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_all_volumes_exact :: Cube -> Bool
  34 prop_all_volumes_exact cube =
  35     and [volume t ~~= (1/24)*(delta^(3::Int)) | t <- tetrahedra cube]
  36     where
  37       delta = h cube
  38
  39 -- | All tetrahedron should have their v0 located at the center of the cube.
  40 prop_v0_all_equal :: Cube -> Bool
  41 prop_v0_all_equal cube = (v0 t0) == (v0 t1)
  42     where
  43       t0 = head (tetrahedra cube) -- Doesn't matter which two we choose.
  44       t1 = head $ tail (tetrahedra cube)
  45
  46
  47 -- | This pretty much repeats the prop_all_volumes_positive property,
  48 --   but will let me know which tetrahedrons's vertices are disoriented.
  49 prop_tetrahedron0_volumes_positive :: Cube -> Bool
  50 prop_tetrahedron0_volumes_positive cube =
  51     volume (tetrahedron0 cube) > 0
  52
  53 -- | This pretty much repeats the prop_all_volumes_positive property,
  54 --   but will let me know which tetrahedrons's vertices are disoriented.
  55 prop_tetrahedron1_volumes_positive :: Cube -> Bool
  56 prop_tetrahedron1_volumes_positive cube =
  57     volume (tetrahedron1 cube) > 0
  58
  59 -- | This pretty much repeats the prop_all_volumes_positive property,
  60 --   but will let me know which tetrahedrons's vertices are disoriented.
  61 prop_tetrahedron2_volumes_positive :: Cube -> Bool
  62 prop_tetrahedron2_volumes_positive cube =
  63     volume (tetrahedron2 cube) > 0
  64
  65 -- | This pretty much repeats the prop_all_volumes_positive property,
  66 --   but will let me know which tetrahedrons's vertices are disoriented.
  67 prop_tetrahedron3_volumes_positive :: Cube -> Bool
  68 prop_tetrahedron3_volumes_positive cube =
  69     volume (tetrahedron3 cube) > 0
  70
  71 -- | This pretty much repeats the prop_all_volumes_positive property,
  72 --   but will let me know which tetrahedrons's vertices are disoriented.
  73 prop_tetrahedron4_volumes_positive :: Cube -> Bool
  74 prop_tetrahedron4_volumes_positive cube =
  75     volume (tetrahedron4 cube) > 0
  76
  77 -- | This pretty much repeats the prop_all_volumes_positive property,
  78 --   but will let me know which tetrahedrons's vertices are disoriented.
  79 prop_tetrahedron5_volumes_positive :: Cube -> Bool
  80 prop_tetrahedron5_volumes_positive cube =
  81     volume (tetrahedron5 cube) > 0
  82
  83 -- | This pretty much repeats the prop_all_volumes_positive property,
  84 --   but will let me know which tetrahedrons's vertices are disoriented.
  85 prop_tetrahedron6_volumes_positive :: Cube -> Bool
  86 prop_tetrahedron6_volumes_positive cube =
  87     volume (tetrahedron6 cube) > 0
  88
  89 -- | This pretty much repeats the prop_all_volumes_positive property,
  90 --   but will let me know which tetrahedrons's vertices are disoriented.
  91 prop_tetrahedron7_volumes_positive :: Cube -> Bool
  92 prop_tetrahedron7_volumes_positive cube =
  93     volume (tetrahedron7 cube) > 0
  94
  95 -- | This pretty much repeats the prop_all_volumes_positive property,
  96 --   but will let me know which tetrahedrons's vertices are disoriented.
  97 prop_tetrahedron8_volumes_positive :: Cube -> Bool
  98 prop_tetrahedron8_volumes_positive cube =
  99     volume (tetrahedron8 cube) > 0
 100
 101 -- | This pretty much repeats the prop_all_volumes_positive property,
 102 --   but will let me know which tetrahedrons's vertices are disoriented.
