1 #LyX 1.6.8 created this file. For more info see http://www.lyx.org/
6 \use_default_options false
16 \font_typewriter default
17 \font_default_family default
24 \paperfontsize default
27 \papersize letterpaper
33 \paperorientation portrait
36 \paragraph_separation skip
38 \quotes_language english
41 \paperpagestyle default
42 \tracking_changes false
58 \begin_layout Example*
59 First, will will define our grid
60 \begin_inset Formula $G$
64 \begin_inset Formula $3\times3\times3$
68 \begin_inset Formula $h=1$
74 \begin_layout Example*
75 To test the reproduction of trilinears, we will take the function (per Sorokina
80 \begin_layout Example*
81 \begin_inset Formula \[
82 f\left(x,y,z\right)=1+x+xy+xyz\]
89 \begin_layout Example*
90 and compute its value at the 27 points,
91 \begin_inset Formula $\left(0,0,0\right),\left(0,0,1\right),\dots\left(2,2,2\right)$
97 \begin_layout Example*
102 \begin_layout Plain Layout
104 sage: x,y,z = var('x,y,z')
107 \begin_layout Plain Layout
109 sage: f = 1 + x + x*y + x*y*z
112 \begin_layout Plain Layout
114 sage: f(x=0, y=0, z=0)
117 \begin_layout Plain Layout
122 \begin_layout Plain Layout
124 sage: f(x=1, y=0, z=0)
127 \begin_layout Plain Layout
132 \begin_layout Plain Layout
134 sage: f(x=2, y=0, z=0)
137 \begin_layout Plain Layout
142 \begin_layout Plain Layout
144 sage: f(x=0, y=1, z=0)
147 \begin_layout Plain Layout
152 \begin_layout Plain Layout
154 sage: f(x=1, y=1, z=0)
157 \begin_layout Plain Layout
162 \begin_layout Plain Layout
164 sage: f(x=2, y=1, z=0)
167 \begin_layout Plain Layout
172 \begin_layout Plain Layout
174 sage: f(x=0, y=2, z=0)
177 \begin_layout Plain Layout
182 \begin_layout Plain Layout
184 sage: f(x=1, y=2, z=0)
187 \begin_layout Plain Layout
192 \begin_layout Plain Layout
194 sage: f(x=2, y=2, z=0)
197 \begin_layout Plain Layout
202 \begin_layout Plain Layout
204 sage: f(x=0, y=0, z=1)
207 \begin_layout Plain Layout
212 \begin_layout Plain Layout
214 sage: f(x=1, y=0, z=1)
217 \begin_layout Plain Layout
222 \begin_layout Plain Layout
224 sage: f(x=2, y=0, z=1)
227 \begin_layout Plain Layout
232 \begin_layout Plain Layout
234 sage: f(x=0, y=1, z=1)
237 \begin_layout Plain Layout
242 \begin_layout Plain Layout
244 sage: f(x=1, y=1, z=1)
247 \begin_layout Plain Layout
252 \begin_layout Plain Layout
254 sage: f(x=2, y=1, z=1)
257 \begin_layout Plain Layout
262 \begin_layout Plain Layout
264 sage: f(x=0, y=2, z=1)
267 \begin_layout Plain Layout
272 \begin_layout Plain Layout
274 sage: f(x=1, y=2, z=1)
277 \begin_layout Plain Layout
282 \begin_layout Plain Layout
284 sage: f(x=2, y=2, z=1)
287 \begin_layout Plain Layout
292 \begin_layout Plain Layout
294 sage: f(x=0, y=0, z=2)
297 \begin_layout Plain Layout
302 \begin_layout Plain Layout
304 sage: f(x=1, y=0, z=2)
307 \begin_layout Plain Layout
312 \begin_layout Plain Layout
314 sage: f(x=2, y=0, z=2)
317 \begin_layout Plain Layout
322 \begin_layout Plain Layout
324 sage: f(x=0, y=1, z=2)
327 \begin_layout Plain Layout
332 \begin_layout Plain Layout
334 sage: f(x=1, y=1, z=2)
337 \begin_layout Plain Layout
