src/Tests/Cube.hs

   1 module Tests.Cube
   2 where
   3
   4 import Prelude hiding (LT)
   5
   6 import Cardinal
   7 import Comparisons
   8 import Cube hiding (i, j, k)
   9 import FunctionValues
  10 import Misc (all_equal)
  11 import Tests.FunctionValues ()
  12 import Tetrahedron (b0, b1, b2, b3, c, fv,
  13                     v0, v1, v2, v3, volume)
  14
  15
  16
  17 -- Quickcheck tests.
  18
  19 -- | Since the grid size is necessarily positive, all tetrahedra
  20 --   (which comprise cubes of positive volume) must have positive volume
  21 --   as well.
  22 prop_all_volumes_positive :: Cube -> Bool
  23 prop_all_volumes_positive cube =
  24     null nonpositive_volumes
  25     where
  26       ts = tetrahedra cube
  27       volumes = map volume ts
  28       nonpositive_volumes = filter (<= 0) volumes
  29
  30 -- | In fact, since all of the tetrahedra are identical, we should
  31 --   already know their volumes. There's 24 tetrahedra to a cube, so
  32 --   we'd expect the volume of each one to be (1/24)*h^3.
  33 prop_tetrahedron0_volumes_exact :: Cube -> Bool
  34 prop_tetrahedron0_volumes_exact cube =
  35     volume (tetrahedron0 cube) ~~= (1/24)*(delta^(3::Int))
  36     where
  37       delta = h cube
  38
  39
  40 -- | In fact, since all of the tetrahedra are identical, we should
  41 --   already know their volumes. There's 24 tetrahedra to a cube, so
  42 --   we'd expect the volume of each one to be (1/24)*h^3.
  43 prop_tetrahedron1_volumes_exact :: Cube -> Bool
  44 prop_tetrahedron1_volumes_exact cube =
  45     volume (tetrahedron1 cube) ~~= (1/24)*(delta^(3::Int))
  46     where
  47       delta = h cube
  48
  49 -- | In fact, since all of the tetrahedra are identical, we should
  50 --   already know their volumes. There's 24 tetrahedra to a cube, so
  51 --   we'd expect the volume of each one to be (1/24)*h^3.
  52 prop_tetrahedron2_volumes_exact :: Cube -> Bool
  53 prop_tetrahedron2_volumes_exact cube =
  54     volume (tetrahedron2 cube) ~~= (1/24)*(delta^(3::Int))
  55     where
  56       delta = h cube
  57
  58 -- | In fact, since all of the tetrahedra are identical, we should
  59 --   already know their volumes. There's 24 tetrahedra to a cube, so
  60 --   we'd expect the volume of each one to be (1/24)*h^3.
  61 prop_tetrahedron3_volumes_exact :: Cube -> Bool
  62 prop_tetrahedron3_volumes_exact cube =
  63     volume (tetrahedron3 cube) ~~= (1/24)*(delta^(3::Int))
  64     where
  65       delta = h cube
  66
  67 -- | In fact, since all of the tetrahedra are identical, we should
  68 --   already know their volumes. There's 24 tetrahedra to a cube, so
  69 --   we'd expect the volume of each one to be (1/24)*h^3.
  70 prop_tetrahedron4_volumes_exact :: Cube -> Bool
  71 prop_tetrahedron4_volumes_exact cube =
  72     volume (tetrahedron4 cube) ~~= (1/24)*(delta^(3::Int))
  73     where
  74       delta = h cube
  75
  76 -- | In fact, since all of the tetrahedra are identical, we should
  77 --   already know their volumes. There's 24 tetrahedra to a cube, so
  78 --   we'd expect the volume of each one to be (1/24)*h^3.
  79 prop_tetrahedron5_volumes_exact :: Cube -> Bool
  80 prop_tetrahedron5_volumes_exact cube =
  81     volume (tetrahedron5 cube) ~~= (1/24)*(delta^(3::Int))
  82     where
  83       delta = h cube
  84
  85 -- | In fact, since all of the tetrahedra are identical, we should
  86 --   already know their volumes. There's 24 tetrahedra to a cube, so
  87 --   we'd expect the volume of each one to be (1/24)*h^3.
  88 prop_tetrahedron6_volumes_exact :: Cube -> Bool
  89 prop_tetrahedron6_volumes_exact cube =
  90     volume (tetrahedron6 cube) ~~= (1/24)*(delta^(3::Int))
  91     where
  92       delta = h cube
  93
  94 -- | In fact, since all of the tetrahedra are identical, we should
  95 --   already know their volumes. There's 24 tetrahedra to a cube, so
  96 --   we'd expect the volume of each one to be (1/24)*h^3.
  97 prop_tetrahedron7_volumes_exact :: Cube -> Bool
  98 prop_tetrahedron7_volumes_exact cube =
  99     volume (tetrahedron7 cube) ~~= (1/24)*(delta^(3::Int))
 100     where
 101       delta = h cube
 102
 103 -- | In fact, since all of the tetrahedra are identical, we should
 104 --   already know their volumes. There's 24 tetrahedra to a cube, so
 105 --   we'd expect the volume of each one to be (1/24)*h^3.
 106 prop_tetrahedron8_volumes_exact :: Cube -> Bool
 107 prop_tetrahedron8_volumes_exact cube =
 108     volume (tetrahedron8 cube) ~~= (1/24)*(delta^(3::Int))
 109     where
 110       delta = h cube
 111
 112 -- | In fact, since all of the tetrahedra are identical, we should
 113 --   already know their volumes. There's 24 tetrahedra to a cube, so
 114 --   we'd expect the volume of each one to be (1/24)*h^3.