 103 prop_tetrahedron9_volumes_positive :: Cube -> Bool
 104 prop_tetrahedron9_volumes_positive cube =
 105     volume (tetrahedron9 cube) > 0
 106
 107 -- | This pretty much repeats the prop_all_volumes_positive property,
 108 --   but will let me know which tetrahedrons's vertices are disoriented.
 109 prop_tetrahedron10_volumes_positive :: Cube -> Bool
 110 prop_tetrahedron10_volumes_positive cube =
 111     volume (tetrahedron10 cube) > 0
 112
 113 -- | This pretty much repeats the prop_all_volumes_positive property,
 114 --   but will let me know which tetrahedrons's vertices are disoriented.
 115 prop_tetrahedron11_volumes_positive :: Cube -> Bool
 116 prop_tetrahedron11_volumes_positive cube =
 117     volume (tetrahedron11 cube) > 0
 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_tetrahedron12_volumes_positive :: Cube -> Bool
 122 prop_tetrahedron12_volumes_positive cube =
 123     volume (tetrahedron12 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_tetrahedron13_volumes_positive :: Cube -> Bool
 128 prop_tetrahedron13_volumes_positive cube =
 129     volume (tetrahedron13 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_tetrahedron14_volumes_positive :: Cube -> Bool
 134 prop_tetrahedron14_volumes_positive cube =
 135     volume (tetrahedron14 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_tetrahedron15_volumes_positive :: Cube -> Bool
 140 prop_tetrahedron15_volumes_positive cube =
 141     volume (tetrahedron15 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_tetrahedron16_volumes_positive :: Cube -> Bool
 146 prop_tetrahedron16_volumes_positive cube =
 147     volume (tetrahedron16 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_tetrahedron17_volumes_positive :: Cube -> Bool
 152 prop_tetrahedron17_volumes_positive cube =
 153     volume (tetrahedron17 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_tetrahedron18_volumes_positive :: Cube -> Bool
 158 prop_tetrahedron18_volumes_positive cube =
 159     volume (tetrahedron18 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_tetrahedron19_volumes_positive :: Cube -> Bool
 164 prop_tetrahedron19_volumes_positive cube =
 165     volume (tetrahedron19 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_tetrahedron20_volumes_positive :: Cube -> Bool
 170 prop_tetrahedron20_volumes_positive cube =
 171     volume (tetrahedron20 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_tetrahedron21_volumes_positive :: Cube -> Bool
 176 prop_tetrahedron21_volumes_positive cube =
 177     volume (tetrahedron21 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_tetrahedron22_volumes_positive :: Cube -> Bool
 182 prop_tetrahedron22_volumes_positive cube =
 183     volume (tetrahedron22 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_tetrahedron23_volumes_positive :: Cube -> Bool
 188 prop_tetrahedron23_volumes_positive cube =
 189     volume (tetrahedron23 cube) > 0
 190
 191
 192 -- | Given in Sorokina and Zeilfelder, p. 79, (2.6). Note that the
 193 --   third and fourth indices of c-t1 have been switched. This is
 194 --   because we store the triangles oriented such that their volume is
 195 --   positive. If T and T-tilde share \<v0,v1,v2\> and v3,v3-tilde point
 196 --   in opposite directions, one of them has to have negative volume!
 197 prop_c0120_identity1 :: Cube -> Bool
 198 prop_c0120_identity1 cube =
 199    c t0 0 1 2 0 ~= (c t0 0 0 2 1 + c t3 0 0 1 2) / 2
 200      where
 201        t0 = tetrahedron0 cube
 202        t3 = tetrahedron3 cube
 203
 204
 205 -- | Given in Sorokina and Zeilfelder, p. 79, (2.6). Repeats
 206 --   'prop_c0120_identity1' with tetrahedrons 1 and 2.