342 \begin_layout Plain Layout
344 sage: f(x=2, y=1, z=2)
347 \begin_layout Plain Layout
352 \begin_layout Plain Layout
354 sage: f(x=0, y=2, z=2)
357 \begin_layout Plain Layout
362 \begin_layout Plain Layout
364 sage: f(x=1, y=2, z=2)
367 \begin_layout Plain Layout
372 \begin_layout Plain Layout
374 sage: f(x=2, y=2, z=2)
377 \begin_layout Plain Layout
387 \begin_layout Example*
388 We are most interested in the
389 \begin_inset Quotes eld
393 \begin_inset Quotes erd
397 \begin_inset Formula $\left(1,1,1\right)$
401 \begin_inset Formula $4$
405 We can enter the data above into a list,
408 \begin_layout Example*
409 \begin_inset listings
413 \begin_layout Plain Layout
415 sage: g = [[[ f(x=a, y=b, z=c) for a in range(0,3) ] for b in range(0,3)
416 ] for c in range(0,3) ]
419 \begin_layout Plain Layout
424 \begin_layout Plain Layout
426 [[[1, 2, 3], [1, 3, 5], [1, 4, 7]],
429 \begin_layout Plain Layout
431 [[1, 2, 3], [1, 4, 7], [1, 6, 11]],
434 \begin_layout Plain Layout
436 [[1, 2, 3], [1, 5, 9], [1, 8, 15]]]
444 \begin_layout Example*
445 although the list will be indexed by
446 \begin_inset Formula $\left(z,y,x\right)$
449 so we define a function to access it by
450 \begin_inset Formula $\left(x,y,z\right)$
456 \begin_layout Example*
457 \begin_inset listings
461 \begin_layout Plain Layout
463 sage: def grid(x,y,z):
466 \begin_layout Plain Layout
468 ....: return g[z][y][x]
476 \begin_layout Example*
477 and define directional functions according to Sorokina and Zeilfelder, p.
481 \begin_layout Example*
482 \begin_inset listings
486 \begin_layout Plain Layout
491 \begin_layout Plain Layout
493 ....: return grid(x,y,z)
496 \begin_layout Plain Layout
500 \begin_layout Plain Layout
505 \begin_layout Plain Layout
507 ....: return grid(x-1, y, z)
510 \begin_layout Plain Layout
514 \begin_layout Plain Layout
519 \begin_layout Plain Layout
521 ....: return grid(x+1, y, z)
524 \begin_layout Plain Layout
528 \begin_layout Plain Layout
533 \begin_layout Plain Layout
535 ....: return grid(x, y-1, z)
538 \begin_layout Plain Layout
542 \begin_layout Plain Layout
547 \begin_layout Plain Layout
549 ....: return grid(x, y+1, z)
552 \begin_layout Plain Layout
556 \begin_layout Plain Layout
561 \begin_layout Plain Layout
563 ....: return grid(x, y, z+1)
566 \begin_layout Plain Layout
570 \begin_layout Plain Layout
575 \begin_layout Plain Layout
577 ....: return grid(x, y, z-1)
580 \begin_layout Plain Layout
584 \begin_layout Plain Layout
589 \begin_layout Plain Layout
591 ....: return grid(x-1, y-1, z)
594 \begin_layout Plain Layout
598 \begin_layout Plain Layout
603 \begin_layout Plain Layout
605 ....: return grid(x-1, y+1, z)
608 \begin_layout Plain Layout
612 \begin_layout Plain Layout
617 \begin_layout Plain Layout
619 ....: return grid(x-1, y, z-1)
622 \begin_layout Plain Layout
626 \begin_layout Plain Layout
631 \begin_layout Plain Layout
633 ....: return grid(x-1, y, z+1)
636 \begin_layout Plain Layout
640 \begin_layout Plain Layout
645 \begin_layout Plain Layout
647 ....: return grid(x+1, y-1, z)