 115 prop_tetrahedron9_volumes_exact :: Cube -> Bool
 116 prop_tetrahedron9_volumes_exact cube =
 117     volume (tetrahedron9 cube) ~~= (1/24)*(delta^(3::Int))
 118     where
 119       delta = h cube
 120
 121 -- | In fact, since all of the tetrahedra are identical, we should
 122 --   already know their volumes. There's 24 tetrahedra to a cube, so
 123 --   we'd expect the volume of each one to be (1/24)*h^3.
 124 prop_tetrahedron10_volumes_exact :: Cube -> Bool
 125 prop_tetrahedron10_volumes_exact cube =
 126     volume (tetrahedron10 cube) ~~= (1/24)*(delta^(3::Int))
 127     where
 128       delta = h cube
 129
 130 -- | In fact, since all of the tetrahedra are identical, we should
 131 --   already know their volumes. There's 24 tetrahedra to a cube, so
 132 --   we'd expect the volume of each one to be (1/24)*h^3.
 133 prop_tetrahedron11_volumes_exact :: Cube -> Bool
 134 prop_tetrahedron11_volumes_exact cube =
 135     volume (tetrahedron11 cube) ~~= (1/24)*(delta^(3::Int))
 136     where
 137       delta = h cube
 138
 139 -- | In fact, since all of the tetrahedra are identical, we should
 140 --   already know their volumes. There's 24 tetrahedra to a cube, so
 141 --   we'd expect the volume of each one to be (1/24)*h^3.
 142 prop_tetrahedron12_volumes_exact :: Cube -> Bool
 143 prop_tetrahedron12_volumes_exact cube =
 144     volume (tetrahedron12 cube) ~~= (1/24)*(delta^(3::Int))
 145     where
 146       delta = h cube
 147
 148 -- | In fact, since all of the tetrahedra are identical, we should
 149 --   already know their volumes. There's 24 tetrahedra to a cube, so
 150 --   we'd expect the volume of each one to be (1/24)*h^3.
 151 prop_tetrahedron13_volumes_exact :: Cube -> Bool
 152 prop_tetrahedron13_volumes_exact cube =
 153     volume (tetrahedron13 cube) ~~= (1/24)*(delta^(3::Int))
 154     where
 155       delta = h cube
 156
 157 -- | In fact, since all of the tetrahedra are identical, we should
 158 --   already know their volumes. There's 24 tetrahedra to a cube, so
 159 --   we'd expect the volume of each one to be (1/24)*h^3.
 160 prop_tetrahedron14_volumes_exact :: Cube -> Bool
 161 prop_tetrahedron14_volumes_exact cube =
 162     volume (tetrahedron14 cube) ~~= (1/24)*(delta^(3::Int))
 163     where
 164       delta = h cube
 165
 166 -- | In fact, since all of the tetrahedra are identical, we should
 167 --   already know their volumes. There's 24 tetrahedra to a cube, so
 168 --   we'd expect the volume of each one to be (1/24)*h^3.
 169 prop_tetrahedron15_volumes_exact :: Cube -> Bool
 170 prop_tetrahedron15_volumes_exact cube =
 171     volume (tetrahedron15 cube) ~~= (1/24)*(delta^(3::Int))
 172     where
 173       delta = h cube
 174
 175 -- | In fact, since all of the tetrahedra are identical, we should
 176 --   already know their volumes. There's 24 tetrahedra to a cube, so
 177 --   we'd expect the volume of each one to be (1/24)*h^3.
 178 prop_tetrahedron16_volumes_exact :: Cube -> Bool
 179 prop_tetrahedron16_volumes_exact cube =
 180     volume (tetrahedron16 cube) ~~= (1/24)*(delta^(3::Int))
 181     where
 182       delta = h cube
 183
 184 -- | In fact, since all of the tetrahedra are identical, we should
 185 --   already know their volumes. There's 24 tetrahedra to a cube, so
 186 --   we'd expect the volume of each one to be (1/24)*h^3.
 187 prop_tetrahedron17_volumes_exact :: Cube -> Bool
 188 prop_tetrahedron17_volumes_exact cube =
 189     volume (tetrahedron17 cube) ~~= (1/24)*(delta^(3::Int))
 190     where
 191       delta = h cube
 192
 193 -- | In fact, since all of the tetrahedra are identical, we should
 194 --   already know their volumes. There's 24 tetrahedra to a cube, so
 195 --   we'd expect the volume of each one to be (1/24)*h^3.
 196 prop_tetrahedron18_volumes_exact :: Cube -> Bool
 197 prop_tetrahedron18_volumes_exact cube =
 198     volume (tetrahedron18 cube) ~~= (1/24)*(delta^(3::Int))
 199     where
 200       delta = h cube
 201
 202 -- | In fact, since all of the tetrahedra are identical, we should
 203 --   already know their volumes. There's 24 tetrahedra to a cube, so
 204 --   we'd expect the volume of each one to be (1/24)*h^3.
 205 prop_tetrahedron19_volumes_exact :: Cube -> Bool
 206 prop_tetrahedron19_volumes_exact cube =
 207     volume (tetrahedron19 cube) ~~= (1/24)*(delta^(3::Int))
 208     where
 209       delta = h cube
 210
 211 -- | In fact, since all of the tetrahedra are identical, we should
 212 --   already know their volumes. There's 24 tetrahedra to a cube, so
 213 --   we'd expect the volume of each one to be (1/24)*h^3.
 214 prop_tetrahedron20_volumes_exact :: Cube -> Bool
 215 prop_tetrahedron20_volumes_exact cube =
 216     volume (tetrahedron20 cube) ~~= (1/24)*(delta^(3::Int))
 217     where
 218       delta = h cube
 219
 220 -- | In fact, since all of the tetrahedra are identical, we should
 221 --   already know their volumes. There's 24 tetrahedra to a cube, so
 222 --   we'd expect the volume of each one to be (1/24)*h^3.