 207 prop_c0120_identity2 :: Cube -> Bool
 208 prop_c0120_identity2 cube =
 209    c t1 0 1 2 0 ~= (c t1 0 0 2 1 + c t0 0 0 1 2) / 2
 210      where
 211        t0 = tetrahedron0 cube
 212        t1 = tetrahedron1 cube
 213
 214 -- | Given in Sorokina and Zeilfelder, p. 79, (2.6). Repeats
 215 --   'prop_c0120_identity1' with tetrahedrons 1 and 2.
 216 prop_c0120_identity3 :: Cube -> Bool
 217 prop_c0120_identity3 cube =
 218    c t2 0 1 2 0 ~= (c t2 0 0 2 1 + c t1 0 0 1 2) / 2
 219      where
 220        t1 = tetrahedron1 cube
 221        t2 = tetrahedron2 cube
 222
 223 -- | Given in Sorokina and Zeilfelder, p. 79, (2.6). Repeats
 224 --   'prop_c0120_identity1' with tetrahedrons 2 and 3.
 225 prop_c0120_identity4 :: Cube -> Bool
 226 prop_c0120_identity4 cube =
 227    c t3 0 1 2 0 ~= (c t3 0 0 2 1 + c t2 0 0 1 2) / 2
 228      where
 229        t2 = tetrahedron2 cube
 230        t3 = tetrahedron3 cube
 231
 232
 233 -- | Given in Sorokina and Zeilfelder, p. 79, (2.6). Repeats
 234 --   'prop_c0120_identity1' with tetrahedrons 4 and 5.
 235 prop_c0120_identity5 :: Cube -> Bool
 236 prop_c0120_identity5 cube =
 237     c t5 0 1 2 0 ~= (c t5 0 0 2 1 + c t4 0 0 1 2) / 2
 238     where
 239       t4 = tetrahedron4 cube
 240       t5 = tetrahedron5 cube
 241
 242 -- -- | Given in Sorokina and Zeilfelder, p. 79, (2.6). Repeats
 243 -- --   'prop_c0120_identity1' with tetrahedrons 5 and 6.
 244 prop_c0120_identity6 :: Cube -> Bool
 245 prop_c0120_identity6 cube =
 246    c t6 0 1 2 0 ~= (c t6 0 0 2 1 + c t5 0 0 1 2) / 2
 247      where
 248        t5 = tetrahedron5 cube
 249        t6 = tetrahedron6 cube
 250
 251
 252 -- -- | Given in Sorokina and Zeilfelder, p. 79, (2.6). Repeats
 253 -- --   'prop_c0120_identity1' with tetrahedrons 6 and 7.
 254 prop_c0120_identity7 :: Cube -> Bool
 255 prop_c0120_identity7 cube =
 256    c t7 0 1 2 0 ~= (c t7 0 0 2 1 + c t6 0 0 1 2) / 2
 257      where
 258        t6 = tetrahedron6 cube
 259        t7 = tetrahedron7 cube
 260
 261
 262 -- | Given in Sorokina and Zeilfelder, p. 79, (2.6). See
 263 --   'prop_c0120_identity1'.
 264 prop_c0210_identity1 :: Cube -> Bool
 265 prop_c0210_identity1 cube =
 266     c t0 0 2 1 0 ~= (c t0 0 1 1 1 + c t3 0 1 1 1) / 2
 267       where
 268         t0 = tetrahedron0 cube
 269         t3 = tetrahedron3 cube
 270
 271
 272 -- | Given in Sorokina and Zeilfelder, p. 79, (2.6). See
 273 --   'prop_c0120_identity1'.
 274 prop_c0300_identity1 :: Cube -> Bool
 275 prop_c0300_identity1 cube =
 276     c t0 0 3 0 0 ~= (c t0 0 2 0 1 + c t3 0 2 1 0) / 2
 277       where
 278         t0 = tetrahedron0 cube
 279         t3 = tetrahedron3 cube
 280
 281
 282 -- | Given in Sorokina and Zeilfelder, p. 79, (2.6). See
 283 --   'prop_c0120_identity1'.