650 \begin_layout Plain Layout
654 \begin_layout Plain Layout
659 \begin_layout Plain Layout
661 ....: return grid(x+1, y+1, z)
664 \begin_layout Plain Layout
668 \begin_layout Plain Layout
673 \begin_layout Plain Layout
675 ....: return grid(x+1, y, z-1)
678 \begin_layout Plain Layout
682 \begin_layout Plain Layout
687 \begin_layout Plain Layout
689 ....: return grid(x+1, y, z+1)
692 \begin_layout Plain Layout
696 \begin_layout Plain Layout
701 \begin_layout Plain Layout
703 ....: return grid(x, y-1, z-1)
706 \begin_layout Plain Layout
710 \begin_layout Plain Layout
715 \begin_layout Plain Layout
717 ....: return grid(x, y-1, z+1)
720 \begin_layout Plain Layout
724 \begin_layout Plain Layout
729 \begin_layout Plain Layout
731 ....: return grid(x, y+1, z-1)
734 \begin_layout Plain Layout
738 \begin_layout Plain Layout
743 \begin_layout Plain Layout
745 ....: return grid(x, y+1, z+1)
748 \begin_layout Plain Layout
752 \begin_layout Plain Layout
754 sage: def FLD(x,y,z):
757 \begin_layout Plain Layout
759 ....: return grid(x-1, y-1, z-1)
762 \begin_layout Plain Layout
766 \begin_layout Plain Layout
768 sage: def FLT(x,y,z):
771 \begin_layout Plain Layout
773 ....: return grid(x-1, y-1, z+1)
776 \begin_layout Plain Layout
780 \begin_layout Plain Layout
782 sage: def FRD(x,y,z):
785 \begin_layout Plain Layout
787 ....: return grid(x-1, y+1, z-1)
790 \begin_layout Plain Layout
794 \begin_layout Plain Layout
796 sage: def FRT(x,y,z):
799 \begin_layout Plain Layout
801 ....: return grid(x-1, y+1, z+1)
804 \begin_layout Plain Layout
808 \begin_layout Plain Layout
810 sage: def BLD(x,y,z):
813 \begin_layout Plain Layout
815 ....: return grid(x+1, y-1, z-1)
818 \begin_layout Plain Layout
822 \begin_layout Plain Layout
824 sage: def BLT(x,y,z):
827 \begin_layout Plain Layout
829 ....: return grid(x+1, y-1, z+1)
832 \begin_layout Plain Layout
836 \begin_layout Plain Layout
838 sage: def BRD(x,y,z):
841 \begin_layout Plain Layout
843 ....: return grid(x+1, y+1, z-1)
846 \begin_layout Plain Layout
850 \begin_layout Plain Layout
852 sage: def BRT(x,y,z):
855 \begin_layout Plain Layout
857 ....: return grid(x+1, y+1, z+1)
865 \begin_layout Example*
866 Next, we define the coefficients for the cube centered on
867 \begin_inset Formula $\left(1,1,1\right)$
870 based on these directional functions.
873 \begin_layout Example*
874 \begin_inset listings
878 \begin_layout Plain Layout
880 sage: c0030 = (1/8)*( I(1,1,1) + F(1,1,1) + L(1,1,1) + T(1,1,1) +
883 \begin_layout Plain Layout
885 LT(1,1,1) + FL(1,1,1) + FT(1,1,1) + FLT(1,1,1) )
888 \begin_layout Plain Layout
892 \begin_layout Plain Layout
894 sage: c0003 = (1/8)*( I(1,1,1) + F(1,1,1) + R(1,1,1) + T(1,1,1) +
897 \begin_layout Plain Layout
899 RT(1,1,1) + FR(1,1,1) + FT(1,1,1) + FRT(1,1,1) )
902 \begin_layout Plain Layout
906 \begin_layout Plain Layout
908 sage: c0021 = (5/24)*(I(1,1,1) + F(1,1,1) + T(1,1,1) + FT(1,1,1)) +
911 \begin_layout Plain Layout
913 (1/24)*(L(1,1,1) + FL(1,1,1) + LT(1,1,1) + FLT(1,1,1))
916 \begin_layout Plain Layout
920 \begin_layout Plain Layout