 223 prop_tetrahedron21_volumes_exact :: Cube -> Bool
 224 prop_tetrahedron21_volumes_exact cube =
 225     volume (tetrahedron21 cube) ~~= (1/24)*(delta^(3::Int))
 226     where
 227       delta = h cube
 228
 229 -- | In fact, since all of the tetrahedra are identical, we should
 230 --   already know their volumes. There's 24 tetrahedra to a cube, so
 231 --   we'd expect the volume of each one to be (1/24)*h^3.
 232 prop_tetrahedron22_volumes_exact :: Cube -> Bool
 233 prop_tetrahedron22_volumes_exact cube =
 234     volume (tetrahedron22 cube) ~~= (1/24)*(delta^(3::Int))
 235     where
 236       delta = h cube
 237
 238 -- | In fact, since all of the tetrahedra are identical, we should
 239 --   already know their volumes. There's 24 tetrahedra to a cube, so
 240 --   we'd expect the volume of each one to be (1/24)*h^3.
 241 prop_tetrahedron23_volumes_exact :: Cube -> Bool
 242 prop_tetrahedron23_volumes_exact cube =
 243     volume (tetrahedron23 cube) ~~= (1/24)*(delta^(3::Int))
 244     where
 245       delta = h cube
 246
 247 -- | All tetrahedron should have their v0 located at the center of the cube.
 248 prop_v0_all_equal :: Cube -> Bool
 249 prop_v0_all_equal cube = (v0 t0) == (v0 t1)
 250     where
 251       t0 = head (tetrahedra cube) -- Doesn't matter which two we choose.
 252       t1 = head $ tail (tetrahedra cube)
 253
 254
 255 -- | This pretty much repeats the prop_all_volumes_positive property,
 256 --   but will let me know which tetrahedrons's vertices are disoriented.
 257 prop_tetrahedron0_volumes_positive :: Cube -> Bool
 258 prop_tetrahedron0_volumes_positive cube =
 259     volume (tetrahedron0 cube) > 0
 260
 261 -- | This pretty much repeats the prop_all_volumes_positive property,
 262 --   but will let me know which tetrahedrons's vertices are disoriented.
 263 prop_tetrahedron1_volumes_positive :: Cube -> Bool
 264 prop_tetrahedron1_volumes_positive cube =
 265     volume (tetrahedron1 cube) > 0
 266
 267 -- | This pretty much repeats the prop_all_volumes_positive property,
 268 --   but will let me know which tetrahedrons's vertices are disoriented.
 269 prop_tetrahedron2_volumes_positive :: Cube -> Bool
 270 prop_tetrahedron2_volumes_positive cube =
 271     volume (tetrahedron2 cube) > 0
 272
 273 -- | This pretty much repeats the prop_all_volumes_positive property,
 274 --   but will let me know which tetrahedrons's vertices are disoriented.
 275 prop_tetrahedron3_volumes_positive :: Cube -> Bool
 276 prop_tetrahedron3_volumes_positive cube =
 277     volume (tetrahedron3 cube) > 0
 278
 279 -- | This pretty much repeats the prop_all_volumes_positive property,
 280 --   but will let me know which tetrahedrons's vertices are disoriented.
 281 prop_tetrahedron4_volumes_positive :: Cube -> Bool
 282 prop_tetrahedron4_volumes_positive cube =
 283     volume (tetrahedron4 cube) > 0
 284
 285 -- | This pretty much repeats the prop_all_volumes_positive property,
 286 --   but will let me know which tetrahedrons's vertices are disoriented.
 287 prop_tetrahedron5_volumes_positive :: Cube -> Bool
 288 prop_tetrahedron5_volumes_positive cube =
 289     volume (tetrahedron5 cube) > 0
 290
 291 -- | This pretty much repeats the prop_all_volumes_positive property,
 292 --   but will let me know which tetrahedrons's vertices are disoriented.
 293 prop_tetrahedron6_volumes_positive :: Cube -> Bool
 294 prop_tetrahedron6_volumes_positive cube =
 295     volume (tetrahedron6 cube) > 0
 296
 297 -- | This pretty much repeats the prop_all_volumes_positive property,
 298 --   but will let me know which tetrahedrons's vertices are disoriented.
 299 prop_tetrahedron7_volumes_positive :: Cube -> Bool
 300 prop_tetrahedron7_volumes_positive cube =
 301     volume (tetrahedron7 cube) > 0
 302
 303 -- | This pretty much repeats the prop_all_volumes_positive property,
 304 --   but will let me know which tetrahedrons's vertices are disoriented.
 305 prop_tetrahedron8_volumes_positive :: Cube -> Bool
 306 prop_tetrahedron8_volumes_positive cube =
 307     volume (tetrahedron8 cube) > 0
 308
 309 -- | This pretty much repeats the prop_all_volumes_positive property,
 310 --   but will let me know which tetrahedrons's vertices are disoriented.
 311 prop_tetrahedron9_volumes_positive :: Cube -> Bool
 312 prop_tetrahedron9_volumes_positive cube =
 313     volume (tetrahedron9 cube) > 0
 314
 315 -- | This pretty much repeats the prop_all_volumes_positive property,
 316 --   but will let me know which tetrahedrons's vertices are disoriented.
 317 prop_tetrahedron10_volumes_positive :: Cube -> Bool
 318 prop_tetrahedron10_volumes_positive cube =
 319     volume (tetrahedron10 cube) > 0
 320
 321 -- | This pretty much repeats the prop_all_volumes_positive property,
 322 --   but will let me know which tetrahedrons's vertices are disoriented.