 284 prop_c1110_identity :: Cube -> Bool
 285 prop_c1110_identity cube =
 286     c t0 1 1 1 0 ~= (c t0 1 0 1 1 + c t3 1 0 1 1) / 2
 287       where
 288         t0 = tetrahedron0 cube
 289         t3 = tetrahedron3 cube
 290
 291
 292 -- | Given in Sorokina and Zeilfelder, p. 79, (2.6). See
 293 --   'prop_c0120_identity1'.
 294 prop_c1200_identity1 :: Cube -> Bool
 295 prop_c1200_identity1 cube =
 296     c t0 1 2 0 0 ~= (c t0 1 1 0 1 + c t3 1 1 1 0) / 2
 297       where
 298         t0 = tetrahedron0 cube
 299         t3 = tetrahedron3 cube
 300
 301
 302 -- | Given in Sorokina and Zeilfelder, p. 79, (2.6). See
 303 --   'prop_c0120_identity1'.
 304 prop_c2100_identity1 :: Cube -> Bool
 305 prop_c2100_identity1 cube =
 306     c t0 2 1 0 0 ~= (c t0 2 0 0 1 + c t3 2 0 1 0) / 2
 307       where
 308         t0 = tetrahedron0 cube
 309         t3 = tetrahedron3 cube
 310
 311
 312
 313 -- | Given in Sorokina and Zeilfelder, p. 79, (2.7). Note that the
 314 --   third and fourth indices of c-t3 have been switched. This is
 315 --   because we store the triangles oriented such that their volume is
 316 --   positive. If T and T-tilde share \<v0,v1,v2\> and v3,v3-tilde
 317 --   point in opposite directions, one of them has to have negative
 318 --   volume!
 319 prop_c0102_identity1 :: Cube -> Bool
 320 prop_c0102_identity1 cube =
 321     c t0 0 1 0 2 ~= (c t0 0 0 1 2 + c t1 0 0 2 1) / 2
 322       where
 323         t0 = tetrahedron0 cube
 324         t1 = tetrahedron1 cube
 325
 326
 327 -- | Given in Sorokina and Zeilfelder, p. 79, (2.7). See
 328 --   'prop_c0102_identity1'.
 329 prop_c0201_identity1 :: Cube -> Bool
 330 prop_c0201_identity1 cube =
 331     c t0 0 2 0 1 ~= (c t0 0 1 1 1 + c t1 0 1 1 1) / 2
 332       where
 333         t0 = tetrahedron0 cube
 334         t1 = tetrahedron1 cube
 335
 336
 337 -- | Given in Sorokina and Zeilfelder, p. 79, (2.7). See
 338 --   'prop_c0102_identity1'.
 339 prop_c0300_identity2 :: Cube -> Bool
 340 prop_c0300_identity2 cube =
 341     c t0 0 3 0 0 ~= (c t0 0 2 1 0 + c t1 0 2 0 1) / 2
 342       where
 343         t0 = tetrahedron0 cube
 344         t1 = tetrahedron1 cube
 345
 346
 347 -- | Given in Sorokina and Zeilfelder, p. 79, (2.7). See
 348 --   'prop_c0102_identity1'.
 349 prop_c1101_identity :: Cube -> Bool
 350 prop_c1101_identity cube =
 351     c t0 1 1 0 1 ~= (c t0 1 0 1 1 + c t1 1 0 1 1) / 2
 352       where
 353         t0 = tetrahedron0 cube
 354         t1 = tetrahedron1 cube
 355
 356
 357 -- | Given in Sorokina and Zeilfelder, p. 79, (2.7). See
 358 --   'prop_c0102_identity1'.