922 sage: c0012 = (5/24)*(I(1,1,1) + F(1,1,1) + T(1,1,1) + FT(1,1,1)) +
925 \begin_layout Plain Layout
927 (1/24)*(R(1,1,1) + FR(1,1,1) + RT(1,1,1) + FRT(1,1,1))
930 \begin_layout Plain Layout
934 \begin_layout Plain Layout
936 sage: c0120 = (5/24)*(I(1,1,1) + F(1,1,1)) +
939 \begin_layout Plain Layout
941 (1/8)*(L(1,1,1) + T(1,1,1) + FL(1,1,1) + FT(1,1,1)) +
944 \begin_layout Plain Layout
946 (1/24)*(LT(1,1,1) + FLT(1,1,1))
949 \begin_layout Plain Layout
953 \begin_layout Plain Layout
955 sage: c0102 = (5/24)*(I(1,1,1) + F(1,1,1)) +
958 \begin_layout Plain Layout
960 (1/8)*(R(1,1,1) + T(1,1,1) + FR(1,1,1) + FT(1,1,1)) +
963 \begin_layout Plain Layout
965 (1/24)*(RT(1,1,1) + FRT(1,1,1))
968 \begin_layout Plain Layout
972 \begin_layout Plain Layout
974 sage: c0111 = (13/48)*(I(1,1,1) + F(1,1,1)) +
977 \begin_layout Plain Layout
979 (7/48)*(T(1,1,1) + FT(1,1,1)) +
982 \begin_layout Plain Layout
984 (1/32)*(L(1,1,1) + R(1,1,1) + FL(1,1,1) + FR(1,1,1)) +
987 \begin_layout Plain Layout
989 (1/96)*(LT(1,1,1) + RT(1,1,1) + FLT(1,1,1) + FRT(1,1,1))
992 \begin_layout Plain Layout
996 \begin_layout Plain Layout
998 sage: c0210 = (13/48)*(I(1,1,1) + F(1,1,1)) +
1001 \begin_layout Plain Layout
1003 (17/192)*(L(1,1,1) + T(1,1,1) + FL(1,1,1) + FT(1,1,1)) +
1006 \begin_layout Plain Layout
1008 (1/96)*(LT(1,1,1) + FLT(1,1,1)) +
1011 \begin_layout Plain Layout
1013 (1/64)*(R(1,1,1) + D(1,1,1) + FR(1,1,1) + FD(1,1,1)) +
1016 \begin_layout Plain Layout
1018 (1/192)*(RT(1,1,1) + LD(1,1,1) + FRT(1,1,1) + FLD(1,1,1))
1021 \begin_layout Plain Layout
1025 \begin_layout Plain Layout
1027 sage: c0201 = (13/48)*(I(1,1,1) + F(1,1,1)) +
1030 \begin_layout Plain Layout
1032 (17/192)*(R(1,1,1) + T(1,1,1) + FR(1,1,1) + FT(1,1,1)) +
1035 \begin_layout Plain Layout
1037 (1/96)*(RT(1,1,1) + FRT(1,1,1)) +
1040 \begin_layout Plain Layout
1042 (1/64)*(L(1,1,1) + D(1,1,1) + FL(1,1,1) + FD(1,1,1)) +
1045 \begin_layout Plain Layout
1047 (1/192)*(RD(1,1,1) + LT(1,1,1) + FLT(1,1,1) + FRD(1,1,1))
1050 \begin_layout Plain Layout
1054 \begin_layout Plain Layout
1056 sage: c0300 = (13/48)*(I(1,1,1) + F(1,1,1)) +
1059 \begin_layout Plain Layout
1061 (5/96)*(L(1,1,1) + R(1,1,1) + T(1,1,1) + D(1,1,1) +
1064 \begin_layout Plain Layout
1066 FL(1,1,1) + FR(1,1,1) + FT(1,1,1) + FD(1,1,1)) +
1069 \begin_layout Plain Layout
1071 (1/192)*(RT(1,1,1) + RD(1,1,1) + LT(1,1,1) + LD(1,1,1) +
1074 \begin_layout Plain Layout
1076 FRT(1,1,1) + FRD(1,1,1) + FLT(1,1,1) + FLD(1,1,1))
1079 \begin_layout Plain Layout
1083 \begin_layout Plain Layout
1085 sage: c1020 = (1/4)*I(1,1,1) +
1088 \begin_layout Plain Layout
1090 (1/6)*(F(1,1,1) + L(1,1,1) + T(1,1,1)) +
1093 \begin_layout Plain Layout
1095 (1/12)*(LT(1,1,1) + FL(1,1,1) + FT(1,1,1))
1098 \begin_layout Plain Layout
1102 \begin_layout Plain Layout
1104 sage: c1002 = (1/4)*I(1,1,1) +
1107 \begin_layout Plain Layout
1109 (1/6)*(F(1,1,1) + R(1,1,1) + T(1,1,1)) +
1112 \begin_layout Plain Layout
1114 (1/12)*(RT(1,1,1) + FR(1,1,1) + FT(1,1,1))
1117 \begin_layout Plain Layout
1121 \begin_layout Plain Layout
1123 sage: c1011 = (1/3)*I(1,1,1) +