 323 prop_tetrahedron11_volumes_positive :: Cube -> Bool
 324 prop_tetrahedron11_volumes_positive cube =
 325     volume (tetrahedron11 cube) > 0
 326
 327 -- | This pretty much repeats the prop_all_volumes_positive property,
 328 --   but will let me know which tetrahedrons's vertices are disoriented.
 329 prop_tetrahedron12_volumes_positive :: Cube -> Bool
 330 prop_tetrahedron12_volumes_positive cube =
 331     volume (tetrahedron12 cube) > 0
 332
 333 -- | This pretty much repeats the prop_all_volumes_positive property,
 334 --   but will let me know which tetrahedrons's vertices are disoriented.
 335 prop_tetrahedron13_volumes_positive :: Cube -> Bool
 336 prop_tetrahedron13_volumes_positive cube =
 337     volume (tetrahedron13 cube) > 0
 338
 339 -- | This pretty much repeats the prop_all_volumes_positive property,
 340 --   but will let me know which tetrahedrons's vertices are disoriented.
 341 prop_tetrahedron14_volumes_positive :: Cube -> Bool
 342 prop_tetrahedron14_volumes_positive cube =
 343     volume (tetrahedron14 cube) > 0
 344
 345 -- | This pretty much repeats the prop_all_volumes_positive property,
 346 --   but will let me know which tetrahedrons's vertices are disoriented.
 347 prop_tetrahedron15_volumes_positive :: Cube -> Bool
 348 prop_tetrahedron15_volumes_positive cube =
 349     volume (tetrahedron15 cube) > 0
 350
 351 -- | This pretty much repeats the prop_all_volumes_positive property,
 352 --   but will let me know which tetrahedrons's vertices are disoriented.
 353 prop_tetrahedron16_volumes_positive :: Cube -> Bool
 354 prop_tetrahedron16_volumes_positive cube =
 355     volume (tetrahedron16 cube) > 0
 356
 357 -- | This pretty much repeats the prop_all_volumes_positive property,
 358 --   but will let me know which tetrahedrons's vertices are disoriented.
 359 prop_tetrahedron17_volumes_positive :: Cube -> Bool
 360 prop_tetrahedron17_volumes_positive cube =
 361     volume (tetrahedron17 cube) > 0
 362
 363 -- | This pretty much repeats the prop_all_volumes_positive property,
 364 --   but will let me know which tetrahedrons's vertices are disoriented.
 365 prop_tetrahedron18_volumes_positive :: Cube -> Bool
 366 prop_tetrahedron18_volumes_positive cube =
 367     volume (tetrahedron18 cube) > 0
 368
 369 -- | This pretty much repeats the prop_all_volumes_positive property,
 370 --   but will let me know which tetrahedrons's vertices are disoriented.
 371 prop_tetrahedron19_volumes_positive :: Cube -> Bool
 372 prop_tetrahedron19_volumes_positive cube =
 373     volume (tetrahedron19 cube) > 0
 374
 375 -- | This pretty much repeats the prop_all_volumes_positive property,
 376 --   but will let me know which tetrahedrons's vertices are disoriented.
 377 prop_tetrahedron20_volumes_positive :: Cube -> Bool
 378 prop_tetrahedron20_volumes_positive cube =
 379     volume (tetrahedron20 cube) > 0
 380
 381 -- | This pretty much repeats the prop_all_volumes_positive property,
 382 --   but will let me know which tetrahedrons's vertices are disoriented.
 383 prop_tetrahedron21_volumes_positive :: Cube -> Bool
 384 prop_tetrahedron21_volumes_positive cube =
 385     volume (tetrahedron21 cube) > 0
 386
 387 -- | This pretty much repeats the prop_all_volumes_positive property,
 388 --   but will let me know which tetrahedrons's vertices are disoriented.
 389 prop_tetrahedron22_volumes_positive :: Cube -> Bool
 390 prop_tetrahedron22_volumes_positive cube =
 391     volume (tetrahedron22 cube) > 0
 392
 393 -- | This pretty much repeats the prop_all_volumes_positive property,
 394 --   but will let me know which tetrahedrons's vertices are disoriented.
 395 prop_tetrahedron23_volumes_positive :: Cube -> Bool
 396 prop_tetrahedron23_volumes_positive cube =
 397     volume (tetrahedron23 cube) > 0
 398
 399
 400 -- | Given in Sorokina and Zeilfelder, p. 79, (2.6). Note that the
 401 --   third and fourth indices of c-t1 have been switched. This is
 402 --   because we store the triangles oriented such that their volume is
 403 --   positive. If T and T-tilde share \<v0,v1,v2\> and v3,v3-tilde point
 404 --   in opposite directions, one of them has to have negative volume!
 405 prop_c0120_identity1 :: Cube -> Bool
 406 prop_c0120_identity1 cube =
 407    c t0 0 1 2 0 ~= (c t0 0 0 2 1 + c t3 0 0 1 2) / 2
 408      where
 409        t0 = tetrahedron0 cube
 410        t3 = tetrahedron3 cube
 411
 412
 413 -- | Given in Sorokina and Zeilfelder, p. 79, (2.6). Repeats
 414 --   'prop_c0120_identity1' with tetrahedrons 1 and 2.
 415 prop_c0120_identity2 :: Cube -> Bool
 416 prop_c0120_identity2 cube =
 417    c t1 0 1 2 0 ~= (c t1 0 0 2 1 + c t0 0 0 1 2) / 2
 418      where
 419        t0 = tetrahedron0 cube
 420        t1 = tetrahedron1 cube
 421
 422 -- | Given in Sorokina and Zeilfelder, p. 79, (2.6). Repeats
 423 --   'prop_c0120_identity1' with tetrahedrons 1 and 2.