 359 prop_c1200_identity2 :: Cube -> Bool
 360 prop_c1200_identity2 cube =
 361     c t0 1 2 0 0 ~= (c t0 1 1 1 0 + c t1 1 1 0 1) / 2
 362       where
 363         t0 = tetrahedron0 cube
 364         t1 = tetrahedron1 cube
 365
 366
 367 -- | Given in Sorokina and Zeilfelder, p. 79, (2.7). See
 368 --   'prop_c0102_identity1'.
 369 prop_c2100_identity2 :: Cube -> Bool
 370 prop_c2100_identity2 cube =
 371     c t0 2 1 0 0 ~= (c t0 2 0 1 0 + c t1 2 0 0 1) / 2
 372       where
 373         t0 = tetrahedron0 cube
 374         t1 = tetrahedron1 cube
 375
 376
 377 -- | Given in Sorokina and Zeilfelder, p. 79, (2.8). The third and
 378 --   fourth indices of c-t6 have been switched. This is because we
 379 --   store the triangles oriented such that their volume is
 380 --   positive. If T and T-tilde share \<v0,v1,v2\> and v3,v3-tilde
 381 --   point in opposite directions, one of them has to have negative
 382 --   volume!
 383 prop_c3000_identity :: Cube -> Bool
 384 prop_c3000_identity cube =
 385     c t0 3 0 0 0 ~= c t0 2 1 0 0 + c t6 2 1 0 0
 386                     - ((c t0 2 0 1 0 + c t0 2 0 0 1)/ 2)
 387       where
 388         t0 = tetrahedron0 cube
 389         t6 = tetrahedron6 cube
 390
 391
 392 -- | Given in Sorokina and Zeilfelder, p. 79, (2.8). See
 393 --   'prop_c3000_identity'.
 394 prop_c2010_identity :: Cube -> Bool
 395 prop_c2010_identity cube =
 396     c t0 2 0 1 0 ~= c t0 1 1 1 0 + c t6 1 1 0 1
 397                     - ((c t0 1 0 2 0 + c t0 1 0 1 1)/ 2)
 398       where
 399         t0 = tetrahedron0 cube
 400         t6 = tetrahedron6 cube
 401
 402
 403 -- | Given in Sorokina and Zeilfelder, p. 79, (2.8). See
 404 --   'prop_c3000_identity'.
 405 prop_c2001_identity :: Cube -> Bool
 406 prop_c2001_identity cube =
 407     c t0 2 0 0 1 ~= c t0 1 1 0 1 + c t6 1 1 1 0
 408                     - ((c t0 1 0 0 2 + c t0 1 0 1 1)/ 2)
 409       where
 410         t0 = tetrahedron0 cube
 411         t6 = tetrahedron6 cube
 412
 413
 414 -- | Given in Sorokina and Zeilfelder, p. 79, (2.8). See
 415 --   'prop_c3000_identity'.
 416 prop_c1020_identity :: Cube -> Bool
 417 prop_c1020_identity cube =
 418     c t0 1 0 2 0 ~= c t0 0 1 2 0 + c t6 0 1 0 2
 419                     - ((c t0 0 0 3 0 + c t0 0 0 2 1)/ 2)
 420       where
 421         t0 = tetrahedron0 cube
 422         t6 = tetrahedron6 cube
 423
 424
 425 -- | Given in Sorokina and Zeilfelder, p. 79, (2.8). See
 426 --   'prop_c3000_identity'.
 427 prop_c1002_identity :: Cube -> Bool
 428 prop_c1002_identity cube =
 429     c t0 1 0 0 2 ~= c t0 0 1 0 2 + c t6 0 1 2 0
 430                     - ((c t0 0 0 0 3 + c t0 0 0 1 2)/ 2)
 431       where
 432         t0 = tetrahedron0 cube
 433         t6 = tetrahedron6 cube
 434
 435
 436 -- | Given in Sorokina and Zeilfelder, p. 79, (2.8). See
 437 --   'prop_c3000_identity'.