1126 \begin_layout Plain Layout
1128 (5/24)*(F(1,1,1) + T(1,1,1)) +
1131 \begin_layout Plain Layout
1133 (1/12)*FT(1,1,1) + (1/24)*(L(1,1,1) + R(1,1,1)) +
1136 \begin_layout Plain Layout
1138 (1/48)*(LT(1,1,1) + RT(1,1,1) + FL(1,1,1) + FR(1,1,1))
1141 \begin_layout Plain Layout
1145 \begin_layout Plain Layout
1147 sage: c1110 = (1/3)*I(1,1,1) +
1150 \begin_layout Plain Layout
1155 \begin_layout Plain Layout
1157 (1/8)*(L(1,1,1) + T(1,1,1)) +
1160 \begin_layout Plain Layout
1162 (5/96)*(FL(1,1,1) + FT(1,1,1)) +
1165 \begin_layout Plain Layout
1167 (1/48)*(D(1,1,1) + R(1,1,1) + LT(1,1,1)) +
1170 \begin_layout Plain Layout
1172 (1/96)*(FD(1,1,1) + LD(1,1,1) + RT(1,1,1) + FR(1,1,1))
1175 \begin_layout Plain Layout
1179 \begin_layout Plain Layout
1181 sage: c1101 = (1/3)*I(1,1,1) +
1184 \begin_layout Plain Layout
1189 \begin_layout Plain Layout
1191 (1/8)*(R(1,1,1) + T(1,1,1)) +
1194 \begin_layout Plain Layout
1196 (5/96)*(FR(1,1,1) + FT(1,1,1)) +
1199 \begin_layout Plain Layout
1201 (1/48)*(D(1,1,1) + L(1,1,1) + RT(1,1,1)) +
1204 \begin_layout Plain Layout
1206 (1/96)*(FD(1,1,1) + LT(1,1,1) + RD(1,1,1) + FL(1,1,1))
1209 \begin_layout Plain Layout
1213 \begin_layout Plain Layout
1215 sage: c1200 = (1/3)*I(1,1,1) +
1218 \begin_layout Plain Layout
1223 \begin_layout Plain Layout
1225 (7/96)*(L(1,1,1) + R(1,1,1) + T(1,1,1) + D(1,1,1)) +
1228 \begin_layout Plain Layout
1230 (1/32)*(FL(1,1,1) + FR(1,1,1) + FT(1,1,1) + FD(1,1,1)) +
1233 \begin_layout Plain Layout
1235 (1/96)*(RT(1,1,1) + RD(1,1,1) + LT(1,1,1) + LD(1,1,1))
1238 \begin_layout Plain Layout
1242 \begin_layout Plain Layout
1244 sage: c2010 = (3/8)*I(1,1,1) +
1247 \begin_layout Plain Layout
1249 (7/48)*(F(1,1,1) + T(1,1,1) + L(1,1,1)) +
1252 \begin_layout Plain Layout
1254 (1/48)*(R(1,1,1) + D(1,1,1) + B(1,1,1) + LT(1,1,1) + FL(1,1,1)
1258 \begin_layout Plain Layout
1260 (1/96)*(RT(1,1,1) + BT(1,1,1) + FR(1,1,1) + FD(1,1,1) + LD(1,1,1)
1264 \begin_layout Plain Layout
1268 \begin_layout Plain Layout
1270 sage: c2001 = (3/8)*I(1,1,1) +
1273 \begin_layout Plain Layout
1275 (7/48)*(F(1,1,1) + T(1,1,1) + R(1,1,1)) +
1278 \begin_layout Plain Layout
1280 (1/48)*(L(1,1,1) + D(1,1,1) + B(1,1,1) + RT(1,1,1) + FR(1,1,1)
1284 \begin_layout Plain Layout
1286 (1/96)*(LT(1,1,1) + BT(1,1,1) + FL(1,1,1) + FD(1,1,1) + RD(1,1,1)
1290 \begin_layout Plain Layout
1294 \begin_layout Plain Layout
1296 sage: c2100 = (3/8)*I(1,1,1) +
1299 \begin_layout Plain Layout
1301 (1/12)*(T(1,1,1) + R(1,1,1) + L(1,1,1) + D(1,1,1)) +
1304 \begin_layout Plain Layout
1306 (1/64)*(FT(1,1,1) + FR(1,1,1) + FL(1,1,1) + FD(1,1,1)) +
1309 \begin_layout Plain Layout
1314 \begin_layout Plain Layout
1319 \begin_layout Plain Layout
1321 (1/96)*(RT(1,1,1) + LD(1,1,1) + LT(1,1,1) + RD(1,1,1)) +
1324 \begin_layout Plain Layout
1326 (1/192)*(BT(1,1,1) + BR(1,1,1) + BL(1,1,1) + BD(1,1,1))
1329 \begin_layout Plain Layout
1333 \begin_layout Plain Layout
1335 sage: c3000 = (3/8)*I(1,1,1) +
1338 \begin_layout Plain Layout
1340 (1/12)*(T(1,1,1) + F(1,1,1) + L(1,1,1) + R(1,1,1) + D(1,1,1)
1344 \begin_layout Plain Layout
