t4_shares_edge_with_t10 :: Cube -> Bool
 710 prop_t4_shares_edge_with_t10 cube =
 711     (v2 t4) == (v3 t10) && (v3 t4) == (v2 t10)
 712       where
 713         t4 = tetrahedron4 cube
 714         t10 = tetrahedron10 cube
 715
 716 prop_t5_shares_edge_with_t6 :: Cube -> Bool
 717 prop_t5_shares_edge_with_t6 cube =
 718     (v1 t5) == (v1 t6) && (v3 t5) == (v2 t6)
 719       where
 720         t5 = tetrahedron5 cube
 721         t6 = tetrahedron6 cube
 722
 723 prop_t5_shares_edge_with_t16 :: Cube -> Bool
 724 prop_t5_shares_edge_with_t16 cube =
 725     (v2 t5) == (v3 t16) && (v3 t5) == (v2 t16)
 726       where
 727         t5 = tetrahedron5 cube
 728         t16 = tetrahedron16 cube
 729
 730 prop_t6_shares_edge_with_t7 :: Cube -> Bool
 731 prop_t6_shares_edge_with_t7 cube =
 732     (v1 t6) == (v1 t7) && (v3 t6) == (v2 t7)
 733       where
 734         t6 = tetrahedron6 cube
 735         t7 = tetrahedron7 cube
 736
 737 prop_t7_shares_edge_with_t20 :: Cube -> Bool
 738 prop_t7_shares_edge_with_t20 cube =
 739     (v2 t7) == (v3 t20) && (v2 t7) == (v3 t20)
 740       where
 741         t7 = tetrahedron7 cube
 742         t20 = tetrahedron20 cube