 438 prop_c1011_identity :: Cube -> Bool
 439 prop_c1011_identity cube =
 440     c t0 1 0 1 1 ~= c t0 0 1 1 1 + c t6 0 1 1 1 -
 441                     ((c t0 0 0 1 2 + c t0 0 0 2 1)/ 2)
 442       where
 443         t0 = tetrahedron0 cube
 444         t6 = tetrahedron6 cube
 445
 446
 447
 448 -- | Given in Sorokina and Zeilfelder, p. 78.
 449 prop_cijk1_identity :: Cube -> Bool
 450 prop_cijk1_identity cube =
 451      and [ c t0 i j k 1 ~=
 452                  (c t1 (i+1) j k 0) * ((b0 t0) (v3 t1)) +
 453                  (c t1 i (j+1) k 0) * ((b1 t0) (v3 t1)) +
 454                  (c t1 i j (k+1) 0) * ((b2 t0) (v3 t1)) +
 455                  (c t1 i j k 1) * ((b3 t0) (v3 t1)) | i <- [0..2],
 456                                                       j <- [0..2],
 457                                                       k <- [0..2],
 458                                                       i + j + k == 2]
 459       where
 460         t0 = tetrahedron0 cube
 461         t1 = tetrahedron1 cube
 462
 463
 464 -- | The function values at the interior should be the same for all tetrahedra.
 465 prop_interior_values_all_identical :: Cube -> Bool
 466 prop_interior_values_all_identical cube =
 467     all_equal [i0, i1, i2, i3, i4, i5, i6, i7, i8,
 468                i9, i10, i11, i12, i13, i14, i15, i16,
 469                i17, i18, i19, i20, i21, i22, i23]
 470     where
 471       i0 = eval (Tetrahedron.fv (tetrahedron0 cube)) I
 472       i1 = eval (Tetrahedron.fv (tetrahedron1 cube)) I
 473       i2 = eval (Tetrahedron.fv (tetrahedron2 cube)) I
 474       i3 = eval (Tetrahedron.fv (tetrahedron3 cube)) I
 475       i4 = eval (Tetrahedron.fv (tetrahedron4 cube)) I
 476       i5 = eval (Tetrahedron.fv (tetrahedron5 cube)) I
 477       i6 = eval (Tetrahedron.fv (tetrahedron6 cube)) I
 478       i7 = eval (Tetrahedron.fv (tetrahedron7 cube)) I
 479       i8 = eval (Tetrahedron.fv (tetrahedron8 cube)) I
 480       i9 = eval (Tetrahedron.fv (tetrahedron9 cube)) I
 481       i10 = eval (Tetrahedron.fv (tetrahedron10 cube)) I
 482       i11 = eval (Tetrahedron.fv (tetrahedron11 cube)) I
 483       i12 = eval (Tetrahedron.fv (tetrahedron12 cube)) I
 484       i13 = eval (Tetrahedron.fv (tetrahedron13 cube)) I
 485       i14 = eval (Tetrahedron.fv (tetrahedron14 cube)) I
 486       i15 = eval (Tetrahedron.fv (tetrahedron15 cube)) I
 487       i16 = eval (Tetrahedron.fv (tetrahedron16 cube)) I
 488       i17 = eval (Tetrahedron.fv (tetrahedron17 cube)) I
 489       i18 = eval (Tetrahedron.fv (tetrahedron18 cube)) I
 490       i19 = eval (Tetrahedron.fv (tetrahedron19 cube)) I
 491       i20 = eval (Tetrahedron.fv (tetrahedron20 cube)) I
 492       i21 = eval (Tetrahedron.fv (tetrahedron21 cube)) I
 493       i22 = eval (Tetrahedron.fv (tetrahedron22 cube)) I
 494       i23 = eval (Tetrahedron.fv (tetrahedron23 cube)) I
 495
 496
 497 -- | We know what (c t6 2 1 0 0) should be from Sorokina and Zeilfelder, p. 87.