1346 (1/96)*(LT(1,1,1) + FL(1,1,1) + FT(1,1,1) + RT(1,1,1) + BT(1,1,1)
1350 \begin_layout Plain Layout
1352 FD(1,1,1) + LD(1,1,1) + BD(1,1,1) + BR(1,1,1) + RD(1,1,1)
1361 \begin_layout Example*
1362 We can see what the constant values are now:
1365 \begin_layout Example*
1366 \begin_inset listings
1370 \begin_layout Plain Layout
1375 \begin_layout Plain Layout
1380 \begin_layout Plain Layout
1384 \begin_layout Plain Layout
1389 \begin_layout Plain Layout
1394 \begin_layout Plain Layout
1398 \begin_layout Plain Layout
1403 \begin_layout Plain Layout
1408 \begin_layout Plain Layout
1412 \begin_layout Plain Layout
1417 \begin_layout Plain Layout
1422 \begin_layout Plain Layout
1426 \begin_layout Plain Layout
1431 \begin_layout Plain Layout
1436 \begin_layout Plain Layout
1440 \begin_layout Plain Layout
1445 \begin_layout Plain Layout
1450 \begin_layout Plain Layout
1454 \begin_layout Plain Layout
1459 \begin_layout Plain Layout
1464 \begin_layout Plain Layout
1468 \begin_layout Plain Layout
1473 \begin_layout Plain Layout
1478 \begin_layout Plain Layout
1482 \begin_layout Plain Layout
1487 \begin_layout Plain Layout
1492 \begin_layout Plain Layout
1496 \begin_layout Plain Layout
1501 \begin_layout Plain Layout
1506 \begin_layout Plain Layout
1510 \begin_layout Plain Layout
1515 \begin_layout Plain Layout
1520 \begin_layout Plain Layout
1524 \begin_layout Plain Layout
1529 \begin_layout Plain Layout
1534 \begin_layout Plain Layout
1538 \begin_layout Plain Layout
1543 \begin_layout Plain Layout
1548 \begin_layout Plain Layout
1552 \begin_layout Plain Layout
1557 \begin_layout Plain Layout
1562 \begin_layout Plain Layout
1566 \begin_layout Plain Layout
1571 \begin_layout Plain Layout
1576 \begin_layout Plain Layout
1580 \begin_layout Plain Layout
1585 \begin_layout Plain Layout
1590 \begin_layout Plain Layout
1594 \begin_layout Plain Layout
1599 \begin_layout Plain Layout
1604 \begin_layout Plain Layout
1608 \begin_layout Plain Layout
1613 \begin_layout Plain Layout
1618 \begin_layout Plain Layout
1622 \begin_layout Plain Layout
1627 \begin_layout Plain Layout
1632 \begin_layout Plain Layout
1636 \begin_layout Plain Layout
1641 \begin_layout Plain Layout
1651 \begin_layout Example*
1652 Now that we have the coefficients, we'll choose a particular tetrahedron
1653 and compute the polynomial over it.
1655 \begin_inset Quotes eld
1659 \begin_inset Quotes erd
1662 face of the cube (in the positive
1663 \begin_inset Formula $z$
1666 direction), there are only four tetrahedra to choose from.
1667 We'll be consider the
1668 \begin_inset Quotes eld
1672 \begin_inset Quotes erd
1675 tetrahedron; that is, the one with vertices,
1678 \begin_layout Example*
1679 \begin_inset Formula \begin{eqnarray*}
1680 v_{0}=\left(0.5,1.5,1.5\right) & & \mbox{at the front-right of the cube}\\
1681 v_{1}=\left(1.5,1.5,1.5\right) & & \mbox{at the back-right of the cube}\\
1682 v_{2}=\left(1,1,1.5\right) & & \mbox{at the center of the top face of the cube}\\
1683 v_{3}=\left(1,1,1\right) & & \mbox{at the center of the cube}\end{eqnarray*}