 424 prop_c0120_identity3 :: Cube -> Bool
 425 prop_c0120_identity3 cube =
 426    c t2 0 1 2 0 ~= (c t2 0 0 2 1 + c t1 0 0 1 2) / 2
 427      where
 428        t1 = tetrahedron1 cube
 429        t2 = tetrahedron2 cube
 430
 431 -- | Given in Sorokina and Zeilfelder, p. 79, (2.6). Repeats
 432 --   'prop_c0120_identity1' with tetrahedrons 2 and 3.
 433 prop_c0120_identity4 :: Cube -> Bool
 434 prop_c0120_identity4 cube =
 435    c t3 0 1 2 0 ~= (c t3 0 0 2 1 + c t2 0 0 1 2) / 2
 436      where
 437        t2 = tetrahedron2 cube
 438        t3 = tetrahedron3 cube
 439
 440
 441 -- | Given in Sorokina and Zeilfelder, p. 79, (2.6). Repeats
 442 --   'prop_c0120_identity1' with tetrahedrons 4 and 5.
 443 prop_c0120_identity5 :: Cube -> Bool
 444 prop_c0120_identity5 cube =
 445     c t5 0 1 2 0 ~= (c t5 0 0 2 1 + c t4 0 0 1 2) / 2
 446     where
 447       t4 = tetrahedron4 cube
 448       t5 = tetrahedron5 cube
 449
 450 -- -- | Given in Sorokina and Zeilfelder, p. 79, (2.6). Repeats
 451 -- --   'prop_c0120_identity1' with tetrahedrons 5 and 6.
 452 prop_c0120_identity6 :: Cube -> Bool
 453 prop_c0120_identity6 cube =
 454    c t6 0 1 2 0 ~= (c t6 0 0 2 1 + c t5 0 0 1 2) / 2
 455      where
 456        t5 = tetrahedron5 cube
 457        t6 = tetrahedron6 cube
 458
 459
 460 -- -- | Given in Sorokina and Zeilfelder, p. 79, (2.6). Repeats
 461 -- --   'prop_c0120_identity1' with tetrahedrons 6 and 7.
 462 prop_c0120_identity7 :: Cube -> Bool
 463 prop_c0120_identity7 cube =
 464    c t7 0 1 2 0 ~= (c t7 0 0 2 1 + c t6 0 0 1 2) / 2
 465      where
 466        t6 = tetrahedron6 cube
 467        t7 = tetrahedron7 cube
 468
 469
 470 -- | Given in Sorokina and Zeilfelder, p. 79, (2.6). See
 471 --   'prop_c0120_identity1'.
 472 prop_c0210_identity1 :: Cube -> Bool
 473 prop_c0210_identity1 cube =
 474     c t0 0 2 1 0 ~= (c t0 0 1 1 1 + c t3 0 1 1 1) / 2
 475       where
 476         t0 = tetrahedron0 cube
 477         t3 = tetrahedron3 cube
 478
 479
 480 -- | Given in Sorokina and Zeilfelder, p. 79, (2.6). See
 481 --   'prop_c0120_identity1'.
 482 prop_c0300_identity1 :: Cube -> Bool
 483 prop_c0300_identity1 cube =
 484     c t0 0 3 0 0 ~= (c t0 0 2 0 1 + c t3 0 2 1 0) / 2
 485       where
 486         t0 = tetrahedron0 cube
 487         t3 = tetrahedron3 cube
 488
 489
 490 -- | Given in Sorokina and Zeilfelder, p. 79, (2.6). See
 491 --   'prop_c0120_identity1'.
 492 prop_c1110_identity :: Cube -> Bool
 493 prop_c1110_identity cube =
 494     c t0 1 1 1 0 ~= (c t0 1 0 1 1 + c t3 1 0 1 1) / 2
 495       where
 496         t0 = tetrahedron0 cube
 497         t3 = tetrahedron3 cube
 498
 499
 500 -- | Given in Sorokina and Zeilfelder, p. 79, (2.6). See
 501 --   'prop_c0120_identity1'.
 502 prop_c1200_identity1 :: Cube -> Bool
 503 prop_c1200_identity1 cube =
 504     c t0 1 2 0 0 ~= (c t0 1 1 0 1 + c t3 1 1 1 0) / 2
 505       where
 506         t0 = tetrahedron0 cube
 507         t3 = tetrahedron3 cube
 508
 509
 510 -- | Given in Sorokina and Zeilfelder, p. 79, (2.6). See
 511 --   'prop_c0120_identity1'.
 512 prop_c2100_identity1 :: Cube -> Bool
 513 prop_c2100_identity1 cube =
 514     c t0 2 1 0 0 ~= (c t0 2 0 0 1 + c t3 2 0 1 0) / 2
 515       where
 516         t0 = tetrahedron0 cube
 517         t3 = tetrahedron3 cube
 518
 519
 520
 521 -- | Given in Sorokina and Zeilfelder, p. 79, (2.7). Note that the
 522 --   third and fourth indices of c-t3 have been switched. This is
 523 --   because we store the triangles oriented such that their volume is
 524 --   positive. If T and T-tilde share \<v0,v1,v2\> and v3,v3-tilde
 525 --   point in opposite directions, one of them has to have negative
 526 --   volume!
 527 prop_c0102_identity1 :: Cube -> Bool
 528 prop_c0102_identity1 cube =
 529     c t0 0 1 0 2 ~= (c t0 0 0 1 2 + c t1 0 0 2 1) / 2
 530       where
 531         t0 = tetrahedron0 cube
 532         t1 = tetrahedron1 cube
 533
 534
 535 -- | Given in Sorokina and Zeilfelder, p. 79, (2.7). See
 536 --   'prop_c0102_identity1'.