 498 --   This test checks the rotation works as expected.
 499 prop_c_tilde_2100_rotation_correct :: Cube -> Bool
 500 prop_c_tilde_2100_rotation_correct cube =
 501     expr1 == expr2
 502     where
 503       t0 = tetrahedron0 cube
 504       t6 = tetrahedron6 cube
 505
 506       -- What gets computed for c2100 of t6.
 507       expr1 = eval (Tetrahedron.fv t6) $
 508           (3/8)*I +
 509           (1/12)*(T + R + L + D) +
 510           (1/64)*(FT + FR + FL + FD) +
 511           (7/48)*F +
 512           (1/48)*B +
 513           (1/96)*(RT + LD + LT + RD) +
 514           (1/192)*(BT + BR + BL + BD)
 515
 516       -- What should be computed for c2100 of t6.
 517       expr2 = eval (Tetrahedron.fv t0) $
 518               (3/8)*I +
 519               (1/12)*(F + R + L + B) +
 520               (1/64)*(FT + RT + LT + BT) +
 521               (7/48)*T +
 522               (1/48)*D +
 523               (1/96)*(FR + FL + BR + BL) +
 524               (1/192)*(FD + RD + LD + BD)
 525
 526
 527 -- | We know what (c t6 2 1 0 0) should be from Sorokina and Zeilfelder, p. 87.
 528 --   This test checks the actual value based on the FunctionValues of the cube.
 529 prop_c_tilde_2100_correct :: Cube -> Bool
 530 prop_c_tilde_2100_correct cube =
 531     c t6 2 1 0 0 == (3/8)*int
 532                     + (1/12)*(f + r + l + b)
 533                     + (1/64)*(ft + rt + lt + bt)
 534                     + (7/48)*t + (1/48)*d + (1/96)*(fr + fl + br + bl)
 535                     + (1/192)*(fd + rd + ld + bd)
 536     where
 537       t0 = tetrahedron0 cube
 538       t6 = tetrahedron6 cube
 539       fvs = Tetrahedron.fv t0
 540       int = interior fvs
 541       f = front fvs
 542       r = right fvs
 543       l = left fvs
 544       b = back fvs
 545       ft = front_top fvs
 546       rt = right_top fvs
 547       lt = left_top fvs
 548       bt = back_top fvs
 549       t = top fvs
 550       d = down fvs
 551       fr = front_right fvs
 552       fl = front_left fvs
 553       br = back_right fvs
 554       bl = back_left fvs
 555       fd = front_down fvs
 556       rd = right_down fvs
 557       ld = left_down fvs
 558       bd = back_down fvs
 559
 560 -- Tests to check that the correct edges are incidental.
 561 prop_t0_shares_edge_with_t1 :: Cube -> Bool
 562 prop_t0_shares_edge_with_t1 cube =
 563     (v1 t0) == (v1 t1) && (v3 t0) == (v2 t1)
 564       where
 565         t0 = tetrahedron0 cube
 566         t1 = tetrahedron1 cube
 567
 568 prop_t0_shares_edge_with_t3 :: Cube -> Bool
 569 prop_t0_shares_edge_with_t3 cube =
 570     (v1 t0) == (v1 t3) && (v2 t0) == (v3 t3)
 571       where
 572         t0 = tetrahedron0 cube
 573         t3 = tetrahedron3 cube
 574
 575 prop_t0_shares_edge_with_t6 :: Cube -> Bool
 576 prop_t0_shares_edge_with_t6 cube =
 577     (v2 t0) == (v3 t6) && (v3 t0) == (v2 t6)
 578       where
 579         t0 = tetrahedron0 cube
 580         t6 = tetrahedron6 cube
 581
 582 prop_t1_shares_edge_with_t2 :: Cube -> Bool
 583 prop_t1_shares_edge_with_t2 cube =
 584     (v1 t1) == (v1 t2) && (v3 t1) == (v2 t2)
 585       where
 586         t1 = tetrahedron1 cube
 587         t2 = tetrahedron2 cube
 588
 589 prop_t1_shares_edge_with_t19 :: Cube -> Bool
 590 prop_t1_shares_edge_with_t19 cube =
 591     (v2 t1) == (v3 t19) && (v3 t1) == (v2 t19)
 592       where
 593         t1 = tetrahedron1 cube
 594         t19 = tetrahedron19 cube
 595
 596 prop_t2_shares_edge_with_t3 :: Cube -> Bool
 597 prop_t2_shares_edge_with_t3 cube =
 598     (v1 t1) == (v1 t2) && (v3 t1) == (v2 t2)
 599       where
 600         t1 = tetrahedron1 cube
 601         t2 = tetrahedron2 cube
 602
 603 prop_t2_shares_edge_with_t12 :: Cube -> Bool
 604 prop_t2_shares_edge_with_t12 cube =
 605     (v2 t2) == (v3 t12) && (v3 t2) == (v2 t12)
 606       where
 607         t2 = tetrahedron2 cube
 608         t12 = tetrahedron12 cube
 609
 610 prop_t3_shares_edge_with_t21 :: Cube -> Bool
 611 prop_t3_shares_edge_with_t21 cube =
 612     (v2 t3) == (v3 t21) && (v3 t3) == (v2 t21)
 613       where
 614         t3 = tetrahedron3 cube
 615         t21 = tetrahedron21 cube
 616
 617 prop_t4_shares_edge_with_t5 :: Cube -> Bool
 618 prop_t4_shares_edge_with_t5 cube =
 619     (v1 t4) == (v1 t5) && (v3 t4) == (v2 t5)
 620       where
 621         t4 = tetrahedron4 cube
 622         t5 = tetrahedron5 cube
 623
 624 prop_t4_shares_edge_with_t7 :: Cube -> Bool
 625 prop_t4_shares_edge_with_t7 cube =
 626     (v1 t4) == (v1 t7) && (v2 t4) == (v3 t7)
 627       where
 628         t4 = tetrahedron4 cube
 629         t7 = tetrahedron7 cube
 630
 631 prop_t4_shares_edge_with_t10 :: Cube -> Bool
 632 prop_t4_shares_edge_with_t10 cube =
 633     (v2 t4) == (v3 t10) && (v3 t4) == (v2 t10)
 634       where
 635         t4 = tetrahedron4 cube
 636         t10 = tetrahedron10 cube
 637
 638 prop_t5_shares_edge_with_t6 :: Cube -> Bool
 639 prop_t5_shares_edge_with_t6 cube =
 640     (v1 t5) == (v1 t6) && (v3 t5) == (v2 t6)
 641       where
 642         t5 = tetrahedron5 cube
 643         t6 = tetrahedron6 cube
 644
 645 prop_t5_shares_edge_with_t16 :: Cube -> Bool
 646 prop_t5_shares_edge_with_t16 cube =
 647     (v2 t5) == (v3 t16) && (v3 t5) == (v2 t16)
 648       where
 649         t5 = tetrahedron5 cube
 650         t16 = tetrahedron16 cube
 651
 652 prop_t6_shares_edge_with_t7 :: Cube -> Bool
 653 prop_t6_shares_edge_with_t7 cube =
 654     (v1 t6) == (v1 t7) && (v3 t6) == (v2 t7)
 655       where
 656         t6 = tetrahedron6 cube
 657         t7 = tetrahedron7 cube
 658
 659 prop_t7_shares_edge_with_t20 :: Cube -> Bool
 660 prop_t7_shares_edge_with_t20 cube =
 661     (v2 t7) == (v3 t20) && (v2 t7) == (v3 t20)
 662       where
 663         t7 = tetrahedron7 cube
 664         t20 = tetrahedron20 cube