 537 prop_c0201_identity1 :: Cube -> Bool
 538 prop_c0201_identity1 cube =
 539     c t0 0 2 0 1 ~= (c t0 0 1 1 1 + c t1 0 1 1 1) / 2
 540       where
 541         t0 = tetrahedron0 cube
 542         t1 = tetrahedron1 cube
 543
 544
 545 -- | Given in Sorokina and Zeilfelder, p. 79, (2.7). See
 546 --   'prop_c0102_identity1'.
 547 prop_c0300_identity2 :: Cube -> Bool
 548 prop_c0300_identity2 cube =
 549     c t0 0 3 0 0 ~= (c t0 0 2 1 0 + c t1 0 2 0 1) / 2
 550       where
 551         t0 = tetrahedron0 cube
 552         t1 = tetrahedron1 cube
 553
 554
 555 -- | Given in Sorokina and Zeilfelder, p. 79, (2.7). See
 556 --   'prop_c0102_identity1'.
 557 prop_c1101_identity :: Cube -> Bool
 558 prop_c1101_identity cube =
 559     c t0 1 1 0 1 ~= (c t0 1 0 1 1 + c t1 1 0 1 1) / 2
 560       where
 561         t0 = tetrahedron0 cube
 562         t1 = tetrahedron1 cube
 563
 564
 565 -- | Given in Sorokina and Zeilfelder, p. 79, (2.7). See
 566 --   'prop_c0102_identity1'.
 567 prop_c1200_identity2 :: Cube -> Bool
 568 prop_c1200_identity2 cube =
 569     c t0 1 2 0 0 ~= (c t0 1 1 1 0 + c t1 1 1 0 1) / 2
 570       where
 571         t0 = tetrahedron0 cube
 572         t1 = tetrahedron1 cube
 573
 574
 575 -- | Given in Sorokina and Zeilfelder, p. 79, (2.7). See
 576 --   'prop_c0102_identity1'.
 577 prop_c2100_identity2 :: Cube -> Bool
 578 prop_c2100_identity2 cube =
 579     c t0 2 1 0 0 ~= (c t0 2 0 1 0 + c t1 2 0 0 1) / 2
 580       where
 581         t0 = tetrahedron0 cube
 582         t1 = tetrahedron1 cube
 583
 584
 585 -- | Given in Sorokina and Zeilfelder, p. 79, (2.8). The third and
 586 --   fourth indices of c-t6 have been switched. This is because we
 587 --   store the triangles oriented such that their volume is
 588 --   positive. If T and T-tilde share \<v0,v1,v2\> and v3,v3-tilde
 589 --   point in opposite directions, one of them has to have negative
 590 --   volume!
 591 prop_c3000_identity :: Cube -> Bool
 592 prop_c3000_identity cube =
 593     c t0 3 0 0 0 ~= c t0 2 1 0 0 + c t6 2 1 0 0
 594                     - ((c t0 2 0 1 0 + c t0 2 0 0 1)/ 2)
 595       where
 596         t0 = tetrahedron0 cube
 597         t6 = tetrahedron6 cube
 598
 599
 600 -- | Given in Sorokina and Zeilfelder, p. 79, (2.8). See
 601 --   'prop_c3000_identity'.
 602 prop_c2010_identity :: Cube -> Bool
 603 prop_c2010_identity cube =
 604     c t0 2 0 1 0 ~= c t0 1 1 1 0 + c t6 1 1 0 1
 605                     - ((c t0 1 0 2 0 + c t0 1 0 1 1)/ 2)
 606       where
 607         t0 = tetrahedron0 cube
 608         t6 = tetrahedron6 cube
 609
 610
 611 -- | Given in Sorokina and Zeilfelder, p. 79, (2.8). See
 612 --   'prop_c3000_identity'.
 613 prop_c2001_identity :: Cube -> Bool
 614 prop_c2001_identity cube =
 615     c t0 2 0 0 1 ~= c t0 1 1 0 1 + c t6 1 1 1 0
 616                     - ((c t0 1 0 0 2 + c t0 1 0 1 1)/ 2)
 617       where
 618         t0 = tetrahedron0 cube
 619         t6 = tetrahedron6 cube
 620
 621
 622 -- | Given in Sorokina and Zeilfelder, p. 79, (2.8). See
 623 --   'prop_c3000_identity'.
 624 prop_c1020_identity :: Cube -> Bool
 625 prop_c1020_identity cube =
 626     c t0 1 0 2 0 ~= c t0 0 1 2 0 + c t6 0 1 0 2
 627                     - ((c t0 0 0 3 0 + c t0 0 0 2 1)/ 2)
 628       where
 629         t0 = tetrahedron0 cube
 630         t6 = tetrahedron6 cube
 631
 632
 633 -- | Given in Sorokina and Zeilfelder, p. 79, (2.8). See
 634 --   'prop_c3000_identity'.
 635 prop_c1002_identity :: Cube -> Bool
 636 prop_c1002_identity cube =
 637     c t0 1 0 0 2 ~= c t0 0 1 0 2 + c t6 0 1 2 0
 638                     - ((c t0 0 0 0 3 + c t0 0 0 1 2)/ 2)
 639       where
 640         t0 = tetrahedron0 cube
 641         t6 = tetrahedron6 cube
 642
 643
 644 -- | Given in Sorokina and Zeilfelder, p. 79, (2.8). See
 645 --   'prop_c3000_identity'.
 646 prop_c1011_identity :: Cube -> Bool
 647 prop_c1011_identity cube =
 648     c t0 1 0 1 1 ~= c t0 0 1 1 1 + c t6 0 1 1 1 -
 649                     ((c t0 0 0 1 2 + c t0 0 0 2 1)/ 2)
 650       where
 651         t0 = tetrahedron0 cube
 652         t6 = tetrahedron6 cube
 653
 654
 655
 656 -- | Given in Sorokina and Zeilfelder, p. 78.
 657 prop_cijk1_identity :: Cube -> Bool
 658 prop_cijk1_identity cube =
 659      and [ c t0 i j k 1 ~=
 660                  (c t1 (i+1) j k 0) * ((b0 t0) (v3 t1)) +
 661                  (c t1 i (j+1) k 0) * ((b1 t0) (v3 t1)) +
 662                  (c t1 i j (k+1) 0) * ((b2 t0) (v3 t1)) +
 663                  (c t1 i j k 1) * ((b3 t0) (v3 t1)) | i <- [0..2],
 664                                                       j <- [0..2],
 665                                                       k <- [0..2],
 666                                                       i + j + k == 2]
 667       where
 668         t0 = tetrahedron0 cube
 669         t1 = tetrahedron1 cube
 670
 671
 672 -- | The function values at the interior should be the same for all tetrahedra.
 673 prop_interior_values_all_identical :: Cube -> Bool
 674 prop_interior_values_all_identical cube =
 675     all_equal [i0, i1, i2, i3, i4, i5, i6, i7, i8,
 676                i9, i10, i11, i12, i13, i14, i15, i16,
 677                i17, i18, i19, i20, i21, i22, i23]
 678     where
 679       i0 = eval (Tetrahedron.fv (tetrahedron0 cube)) I
 680       i1 = eval (Tetrahedron.fv (tetrahedron1 cube)) I
 681       i2 = eval (Tetrahedron.fv (tetrahedron2 cube)) I
 682       i3 = eval (Tetrahedron.fv (tetrahedron3 cube)) I
 683       i4 = eval (Tetrahedron.fv (tetrahedron4 cube)) I
 684       i5 = eval (Tetrahedron.fv (tetrahedron5 cube)) I
 685       i6 = eval (Tetrahedron.fv (tetrahedron6 cube)) I
 686       i7 = eval (Tetrahedron.fv (tetrahedron7 cube)) I
 687       i8 = eval (Tetrahedron.fv (tetrahedron8 cube)) I
 688       i9 = eval (Tetrahedron.fv (tetrahedron9 cube)) I
 689       i10 = eval (Tetrahedron.fv (tetrahedron10 cube)) I
 690       i11 = eval (Tetrahedron.fv (tetrahedron11 cube)) I
 691       i12 = eval (Tetrahedron.fv (tetrahedron12 cube)) I
 692       i13 = eval (Tetrahedron.fv (tetrahedron13 cube)) I
 693       i14 = eval (Tetrahedron.fv (tetrahedron14 cube)) I
 694       i15 = eval (Tetrahedron.fv (tetrahedron15 cube)) I
 695       i16 = eval (Tetrahedron.fv (tetrahedron16 cube)) I
 696       i17 = eval (Tetrahedron.fv (tetrahedron17 cube)) I
 697       i18 = eval (Tetrahedron.fv (tetrahedron18 cube)) I
 698       i19 = eval (Tetrahedron.fv (tetrahedron19 cube)) I
 699       i20 = eval (Tetrahedron.fv (tetrahedron20 cube)) I
 700       i21 = eval (Tetrahedron.fv (tetrahedron21 cube)) I
 701       i22 = eval (Tetrahedron.fv (tetrahedron22 cube)) I
 702       i23 = eval (Tetrahedron.fv (tetrahedron23 cube)) I
 703
 704
 705 -- | We know what (c t6 2 1 0 0) should be from Sorokina and Zeilfelder, p. 87.
 706 --   This test checks the rotation works as expected.
 707 prop_c_tilde_2100_rotation_correct :: Cube -> Bool
 708 prop_c_tilde_2100_rotation_correct cube =
 709     expr1 == expr2
 710     where
 711       t0 = tetrahedron0 cube
 712       t6 = tetrahedron6 cube
 713
 714       -- What gets computed for c2100 of t6.
 715       expr1 = eval (Tetrahedron.fv t6) $
 716           (3/8)*I +
 717           (1/12)*(T + R + L + D) +
 718           (1/64)*(FT + FR + FL + FD) +
 719           (7/48)*F +
 720           (1/48)*B +
 721           (1/96)*(RT + LD + LT + RD) +
 722           (1/192)*(BT + BR + BL + BD)
 723
 724       -- What should be computed for c2100 of t6.
 725       expr2 = eval (Tetrahedron.fv t0) $
 726               (3/8)*I +
 727               (1/12)*(F + R + L + B) +
 728               (1/64)*(FT + RT + LT + BT) +
 729               (7/48)*T +
 730               (1/48)*D +
 731               (1/96)*(FR + FL + BR + BL) +
 732               (1/192)*(FD + RD + LD + BD)
 733
 734
 735 -- | We know what (c t6 2 1 0 0) should be from Sorokina and Zeilfelder, p. 87.
 736 --   This test checks the actual value based on the FunctionValues of the cube.
 737 prop_c_tilde_2100_correct :: Cube -> Bool
 738 prop_c_tilde_2100_correct cube =
 739     c t6 2 1 0 0 == (3/8)*int
 740                     + (1/12)*(f + r + l + b)
 741                     + (1/64)*(ft + rt + lt + bt)
 742                     + (7/48)*t + (1/48)*d + (1/96)*(fr + fl + br + bl)
 743                     + (1/192)*(fd + rd + ld + bd)
 744     where
 745       t0 = tetrahedron0 cube
 746       t6 = tetrahedron6 cube
 747       fvs = Tetrahedron.fv t0
 748       int = interior fvs
 749       f = front fvs
 750       r = right fvs
 751       l = left fvs
 752       b = back fvs
 753       ft = front_top fvs
 754       rt = right_top fvs
 755       lt = left_top fvs
 756       bt = back_top fvs
 757       t = top fvs
 758       d = down fvs
 759       fr = front_right fvs
 760       fl = front_left fvs
 761       br = back_right fvs
 762       bl = back_left fvs
 763       fd = front_down fvs
 764       rd = right_down fvs
 765       ld = left_down fvs
 766       bd = back_down fvs
 767
 768 -- Tests to check that the correct edges are incidental.
 769 prop_t0_shares_edge_with_t1 :: Cube -> Bool
 770 prop_t0_shares_edge_with_t1 cube =
 771     (v1 t0) == (v1 t1) && (v3 t0) == (v2 t1)
 772       where
 773         t0 = tetrahedron0 cube
 774         t1 = tetrahedron1 cube
 775
 776 prop_t0_shares_edge_with_t3 :: Cube -> Bool
 777 prop_t0_shares_edge_with_t3 cube =
 778     (v1 t0) == (v1 t3) && (v2 t0) == (v3 t3)
 779       where
 780         t0 = tetrahedron0 cube
 781         t3 = tetrahedron3 cube
 782
 783 prop_t0_shares_edge_with_t6 :: Cube -> Bool
 784 prop_t0_shares_edge_with_t6 cube =
 785     (v2 t0) == (v3 t6) && (v3 t0) == (v2 t6)
 786       where
 787         t0 = tetrahedron0 cube
 788         t6 = tetrahedron6 cube
 789
 790 prop_t1_shares_edge_with_t2 :: Cube -> Bool
 791 prop_t1_shares_edge_with_t2 cube =
 792     (v1 t1) == (v1 t2) && (v3 t1) == (v2 t2)
 793       where
 794         t1 = tetrahedron1 cube
 795         t2 = tetrahedron2 cube
 796
 797 prop_t1_shares_edge_with_t19 :: Cube -> Bool
 798 prop_t1_shares_edge_with_t19 cube =
 799     (v2 t1) == (v3 t19) && (v3 t1) == (v2 t19)
 800       where
 801         t1 = tetrahedron1 cube
 802         t19 = tetrahedron19 cube
 803
 804 prop_t2_shares_edge_with_t3 :: Cube -> Bool
 805 prop_t2_shares_edge_with_t3 cube =
 806     (v1 t1) == (v1 t2) && (v3 t1) == (v2 t2)
 807       where
 808         t1 = tetrahedron1 cube
 809         t2 = tetrahedron2 cube
 810
 811 prop_t2_shares_edge_with_t12 :: Cube -> Bool
 812 prop_t2_shares_edge_with_t12 cube =
 813     (v2 t2) == (v3 t12) && (v3 t2) == (v2 t12)
 814       where
 815         t2 = tetrahedron2 cube
 816         t12 = tetrahedron12 cube
 817
 818 prop_t3_shares_edge_with_t21 :: Cube -> Bool
 819 prop_t3_shares_edge_with_t21 cube =
 820     (v2 t3) == (v3 t21) && (v3 t3) == (v2 t21)
 821       where
 822         t3 = tetrahedron3 cube
 823         t21 = tetrahedron21 cube
 824
 825 prop_t4_shares_edge_with_t5 :: Cube -> Bool
 826 prop_t4_shares_edge_with_t5 cube =
 827     (v1 t4) == (v1 t5) && (v3 t4) == (v2 t5)
 828       where
 829         t4 = tetrahedron4 cube
 830         t5 = tetrahedron5 cube
 831
 832 prop_t4_shares_edge_with_t7 :: Cube -> Bool
 833 prop_t4_shares_edge_with_t7 cube =
 834     (v1 t4) == (v1 t7) && (v2 t4) == (v3 t7)
 835       where
 836         t4 = tetrahedron4 cube
 837         t7 = tetrahedron7 cube
 838
 839 prop_t4_shares_edge_with_t10 :: Cube -> Bool
 840 prop_t4_shares_edge_with_t10 cube =
 841     (v2 t4) == (v3 t10) && (v3 t4) == (v2 t10)
 842       where
 843         t4 = tetrahedron4 cube
 844         t10 = tetrahedron10 cube
 845
 846 prop_t5_shares_edge_with_t6 :: Cube -> Bool
 847 prop_t5_shares_edge_with_t6 cube =
 848     (v1 t5) == (v1 t6) && (v3 t5) == (v2 t6)
 849       where
 850         t5 = tetrahedron5 cube
 851         t6 = tetrahedron6 cube
 852
 853 prop_t5_shares_edge_with_t16 :: Cube -> Bool
 854 prop_t5_shares_edge_with_t16 cube =
 855     (v2 t5) == (v3 t16) && (v3 t5) == (v2 t16)
 856       where
 857         t5 = tetrahedron5 cube
 858         t16 = tetrahedron16 cube
 859
 860 prop_t6_shares_edge_with_t7 :: Cube -> Bool
 861 prop_t6_shares_edge_with_t7 cube =
 862     (v1 t6) == (v1 t7) && (v3 t6) == (v2 t7)
 863       where
 864         t6 = tetrahedron6 cube
 865         t7 = tetrahedron7 cube
 866
 867 prop_t7_shares_edge_with_t20 :: Cube -> Bool
 868 prop_t7_shares_edge_with_t20 cube =
 869     (v2 t7) == (v3 t20) && (v2 t7) == (v3 t20)
 870       where
 871         t7 = tetrahedron7 cube
 872         t20 = tetrahedron20 cube